link.springer.com978-1-4939-7887-8/1.pdf · APPENDIX § 1. CONSTRUCTIBILITY IN ALGEBRAIC GEOMETRY...

APPENDIX

§ 1. CONSTRUCTIBILITY IN ALGEBRAIC GEOMETRY

1.1. Constructible Subsets of a Scheme

In this appendix, we recall some definitions and results concerningconstructible subsets of a (general) scheme. This paragraph is based on(ÉGA III1, 0, §9), (ÉGA IV1, §1.8, §1.9), and (ÉGA IV3, §8, §9). However,we follow the terminology of (ÉGA Isv).(1.1.1). — Let X be a topological space. We say that a subset Z of X isretrocompact if the inclusion from Z to X is a quasi-compact morphism, thatis, if Z ∩ U is quasi-compact for every quasi-compact open set U of X. If Xis a scheme, a subset Z is retrocompact if and only if Z ∩ U is quasi-compactfor every affine open subscheme U of X.

The space X is retrocompact; the union of two retrocompact subsets of Xis retrocompact; the intersection of two retrocompact open subsets is retro-compact. If X is a noetherian topological space, then every subset of X isretrocompact.(1.1.2). — Let X be a topological space. One says that a subset C of X isglobally constructible (in X) if it belongs to the smallest set of subsets of Xwhich contains all retrocompact open subsets of X and is stable under finiteintersection and complements and hence also under finite unions. Explicitly,C is globally constructible if and only if there exist finite families (U1, . . . , Un)and (V1, . . . , Vn) of retrocompact open sets of X such that

C =n⋃

i=1Ui ∩ (X Vi).

One says that C is constructible if every point of X is contained in anaffine open subscheme V of X such that C ∩ V is globally constructible in V .

© Springer Science+Business Media, LLC, part of Springer Nature 2018A. Chambert-Loir et al., Motivic Integration, Progress in Mathematics 325,https://doi.org/10.1007/978-1-4939-7887-8

465

https://doi.org/10.1007/978-1-4939-7887-8

466 APPENDIX

We define ConsX to be the set of all constructible subsets of X.

Remark 1.1.3. — By definition, every globally constructible subset of atopological space X is a constructible set. The converse holds true if X isquasi-compact and if the topology of X admits a basis consisting of retro-compact open sets.

Lemma 1.1.4. — Let X be a quasi-separated scheme. Let U be an opensubset of X.

a) If U is quasi-compact, then U is retrocompact.b) If X is quasi-compact, then U is retrocompact if and only if U is quasi-

compact.c) If X is quasi-compact, then every constructible set of X is globally

constructible.

Proof. — a) Let V be an affine open subscheme of X. If X is quasi-separated,then, by definition, the diagonal morphism ΔX : X → X ×Z X is quasi-compact. So U ∩ V = Δ−1

X (U ×Z V ) is quasi-compact. That concludes theproof.

b) Let (Xi)i∈I be a finite open covering of X by affine schemes. If Uis retrocompact, then U ∩ Xi is quasi-compact for each i ∈ I. Hence U =⋃

i∈I U ∩ Xi is also quasi-compact. If U is quasi-compact and X quasi-separated, for every affine open subscheme V of X, the open set U∩V is quasi-compact as a finite union of quasi-compact open sets. So U is retrocompact.

c) Let C be a constructible subset of X. By the definition of constructibil-ity and the quasi-compactness of X, there exist a finite open covering (Xi)i∈I

by affine schemes and, for each i ∈ I, a finite number of quasi-compact opensets Uα,i, Vα,i of Xi (by b)) such that

C ∩ Xi =⋃

α

Uα,i ∩ (Xi Vα,i) =⋃

α

Uα,i ∩ (X Vα,i).

But C = C ∩ ⋃i∈I Xi =

⋃i∈I

⋃α(Uα,i ∩ (X Vα,i)). Hence C is globally

constructible in X.

Proposition 1.1.5. — The set ConsX of the constructible subsets of ascheme X is a Boolean algebra, i.e., ConsX is stable under finite unions,finite intersections, and complements.

Proof. — See (ÉGA III1, 0III, §9.1).

Remark 1.1.6. — Let X be a noetherian scheme. By Lemma 1.1.4, b),every open set of X is retrocompact. Consequently, the following propertiesare equivalent, for a subset C of X:

(i) C is constructible;(ii) C is globally constructible in X;(iii) There exist locally closed subsets C1, . . . , Cm of X such that C =

C1 ∪ · · · ∪ Cm.Moreover, introducing the irreducible components of the locally closed sub-sets Ci in (iii), we see that they are equivalent to the following:

§ 1. CONSTRUCTIBILITY IN ALGEBRAIC GEOMETRY 467

(iv) There exist irreducible, locally closed, and pairwise disjoint subsetsC1, . . . , Cm of X such that C = C1 ∪ · · · ∪ Cm.

1.2. The Constructible Topology

(1.2.1). — Let X be a scheme. A subset F of X is proconstructible if, forall x ∈ X, there exists an affine open subscheme U of X, containing x, suchthat F ∩ U is an intersection of constructible subsets of U . A subset E of Xis indconstructible if, for all x ∈ X, there exists an affine open subscheme Uof X, containing x, such that F ∩ U is a union of constructible subsets of U .

The family of the indconstructible subsets of X forms the open sets ofa topology on X that we call constructible topology. The closed sets of thistopology are exactly the proconstructible subsets of X, by (ÉGA IV1, 1.9.13).So, by the definition of indconstructibility, this topology is generated by theconstructible subsets of X, which are open and closed in X.

Remark 1.2.2. — When X is a noetherian scheme, this topology is alsogenerated by the (locally closed) subschemes of X.

Remark 1.2.3. — The constructible topology is different, in general, fromthe discrete topology (even in the noetherian case). Consider, for example,X = Spec(Z). For the constructible topology, the prime ideal (p) in Z, with pa prime number, corresponds to an open subset of X; but the generic point η,corresponding to the ideal (0), is closed (since it corresponds to X

⋃p V (p),

where p runs over the set of the prime numbers of Z) and not open in X.Indeed, if η is indconstructible, then {η} =

⋃i∈I Ci, where Ci is locally closed

in X. So there exists i0 such that {η} = Ci0 . Since η is the generic pointof X (for the Zariski topology), Ci0 contains necessarily a nonempty opensubset of X (for the Zariski topology). This is a contradiction.

Theorem 1.2.4. — a) Let X be a quasi-compact scheme. Then X isquasi-compact for the constructible topology. In particular, let F be a procon-structible subset of X, and let (Oi)i∈I be a family of indconstructible subsetsof X such that F ⊂ ⋃

i∈I Oi; then there exists a finite set J ⊂ I such thatF ⊂ ⋃

j∈J Oij .b) (Chevalley) Let f : X → Y be a morphism of schemes which is of finite

type, and let C be a constructible subset of X. Then the subset f(C) of Yis a constructible subset of Y . If Y is noetherian, and if Z is a globallyconstructible subset of X, then f(Z) is a globally constructible subset of Y .

c) Let f : X → Y be a morphism of schemes, and let C be a (resp.globally) constructible subset of Y . Then the subset f−1(C) of X is a (resp.globally) constructible subset of X.

Proof. — a) For the second assertion, see (ÉGA IV1, 1.9.9), and the firstassertion follows from it, see (ÉGA IV1, 1.9.15).

Assertion b) is Chevalley’s theorem. See (ÉGA IV1, 1.8.4).

468 APPENDIX

For c), see (ÉGA IV1, 1.8.2).

1.3. Constructible Subsets of Projective Limits

The arc schemes and the Greenberg schemes used in motivic integrationare non-noetherian in general but are naturally written as a projective limitof a sequence of noetherian schemes.

We recall in this section results of (ÉGA IV1, 8.3) that describe the con-structible subsets of such projective limits.(1.3.1). — Let (I,�) be a filtrant ordered set, and let ((Xi)i∈I , (uij)i�j) bea projective system of noetherian schemes, such that for (i, j) ∈ I2, i � j,uij is an affine morphism of schemes. By (ÉGA IV3, Proposition 8.2.3), theprojective limit X = lim←− Xi exists in the category of schemes. Moreover, forevery i ∈ I, the canonical morphism of schemes ui : X → Xi is affine. Inparticular, X is quasi-compact and quasi-separated, since the Xi are quasi-compact and quasi-separated.(1.3.2). — By theorem 1.2.4/c), the family

((ConsXi

)i∈I , (u−1ij )i�j)

)

forms an inductive system of sets. Moreover, for every i ∈ I, the morphismui : X → Xi induces a map u−1

i : ConsXi→ ConsX , compatibly with the

maps u−1ij . We thus obtain a canonical map:

(1.3.2.1) v : lim−→i∈I

ConsXi→ ConsX .

By restriction to the open, resp. closed constructible subsets, this map in-duces maps

v′ : lim−→i∈I

OpenXi→ OpenConsX ,(1.3.2.2)

v′′ : lim−→i∈I

ClosedXi→ ClosedConsX .(1.3.2.3)

Proposition 1.3.3. — Let ((Xi)i∈I , (uij)i�j) be a projective system ofnoetherian schemes, such that for (i, j) ∈ I2, i � j, uij is an affine mor-phism of schemes. Then the maps v, v′, v′′ defined above are bijections.Moreover, every constructible subset of X is globally constructible.

Proof. — Let i ∈ I. Since Xi is noetherian, constructible and globally con-structible subsets of Xi coincide. By théorème 8.3.11 of (ÉGA IV3), the as-sertion holds when “constructible” is replaced with “globally constructible.”In particular, the image of v lies in the set of globally constructible sub-sets of X. To conclude the proof of the proposition, it suffices to show thatevery constructible subset of X is globally constructible. Let thus A be aconstructible subset of X, and let (Us)s be an open cover of X such thatA ∩ Us is globally constructible in Us for every s. Since X is quasi-compact,we may assume that this cover is finite; by definition of the topology on X,

§ 2. BIRATIONAL GEOMETRY 469

we may then assume that there exist an element i ∈ I and a finite opencover (U ′

s) of Xi such that Us = v−1i (U ′

s) for every j. Fix an index s. Sincev−1

i (U ′s) = lim←−j�i

v−1ij (U ′

i), the first part of the proof implies that there existan element j ∈ I such that j � i and a constructible subset A′

s of v−1ij (U ′

i)and A ∩ Us = v−1

j (As). Then A′ =⋃

A′s is a constructible subset of Xj such

that A = v−1j (A′). This proves that A is globally constructible. If A is closed

(resp. open), we may moreover take A′s to be closed (resp. open) in v−1

ij (U ′i),

so that A′ is closed (resp. open) in Xj . This concludes the proof.

§ 2. BIRATIONAL GEOMETRY

We give a recollection of the main notion of birational geometry that weuse in the book.

2.1. Blow-Ups

Definition 2.1.1. — Let X be a scheme. Let Y be a closed subschemeof X, and let IY ⊂ OX be its sheaf of ideals. The blow-up of X along Y isthe X-scheme BlY (X) = Proj

( ⊕n∈N I n

Y

); the projection p : BlY (X) → X

is called the blowing-up of X along Y , and the subscheme Y is called itscenter.

Let us retain the notation of Definition 2.1.1.(2.1.2). — By construction, the tautological line bundle O(1) on BlY (X) isendowed with an isomorphism to p∗IY ; in particular, p∗IY is an invertiblesheaf, and E = p−1(Y ) is a Cartier divisor in BlY (X). It is called theexceptional divisor. In fact, the blowing-up is the universal morphism to Xthat makes IY a line bundle.

In view of the isomorphismsI n

Y ⊗ (OX/IY ) I nY /I n+1

Y ,

one has an isomorphism

E Proj( ⊕

n∈NI n

Y /I n+1Y

)

of E with the projectivized normal cone of Y .Assume that Y is locally defined by a regular sequence of length r in X,

then IY /I 2Y is locally free of rank r as an OY -module, and the canonical

morphismSym• (

IY /I 2Y

) →⊕

n∈NI n

Y /I n+1Y

is an isomorphism of graded algebras. In this case, E = p−1(Y ) is a projectivebundle of rank r − 1 on Y .

470 APPENDIX

(2.1.3). — Above X Y , the ideal sheaf IY is invertible (generated by 1);hence, the blowing-up map p induces an isomorphism from BlY (X) E toX Y . In particular, if Y is nowhere dense, then p is a birational morphism.(2.1.4). — Let Z be a closed subscheme of X. The Zariski closure Zof p−1(Z Y ) in BlY (X) is called the strict transform of Z in BlY (X).The projection Z → Z identifies with the blowing-up of Z along Y ∩ Z.

Example 2.1.5. — The following example is both the simplest one and isthe source of any property of blow-ups.

Let X = Ank = Spec(k[T1, . . . , Tn]) be the affine space over a field k,

and let Y = V (T1, . . . , Tn) be the origin in X. Inside X ×k Pn−1k =

Spec(k[T1, . . . , Tn]) ×k Proj(k[S1, . . . , Sn]), let B be the closed subschemedefined by the equations TiSj − SjTi, for 1 � i < j � n. (Note that they arehomogeneous of degree 1 in S1, . . . , Sn.)

Then the projection p : B → X is the blowing-up of X along Y ; it isprojective.

The fibers of p identify with closed subschemes of Pn−1k . The exceptional

divisor E = p−1(Y ) is Pn−1k , but for x = (x1, . . . , xn) = 0, the fiber p−1(x)

is the single point of Pn−1k with homogeneous coordinates [x1 : . . . : xn].

Proposition 2.1.6. — Let k be a field, and let us assume that X is k-smooth and Y is a smooth closed subscheme of Y . Then BlY (X) is k-smooth.

2.2. Resolution of Singularities

(2.2.1). — Let k be a field and let X be an integral k-variety.Let D be a divisor (closed, purely 1-codimensional subscheme) in X, and

let (Di)i∈I be the family of its (reduced) irreducible components. For everysubset J of I, let DJ =

⋂i∈J Di. One says that D has strict normal crossings

if, for every subset J of I, the subscheme DJ of X is smooth and purelyCard(J)-codimensional, in other words, if the irreducible components of Dare smooth and meet transversally.(2.2.2). — A resolution of singularities of X is a proper birational morphismp : Y → X such that Y is smooth.

Let p : Y → X be a resolution of singularities of X. There exists a largestsubscheme U of X above which p is an isomorphism; the complementarysubset E is a closed subscheme of X called the exceptional locus of p.(2.2.3). — Let Z be a closed subscheme of X, for example, an effectivedivisor in X. One says that p is a log resolution of the pair (X, Z), if itsexceptional locus E is a divisor, as well as p−1(Z), and if the divisor E +p−1(Z) on Y has strict normal crossings.

Theorem 2.2.4 (Hironaka 1964). — Let k be a field of characteristiczero. Let X be an integral k-variety and let Z be a closed subscheme of X.Then there exists a log resolution p : Y → X of the pair (X, Z) which is an


isomorphism outside of Xsing ∪ Z. Moreover, the morphism p can be takenas a composition of blowing-ups with smooth centers.

More recent versions show that p can be chosen so as to commute withsmooth morphisms. Even more recently, Temkin (2012) has proved a func-torial version of Theorem 2.2.4 valid for arbitrary quasi-excellent Q-schemes.See also (Kollár 2007) for a comprehensive survey, additional references, aswell as a relatively short proof.

Resolution of singularities is yet unknown in positive characteristic, butpartial results are available in small dimension. This dates from the nine-teenth century in Dimension 1, is due to Abhyankar (1998); Lipman (1978) inDimension � 2, and has been recently proved by Cossart and Piltant (2014)in Dimension 3.(2.2.5). — We will say that a field k allows resolution of singularities (forvarieties of dimension � n) if the conclusion of Theorem 2.2.4 holds for everypair (X, Z), whenever X is a k-variety (of dimension � n).

2.3. Weak Factorization Theorem

(2.3.1). — Let k be a field, and let X, Y be integral k-varieties.A rational map ϕ : X �� Y is the datum of a morphism ϕU : U → Y

defined over a dense open subscheme U of X, called an open subscheme ofdefinition. There exists a largest open subscheme of definition, the domainof ϕ. We identify two rational maps when they coincide over a dense commonopen subscheme of their domains of definition; rational maps X �� Y thuscorrespond to k(X)-points of Y .

Any rational map ϕ has a graph Γϕ which is the smallest closed subschemeof X ×k Y containing the graph of a morphism ϕU : U → Y defining ϕ.

One says that ϕ is dominant if ϕU is dominant for some (equivalently, any)open subscheme of definition U . This means that the projection from Γϕ to Yis dominant or, equivalently, that the image of the associated k(X)-point of Yis the generic point of Y . Dominant rational maps can be composed naturallyand give rise to the rational category.

One says that ϕ is birational if it is invertible in the rational category; thismeans that it is dominant and induces an isomorphism from k(X) to k(Y ).

One says that ϕ is proper if the two projections from Γϕ to X and Y areproper. This is automatic if X and Y are themselves proper over k.

Example 2.3.2. — Let X be an integral k-variety, and let Y be a strictclosed subscheme of X. Then, the blowing-up p : BlY (X) → X of X along Yis a proper birational morphism.

Symmetrically, the rational map p−1 : X �� BlY (X) is a proper birationalmorphism; one often says that p−1 is a blowing-down.

Theorem 2.3.3. — Let k be a field of characteristic zero. Let X, Y besmooth k-varieties, and let ϕ : X �� Y be a proper birational map.

472 APPENDIX

a) There exist a sequence (V0, . . . , Vm) of smooth k-varieties such thatV0 = X, Vm = Y , and, for every i ∈ {1, . . . , m}, a proper rational mapϕi : Vi−1 �� Vi such that either ϕi or its inverse is is a blowing-up along asmooth center, and such that ϕ = ϕm ◦ · · · ◦ ϕ1.

Such a sequence can be chosen so as to satisfy the following additionalrequirements:

b) There exists an integer p such that Vi �� X is defined everywhere fori � p, and Vi �� Y is defined everywhere for i > p.

c) Let U ⊂ X and V ⊂ Y be dense open subschemes such that ϕ isinduced by an isomorphism from U to V ; then there exists a dense opensubscheme Ui of ϕi, with U0 = U and V0 = V , such that, for every i, ϕi

induces an isomorphism from Ui−1 to Ui.d) Assume moreover that D = X U (resp. E = Y V ) has a strict

normal crossings divisor. Then one may assume that the inverse image of D(resp. of E) in Vi has strict normal crossings.

e) Let S be a k-variety. Assume that X and Y are S-schemes and that ϕis defined by an S-morphism (over some open subscheme of definition). Thenone may assume that every Vi is an S-scheme and that the morphisms ϕi arerational maps of S-schemes.

This is the weak factorization theorem of Abramovich et al. (2002); Wło-darczyk (2003). It shows in particular that birational morphisms betweensmooth proper k-varieties are compositions of blowing-ups along smooth cen-ters and their inverses. The case of surfaces goes back to the nineteenthcentury and holds in any characteristic: any birational map between smoothprojective surfaces over a field k is a composition of blowing-up and blowing-downs along points.

The adjective weak is a reference to the strong factorization conjecture,which states that one can even assume that ϕ1, . . . , ϕp are blowing-ups andϕp+1, . . . , ϕm are blowing-downs. This statement is known to hold for sur-faces but is yet unproven in dimension � 3.

2.4. Canonical Divisors and Resolutions

(2.4.1) Canonical Divisors. — Let k be a perfect field, and let X be a normalintegral k-variety. Let U = Xsm be the smooth open subset of X; since k isperfect, U coincides with the regular locus of X, and U is dense in X. Infact, the assumption that X is normal implies that codim(X U) � 2; asa consequence, the restriction map Div(X) → Div(U) is an isomorphism ofabelian groups.

Let d = dim(X); the restriction to U of the canonical sheaf ΩdX is lo-

cally free. One says that a divisor D on X is a canonical divisor if thereexists a nonzero meromorphic d-form ω on U such that div(ω) = D|U . Sincecodim(X U, X) � 2, canonical divisors exist, and two canonical divisorson X differ by a principal divisor.


(2.4.2). — Let KX be a canonical divisor on X. One says that X is Goren-stein if KX is a Cartier divisor. More generally, one says that X is Q-Gorenstein if KX is Q-Cartier, that is, if there exists an integer m � 1 suchthat mKX is a Cartier divisor. Then every canonical divisor is Cartier (resp.is Q-Cartier).(2.4.3) Relative Canonical Divisors. — From now on, we assume that X isQ-Gorenstein. Let KX be a canonical divisor on X.

Let X ′ be a normal integral scheme, and let p : X ′ → X be a birationalmorphism; let V be the maximal open subset of X over which p is an iso-morphism; since X is normal, one has codim(X V, X) � 2. Let KX′ be acanonical divisor on X ′.

Since the divisor KX is Q-Cartier and p is dominant, one may consider thepull-back p∗KX of KX in DivQ(X ′). Let us recall its definition: let U be anopen subset of X, u be a rational function on U , and m be a positive integersuch that mKX |U = div(u); then p∗KX |p−1(U) = div(p∗u). The Q-divisorKX′ − p∗KX on X ′ is called a relative canonical divisor. It is a priori definedup to a principal divisor on X ′.

However, there is a canonical representative KX′/X . Let indeed U be thesmooth locus of X, and let ω be a nonzero meromorphic form of maximaldegree on U ; then p∗ω is a nonzero form of maximal degree on p−1(U).Let U ′ be the smooth locus of X ′; there is a unique meromorphic form ω′

of maximal degree on U ′ whose restriction to U ′ ∩ p−1(U) coincides withp∗(ω). We set KX′/X = div(ω′) − p∗(div(ω)). Let ω1 be any other nonzeromeromorphic form of maximal degree on U ; there exists a unique rationalfunction f ∈ k(X)× such that ω1 = fω. The previous construction leads toω′1 = (p∗f)ω′ and

div(ω′1) − p∗ div(ω1) = div((p∗f)ω′) − p∗(div(fω))

= div(p∗f) + div(ω′) − div(p∗f) − p∗(div(ω))= KX′/X ,

so that the divisor KX′/X is a well-defined relative canonical divisor, inde-pendently of any choice.

This divisor KX′/X is called the relative canonical divisor of X ′/X; byconstruction, its support is contained in the exceptional locus Exc(p) of p.One says that the morphism p : X ′ → X is crepant if KX′/X = 0.

Lemma 2.4.4. — Let X and X ′ be smooth integral k-schemes of finitetype, and let p : X ′ → X be a birational morphism. Then the morphism pinduces an injective morphism p∗Ωd

X → ΩdX′ of line bundles whose image is

ΩdX′(−KX′/X). In particular, the relative canonical divisor is an effective

Cartier divisor, and its support is precisely the exceptional locus of p.

Proof. — The injectivity of this morphism follows from the fact that p isgenerically smooth—it is generically an isomorphism. Let E be the uniqueCartier divisor on X ′ such that p∗Ωd

X maps to ΩdX′(−E). Let U = X ′ E so

474 APPENDIX

that the morphism p|U is étale. By Zariski’s main theorem, see (ÉGA III1,corollaire 4.4.5); the morphism p|U is an open immersion, hence Exc(p) ⊂ E.On the other hand, p is not a local isomorphism at any point of E, so thatE ⊂ Exc(p), and finally |E| = Exc(p). Since p∗Ωd

X ΩdX′(−E), the divisor E

is linearly equivalent to KX′/X , hence equal to KX′/X . This concludes theproof.

Example 2.4.5. — Let X be a smooth integral k-variety, and let Z be asmooth irreducible subset of codimension r in X. Let X ′ be the blow-upof X along Z, and let p : X ′ → X be the canonical map; let E = p−1(Z) bethe exceptional divisor. One has KX′/X = (r − 1)E.

(2.4.6) Discrepancies. — Let X be an integral k-variety; let us assume thatX is Q-Gorenstein. Let p : X ′ → X be a proper birational morphism suchthat X ′ is smooth and Exc(p) is a divisor with strict normal crossings; letKX′/X be the relative canonical divisor of p. Let (Ei)i∈I be the family ofirreducible components of Exc(p); there exists a unique family (νi)i∈I ofrational numbers such that

KX′/X =∑

i∈I

νiEi.

The rational number νi is called the discrepancy of KX′/X along Ei.One says that the singularities of X are canonical (resp. log terminal), or

that X is canonical (resp. log terminal), if one has νi � 0 (resp. νi > −1)for every i ∈ I. One may prove, see Kollár and Mori (1998, corollary 2.31),that this property is independent of the choice of the resolution p, but thiswill also follow from results below.

2.5. K-equivalence

Definition 2.5.1. — Let k be field. We say that two Q-Gorenstein k-varieties X and Y are K-equivalent if there exist a smooth k-variety Z andtwo proper birational morphisms

(2.5.1.1) Zf g

X Y

such that the relative canonical divisors KZ/X and KZ/Y are equal.

Such a diagram is called a K-equivalence between X and Y .

Remark 2.5.2. — By definition, two K-equivalent varieties are birational.Conversely, let X and Y be two Q-Gorenstein k-varieties which are bira-

tional. Assume that resolution of singularities holds for k-varieties of dimen-sion � dim(X).


Let h : X �� Y be a birational map. Let U be a dense open subschemeof X on which h is defined, and let Z ′ ⊂ X ×k Y be the closure of its graph;it does not depend on the choice of U ; denote by f ′, g′ the projections of Z ′

to X, Y . Let p : Z → Z ′ be a resolution of singularities; define f = f ′ ◦ p andg = g′ ◦ p. We thus obtain a diagram as in (2.5.1.1).

If X and Y are proper or, more generally, if the birational map h is proper,then the morphisms f and g are proper. However, the condition KZ/X =KZ/Y might not hold in general.

Proposition 2.5.3. — Let k be a field, and let X, Y , and Z be connected,Q-Gorenstein and proper k-varieties. Let f : Z → X and g : Z → Y beproper birational k-morphisms such that the line bundles f∗ωX and g∗ωY arenumerically equivalent.

If that resolution of singularities holds for varieties of dimension �dim(X), then the relative canonical divisors KZ/X and KZ/Y are equal. Inparticular, the line bundles f∗ωX and g∗ωY are isomorphic, and the diagramX

f←− Zg−→ Y is a K-equivalence.

Proof. — Recall that the relative canonical divisors KZ/X and KZ/Y arethe divisors defined by the Jacobian ideals of f and g. In particular, KZ/X

belongs to the divisor class of ωZ ⊗ f∗ω−1X , and KZ/Y belongs to the di-

visor class of ωZ ⊗ g∗ω−1Y . Thus our assumption implies that the divisor

D = KZ/Y − KZ/X is numerically trivial. Since each prime divisor in KZ/X

is exceptional with respect to the morphism f , we have f∗KZ/X = 0. Con-sequently, f∗D = f∗KZ/Y is both effective and numerically trivial; hence,f∗D = 0. Applying lemma 3.39 of Kollár and Mori (1998) to the morphism fand to the f -nef line bundles ±D, this implies that D and −D are effective;hence, D = 0.

Example 2.5.4. — Let k be a field. We say that a Q-Gorenstein propervariety is a Calabi–Yau variety if its canonical class vanishes modulo numer-ical equivalence. Since proposition 2.5.3 applies when X and X ′ have trivialcanonical class, Remark 2.5.2 implies that birational Calabi–Yau varieties areK-equivalent.

Example 2.5.5. — Let X be a Q-Gorenstein proper complex variety. Letfi : Xi → X, i ∈ {1, 2} be two crepant resolutions of singularities of X, i.e.,which verify

KXi− f∗

i KX = 0,

for every i ∈ {1, 2}. It follows from Proposition 2.5.3 that X1 and X2 areK-equivalent.

Remark 2.5.6. — The minimal model program furnishes other examplesof K-equivalences.

Recall that an integral proper k-variety is said to be minimal if it has ter-minal singularities and if its canonical divisor is numerically effective. Theminimal model program predicts that every integral proper k-variety is bi-rational to a minimal variety. This is known in Dimension 2; in fact, every

476 APPENDIX

integral projective k-surface is birational to a unique minimal variety, whichis moreover smooth.

The results of Birkar et al. (2010) imply that for every projective integralQ-Gorenstein k-variety X, there exist integral k-varieties X ′ with terminalsingularities, endowed with a projective proper birational morphism to Xwhich are “minimal over X,” that is to say, such that KX′/X is f -nef. (Ex-plicitly, this means that for every closed irreducible curve C ⊂ X ′ such thatf(C) is a point, one has KX′/X · C � 0.)

Wang (1998, Theorem 1.4, variant 1.11) has proved that two models of aprojective integral k-variety X which are minimal over X are K-equivalent.

2.6. A Birational Cancellation Lemma

Definition 2.6.1. — Let X and Y be irreducible k-varieties. We say thatX and Y are stably birational if there exist m, n ∈ N such that X ×k Pm

k

and Y ×k Pnk are birational.

Remark 2.6.2. — Obviously, two birational varieties are stably birational.The question whether two stably birational complex varieties of the samedimension are rational had been put forward by O. Zariski; see Segre (1950).

However, by Beauville et al. (1985, §3, example 3), the hypersurface V inA4

C defined by the equationy2 + (t4 + 1)(t6 + t4 + 1)z2 = 2x3 + 3t2x2 + t4 + 1.

is not rational (is not birational to P3), but V ×C A3C is rational. Conse-

quently, P3 and V are stably birational but not birational.This example also shows that one cannot take X = V , Y = W = Z = A3

in Theorem 2.6.3 below.

Theorem 2.6.3 (Liu and Sebag 2010). — Let k be a field, and let X andY be integral k-varieties of the same dimension such that X and Y are notboth uniruled. Consider two geometrically integral, rationally chain connectedk-varieties W and Z and a birational map

f : X ×k W��Y ×k Z.

Then neither X nor Y is uniruled, and there exists a unique birational mapg : X �� Y such that the diagram

X ×k W Y ×k Z

X Y

f

p1 q1

g

commutes (the vertical arrows are the projection morphisms).

The general idea of the proof is the following. Because of the assumptionon X, Y , we can show that the action of the given birational map is constanton the second factors and thus cancellable. That procedure gives rise thebirational map g.


We present below another argument, more conceptual, based on the theoryof maximal rationally connected (MRC) fibrations (see Kollár 1996, IV.5.1and IV.5.4) but over fields of characteristic zero. For the general case, werefer to (Liu and Sebag 2010, Theorem 2).

Proof. — Since the question is symmetric in X and Y , we may assume thatX is not uniruled. As the question only depends on X, Y , Z, and W up tobirational equivalence and k has characteristic zero, we can suppose that allvarieties are projective and smooth, by resolution of singularities. Let

π : X �� R(X), θ : Y �� R(Y )

be the maximal rationally connected (MRC) fibrations of X and Y , respec-tively. Then the MRC fibrations of X ×k W and Y ×k Z are X ×k W →X �� R(X) and Y ×k Z → Y �� R(Y ). Indeed, let T be the MRCfibration of X ×k W . Since X ×k W is dominant, Kollár (1996, ChapterIV, Theorem 5.5) implies that there exists a dominant map T �� R(X).By Kollár (1996, chapter IV, (5.1.2)), we have to verify that the fibers ofX ×k W �� R(X) are rationally chain connected. By Kollár (1996, chapterIV, (5.1.2) and definition 5.3), the claim is implied by the assumption on Wand the property of R(X), because of the assumption on the field k.

By Kollár (1996, IV.5.5), f induces a birational map

g′ : R(X) �� R(Y ).

Since X is not uniruled,π : X �� R(X)

is birational. Thus, it implies in particular that X has the same dimension asR(X), which is also the dimension of R(Y ). Thus Y and R(Y ) have the samedimension. It follows that Y cannot be uniruled, since otherwise R(Y ) wouldhave smaller dimension than Y . Thus θ : Y �� R(Y ) is birational, andg = θ−1 ◦ g′ ◦ π is a birational map from X to Y that satisfies the conditionsof the lemma. Uniqueness of g is obvious.

Corollary 2.6.4. — Let k be a field of characteristic zero, and let X andY be stably birational integral k-varieties such that X is not uniruled anddim(Y ) � dim(X). Then X and Y are birational.

Proof. — Applying Theorem 2.6.3 to X and Y ×k Pnk , with n = dim(X) −

dim(Y ), we find that X and Y ×k Pnk are birational. Since X is not uniruled,

this implies that n = 0.

Corollary 2.6.5. — Let k be an algebraically closed field of characteristiczero, and let X, Y be integral k-varieties of the same dimension d � 2. If Xand Y are stably birational, then X and Y are birational.

Proof. — Up to replacing X and Y by a smooth projective model, we mayassume that X and Y are connected smooth projective k-varieties.

The case d = 0 is trivial.

478 APPENDIX

Let us assume that d = 1. For the case d = 1, it suffices to note thatby Lüroth’s theorem, X is not uniruled unless X ∼= P1

k. Consequently, theresult follows from Corollary 2.6.4.

Let us assume that d = 2. By corollary 2.6.4, we may suppose that X andY are uniruled and thus of Kodaira dimension −∞. Looking at the Enriquesclassification of surfaces, we see that there exist smooth, projective, connectedk-curves C and D such that X is birational to C ×k P1

k and Y is birationalto D ×k P1

k. Then C and D are stably birational and thus isomorphic by thecase d = 1. Hence X and Y are birational.

§ 3. FORMAL AND NON-ARCHIMEDEAN GEOMETRY

The aim of this section is to provide a quick introduction to formal schemesand non-Archimedean analytic spaces, at the level needed to read the chapterson motivic integration on formal schemes and analytic spaces. In particular,we need to deal with formal schemes that are not adic and not noetherian, butonly at the most basic level; so we will not develop this theory any further.Likewise, we will say very little about the foundations of non-Archimedeangeometry, since we will mostly work with non-Archimedean analytic spacesin terms of their formal models. For a more thorough introduction to formaland non-Archimedean geometry, we recommend (ÉGA I, §10), (Fantechi et al.2005, Ch.8), Abbes (2010), Bosch (2014), and Temkin (2015).

3.1. Formal Schemes

The language of formal schemes was developed by Grothendieck toanalyze infinitesimal structures in algebraic geometry. It has proven to beextremely useful in various context, in particular in deformation theory andmore general moduli problems. See, for instance, (Fantechi et al. 2005, §8.5)for a taste of such applications. The main difference with the language ofschemes is that the algebraic building blocks are topological rings.(3.1.1) Admissible and Adic Topological Rings. — Let A be a ring, endowedwith a topology. We say that A is a topological ring if addition and mul-tiplication are continuous. A topological A-algebra is a topological ring Bendowed with a continuous ring morphism A → B.

The topology on a topological ring A is called linear if the zero elementhas a basis of neighborhoods that are ideals. Note that an ideal of A withnonempty interior is automatically open, since every translation is a homeo-morphism on A.

We say that the topological ring A is pre-admissible if there exists anideal I in A such that I is open and such that the powers In tend to zeroas n → ∞; this means that for every open neighborhood V of 0 in A, there

§ 3. FORMAL AND NON-ARCHIMEDEAN GEOMETRY 479

exists a positive integer n0 such that the ideal In is contained in V for alln � n0.

Such an ideal I is called an ideal of definition.A pre-admissible topological ring is called admissible if it is separated and

complete.(3.1.2). — The most important class of topological rings in formal geometryare the adic rings. A pre-admissible topological ring A is called pre-adic ifit has an ideal of definition I such that In is open for every n > 0; thisimplies that the ideals In form a basis of open neighborhoods of 0 in A. Ifthis holds for one ideal of definition I, then every ideal of definition has thesame property.

We say that A is adic if, moreover, A is separated and complete. This isequivalent to saying that the natural morphism

A → lim←−

n>0(A/In)

is an isomorphism of topological rings, where A/In carries the discrete topol-ogy for every n. In that case, we call the topology on A the I-adic topology.

If A is an adic topological ring with ideal of definition I and J is an ideal inA, then J is an ideal of definition if and only if there exist integers m, n > 0such that Jm ⊂ In ⊂ J .

Example 3.1.3. — Let A be a ring of characteristic p > 0, for some primep. Then the ring of Witt vectors W (A) is an admissible topological ring. Itis adic if A = Ap, but not in general.

(3.1.4) The Category of Formal Schemes. — Let X be a scheme. By puttingthe discrete topology on the sheaf of regular functions OX , we obtain apresheaf of topological rings, which is not a sheaf when X is not quasi-compact (since the product topology on an infinite product of discrete spacesis not the discrete topology). Passing to the associated sheaf, we obtaina topologically ringed space Xtop whose underlying ringed space is X andwhose rings of sections are discrete on every quasi-compact open subset of X.This construction gives rise to a full embedding of the category of schemesinto the category of locally topologically ringed spaces. From now on, we willview every scheme as a topologically ringed space in this way.(3.1.5). — Let A be an admissible topological ring. Then one can associatewith A its formal spectrum Spf(A), which is a locally ringed space in topolog-ical rings. Its underlying topological space is the set of open prime ideals inA, endowed with the Zariski topology (the topology induced by the Zariskitopology on Spec(A)). Note that for every ideal of definition I in A, themorphism

Spec(A/I) → Spec(A)is a homeomorphism onto Spf(A), because every open prime ideal of A con-tains In for sufficiently large n and thus I.

480 APPENDIX

The structure sheaf on Spf(A) is characterized by the following property:for every open subset U of Spf(A), we have a natural isomorphism of topo-logical rings

OSpf(A)(U) ∼= lim←−I

OSpec(A/I)(U)

where I runs through a fundamental system of ideals of definition in A,ordered by inclusion, and OSpec(A/I)(U) carries the discrete topology.

In particular, OSpf(A)(Spf(A)) = A, and we have a natural morphism oflocally topologically ringed spaces ι : Spf(A) → Spec(A).

If P is an open prime ideal of A, then the local ring OSpf(A),P is not com-plete, in general, but its separated completion with respect to the maximalideal is isomorphic to the separated completion of the local ring AP. Thisimplies that ι is flat if A is noetherian.

Example 3.1.6. — Let A be a local adic topological ring whose maximalideal is an ideal of definition. Then the formal spectrum Spf(A) consists ofa unique point. Thus the underlying topological space of Spf(A) containsvery little information about A, but we can recover the topological ring A bylooking at the global sections of Spf(A).

(3.1.7). — A formal scheme is a topologically ringed space X that is locallyof the form Spf(A), with A an admissible topological ring. We say that Xis affine if it is isomorphic to Spf(A) for some admissible topological ring A.If U is an open subspace of a formal scheme X, and we denote by OU therestriction of OX to U, then the topologically ringed space (U,OU) is again aformal scheme. We call such a pair an open formal subscheme of X.(3.1.8). — A morphism of formal schemes Y → X is a morphism of locallyringed spaces in topological rings. Thus the formal schemes form a full sub-category (For) of the category of locally ringed spaces in topological rings.

If X is affine, then the correspondence

f : Y → X) �→ (f �, OX(X) → OY(Y))

defines a bijection between the set of morphisms of formal schemes Y → Xand the set of continuous ring morphisms OX(X) → OY(Y) (ÉGA I, 10.4.6).

The category (For) has fibered products (ÉGA I, 10.7.3). If A, B, andC are admissible topological rings and A → B and A → C are continuousring morphisms, then the fibered product of Spf(B) and Spf(C) over Spf(A)is given by the formal spectrum of the completed tensor product B⊗AC(ÉGA I, 0.7.7.5).

A morphism of formal schemes Y → X is called separated if the image ofthe diagonal morphism Y → Y ×X Y is closed. A formal scheme X is calledseparated if the unique morphism X → Spec(Z) is separated.(3.1.9) Locally Noetherian Formal Schemes. — A formal scheme X is calledadic (resp. locally noetherian) if it can be covered by affine open formal sub-schemes U such that OX(U) is an adic topological ring (resp. a noetherian


adic topological ring). We say that X is noetherian if it is locally noetherianand its underlying topological space is quasi-compact.

One can prove that for every affine open formal subscheme U of a locallynoetherian formal scheme X, the topological ring OX(U) is adic and noethe-rian (ÉGA I, 10.6.5).

All the local rings of a locally noetherian formal scheme are noetherian.(3.1.10). — For every locally noetherian formal scheme X, we define thedimension dim(X) of X as the supremum of the Krull dimensions of its localrings.

If X is affine, then the dimension of X is equal to the Krull dimension ofOX(X), because the local ring of X at a point x and the localization of O(X)at the open prime ideal corresponding to x have the same completion, andcompleting a noetherian local ring preserves the dimension.(3.1.11). — Let A be a noetherian adic topological ring. An ideal of defi-nition of X = Spf(A) is an ideal sheaf I on X for which there exists an idealof definition I in A such that the ideal I (U) is generated by the image of Iin OX(U), for every affine open formal subscheme U of X.

If X is any locally noetherian formal scheme, then an ideal of definition of Xis an ideal sheaf I whose restriction to each affine open formal subscheme Uis an ideal of definition of U.(3.1.12). — If X is a locally noetherian formal scheme and I is an ideal ofdefinition of X, then the locally ringed space

V (I n) = (|X|,OX/I n)

is a scheme, for every integer n > 0, and the natural morphism of topologi-cally ringed spaces

X ∼= lim−→n

V (I n)

is an isomorphism. In practice, it is often convenient to describe a locallynoetherian scheme X in terms of the schemes V (I n).(3.1.13). — Every locally noetherian formal scheme X has a largest idealof definition I , which is equal to the radical of each ideal of definition of X.The scheme V (I ) is reduced and is called the reduction of X and denotedby Xred.

If J is any ideal of definition of X and f : Y → X is a morphism oflocally noetherian formal schemes, then JOY must be contained in an idealof definition of Y, by continuity of the morphism OX → f∗OY. Thus thecorrespondence X �→ Xred gives rise to a functor (·)red from the category oflocally noetherian formal schemes to the category of reduced schemes.

A morphism of locally noetherian formal schemes f : Y → X is separatedif and only if fred is a separated morphism of schemes.

Example 3.1.14. — If X is a locally noetherian scheme, then the asso-ciated topologically ringed space is a locally noetherian formal scheme. Its

482 APPENDIX

largest ideal of definition is the nilradical of X, so that Xred is the maximalreduced closed subscheme of X.

(3.1.15). — A morphism of locally noetherian formal schemes Y → X iscalled adic if there exists an ideal of definition I of X such that J = IOY

is an ideal of definition of Y; then this property holds for every ideal ofdefinition of X. If X is a locally noetherian formal scheme, then an X-adicformal scheme is a locally noetherian formal scheme Y endowed with an adicmorphism Y → X. If X = Spf(A), then we also speak of A-adic formalschemes instead of X-adic formal schemes.

Example 3.1.16. — Let X be a locally noetherian formal scheme (for in-stance, a locally noetherian scheme), and let Z be a subscheme of Xred. Thenthe formal completion X/Z of X along Z is defined as follows. First, we choosean open formal subscheme U of X containing Z such that Z is closed in X.We denote by I the defining ideal of Z in U, and we set

X/Z = lim−→

n>0V (I n)

where the limit is taken in the category of topologically ringed spaces. Thisdefinition only depends on X and the underlying space of Z, and not on thechoice of U or the schematic structure of Z.

The topologically ringed space X/Z is a locally noetherian formal scheme,with underlying topological space Z, and the morphism of locally ringedspaces

X/Z → X

is called the completion morphism. Note that IOX/Z

is an ideal of defini-

tion of X/Z and that the reduction of X/Z is the maximal reduced closedsubscheme Zred of Z.

The formal scheme X/Z should be viewed as an infinitesimal tube aroundZ in X; it can be used to study the infinitesimal structure of X around Z.

If X′ is another locally noetherian formal scheme, Z ′ is a subscheme ofX′

red, and f : X′ → X is a morphism of formal schemes such that f(Z ′) iscontained in Z and then f induces a morphism of formal schemes

f : X′/Z ′ → X/Z.

It is adic if f−1(Z) is open in Z ′, but not in general.

(3.1.17) Coherent Sheaves. — Let X be a locally noetherian formal scheme.Then by (ÉGA I, 10.11.1), the structure sheaf OX is coherent, and every idealof definition of X is coherent. Coherent sheaves of OX-modules are definedin the usual way (ÉGA I, 0.5.3).

They can be described in terms of coherent sheaves on schemes as follows(see (ÉGA I, §10.10 and §10.11) for details). Let I be an ideal of definition of


X. For all integers n � m > 0, we denote by ιn the morphism of topologicallyringed spaces

ιn : V (I n) → X

and by ιmn the closed immersion of schemes

ιmn : V (I m) → V (I n).

If F is a coherent sheaf on X, then ι∗nF is a coherent sheaf on the scheme

V (I n). Conversely, suppose that we are given a projective system (Fn, fnm)where Fn is a coherent sheaf on V (I n), for every n > 0, and such that for alln � m > 0, the transition map fnm : Fn → Fm is OV (In)-linear and inducesan isomorphism ι∗

mnFn → Fm. Then the projective limit of (Fn, fnm) inthe category of OX-modules is a coherent sheaf F , and ι∗

nF is isomorphic toFn for all n.

If X = Spf(A) is a noetherian affine formal scheme and M is an A-moduleof finite type, then we can define a coherent sheaf M on X by first takingthe coherent sheaf on Spec(A) associated with M and then pulling it backthrough the completion morphism Spf(A) → Spec(A). For every affine openformal subscheme U of X, we have

M (U) = M ⊗A OX(U).

In particular, M (X) = M . The correspondence M �→ M defines an equiva-lence of categories between the category of A-modules of finite type and thecategory of coherent sheaves on X = Spf(A).(3.1.18) Closed Formal Subschemes. — Let X be a locally noetherian formalscheme, and let J be a coherent ideal sheaf of X. If we denote by Y thesupport of the quotient sheaf OX/J and by OY the restriction of OX/J toY, then the pair (Y,OY) is again a locally noetherian formal scheme. Formalschemes that are constructed in this way are called closed formal subschemesof X.

A closed (resp. open) immersion of locally noetherian formal schemes Y →X is a morphism of formal schemes that factors through an isomorphism ontoa closed (resp. open) formal subscheme of X.

3.2. Morphisms of Finite Type and Morphisms Formally of FiniteType

(3.2.1) Algebras of Convergent Power Series. — Let A be a noetherian adictopological ring with ideal of definition I, and let r be a positive integer. TheA-algebra of convergent power series A{z1, . . . , zr} in the variables z1, . . . , zr

is the sub-A-algebra of A[[z1, . . . , zr]] consisting of the power series whosecoefficients tend to zero:

A{z1, . . . , zr} = {∑

ν∈Nr

aνzν | aν → 0 as ‖ν‖ → ∞}

484 APPENDIX

where we used the usual multi-index notation zν = zν11 . . . zνr

r and we set‖ν‖ = ν1 + . . . + νr. We turn B = A{z1, . . . , zr} into an adic topologicalA-algebra by endowing it with the IB-adic topology.

Another way to describe this topological A-algebra is as a projective limitA{z1, . . . , zr} = lim

←−n>0

(A/In)[z1, . . . , zr]

where (A/In)[z1, . . . , zr] carries the discrete topology, since the ideals In forma basis of open neighborhoods of 0 in A.

The ring A{z1, . . . , zr} is noetherian (ÉGA I, 0.7.5.4).The name “convergent power series” refers to the fact that for every ele-

ment f in A{z1, . . . , zr} and every a in Ar, the series f(a) converges in A.Some authors use the term “restricted power series” instead.

A topological A-algebra is called topologically of finite type if it is isomor-phic to a topological A-algebra of the form C = A{z1, . . . , zr}/J , endowedwith the IC-adic topology, where J is an ideal in A{z1, . . . , zr}.(3.2.2) Morphisms of Finite Type. — We say that a morphism of locallynoetherian formal schemes f : Y → X is locally of finite type if, for everypoint y of Y, we can find an affine open neighborhood U of f(y) in X and anaffine open neighborhood V of y in f−1(U) such that OY(V) is topologicallyof finite type over OX(U). This is equivalent to saying that f is adic and, forsome ideal of definition I of X, the scheme V (IOY) is locally of finite typeover V (I ); then this property holds for all ideals of definition I of X.

We say that f is of finite type if it is locally of finite type and quasi-compact.

The class of morphisms locally of finite type (resp. of finite type) is stableunder composition and base change.

Open and closed immersions of locally noetherian formal schemes are mor-phisms of finite type.

Example 3.2.3. — Let A be a noetherian adic topological ring with idealof definition I. For every A-scheme X locally of finite type, the completionof X along the closed subscheme X ×A (A/I) is a formal A-scheme locally offinite type, which we denote by X. It is called the I-adic completion of X.By construction, we have X ×A (A/In) = X ×A (A/In) for every n > 0.

If X is affine, say,X = Spec(A[z1, . . . , zr]/(f1, . . . , f�)),

then X is given by

X = Spf(A{z1, . . . , zr}/(f1, . . . , f�)).

(3.2.4) Morphisms Formally of Finite Type. — One can relax the definitionof a morphism of finite type and still obtain a class of morphisms with goodproperties.

Let A be a noetherian adic topological ring with ideal of definition I. Atopological A-algebra is called formally of finite type if it is isomorphic to a


quotient of a topological A-algebra of the formB = A{z1, . . . , zr}[[w1, . . . , ws]]

endowed with the IB + (w1, . . . , ws)-adic topology.(3.2.5). — Let X be a locally noetherian scheme. We say that a morphismof formal schemes f : Y → X is locally formally of finite type if, for every pointy of Y, there exist an affine open neighborhood U of f(y) in Y and an affineopen neighborhood V of y in f−1(U) such that OY(V) is formally of finitetype over OX(U). This implies, in particular, that Y is locally noetherian.

It follows from Tarrío et al. (2007, 1.7) that a morphism of formal schemesf : Y → X is locally formally of finite type if and only if there exist an idealof definition I of X and an ideal of definition J of Y such that IOY iscontained in J and the scheme V (J ) is locally of finite type over V (I );then this property holds for all ideals of definition I and J such that IOY

is contained in J .Note that every morphism locally of finite type is locally formally of finite

type and that a morphism locally formally of finite type is locally of finitetype if and only if it is adic.

We say that f is formally of finite type if it is locally formally of finitetype and quasi-compact.

The class of morphisms locally formally of finite type (resp. formally offinite type) is stable under composition and base change.

These morphisms appear under various names in the literature: morphismslocally formally of finite type are called special in Berkovich (1996a), andmorphisms formally of finite type are called morphisms of pseudo-finite typein Tarrío et al. (2007).

Example 3.2.6. — Let A be a noetherian adic topological ring with idealof definition I, let X be an A-scheme locally of finite type, and let Z bea subscheme of X ×A (A/I). Then the completion X/Z of X along Z is aformal A-scheme locally formally of finite type, since its reduction is equalto Zred and thus locally of finite type over Spf(A)red = Spec((A/I)red). Theformal scheme X/Z is locally of finite type over A if Z is open in X ×A (A/I),but not in general. It is separated if and only if Z is separated, and it is flatover A if and only if the scheme X is flat over A at every point of Z.

If X is affine, say,X = Spec(A[z1, . . . , zr]/(f1, . . . , f�)),

and Z is the zero locus of the ideal generated by I and (zq, . . . , zr) for someinteger q > 0, then X is given by

X = Spf(A{z1, . . . , zq−1}[[zq, . . . , zr]]/(f1, . . . , f�)).More generally, if X is a formal scheme locally formally of finite type over

A and Z is a subscheme of Xred, then the completion X/Z is still locallyformally of finite type over A.

486 APPENDIX

3.3. Smoothness and Differentials

In this book, the notion of smoothness of formal schemes and the sheaves ofdifferentials play an important role. The theory can be developed analogouslyto the case of schemes, and we refer to (Tarrío et al. 2007) for a detailedaccount.(3.3.1) Infinitesimal Lifting Criteria. — One can define formally unrami-fied, étale, and smooth morphisms of formal schemes much in the same wayas for schemes, in terms of infinitesimal lifting criteria.

Let f : Y → X be a morphism of locally noetherian formal schemes. Thenwe say that f is formally unramified (resp. formally étale, formally smooth) ifit satisfies the following infinitesimal lifting criterion: for every affine schemeZ over X and every closed subscheme T of Z define by a square zero ideal,the map

Hom(For/X)(Z,Y) → Hom(For/X)(T,Y)is injective (resp. bijective, surjective) (Tarrío et al. 2007, definition 2.1).(3.3.2). — A morphism of locally noetherian formal schemes f : Y → X iscalled unramified (resp. étale, smooth) if it is formally unramified (resp. for-mally étale, formally smooth) and locally formally of finite type (Tarrío et al.2007, definition 2.6).

Each of these classes of morphisms is stable under composition and basechange (Tarrío et al. 2007, proposition 2.9).

We say that f is unramified (resp. étale, smooth) at a point y of Y if thereexists an open neighborhood U of y in Y such that the morphism U → Xinduced by f is unramified (resp. étale, smooth).

Example 3.3.3. — a) If A is a noetherian adic topological ring, thenthe formal scheme

Spf(A{z1, . . . , zr}[[w1, . . . , ws]])

is smooth over Spf(A) for all r, s � 0. Beware that it is not of finite typeover Spf(A) unless s = 0. Assume that A is local and that its maximal idealm is an ideal of definition. We denote by k = A/m the residue field of A.Then a formal A-scheme formally of finite type X is smooth over Spf(A)at a k-rational point of Xred if and only if the completed local ring OX,x isisomorphic, as an A-algebra, to a power series ring A[[w1, . . . , ws]].

b) If X is a locally noetherian formal scheme and Z is a subscheme of Xred,then the completion morphism

X/Z → X

is étale.

(3.3.4) Modules of Differentials. — Analogously to the case of schemes, onecan study differential properties of morphisms of formal schemes by meansof modules of differentials.


Let A be a noetherian adic topological ring, and let B be a topologicalA-algebra formally of finite type. The usual module of Kähler differentialsΩB/A (forgetting the topology on A and B) is not of finite type over B, ingeneral. However, its completion is well behaved; it is defined by

ΩB/A = lim←−

n>0(ΩB/A/JnΩB/A)

where J is any ideal of definition in B. This is a B-module of finite type. Wecall it the module of continuous differentials of B over A.

The mapd : B → ΩB/A, b �→ db

is the universal continuous derivation of B over A into a complete B-module;see Tarrío et al. (2007, 1.10).

Example 3.3.5. — Assume that

B = A{z1, . . . , zr}[[w1, . . . , ws]].

In this case, the elements dz1, . . . , dzr, dw1, . . . , dws form a basis of ΩB/A,which is thus free of rank r + s, and the differential of an element f ∈ B iscomputed formally:

df =r∑

i=1

∂f

∂zidzi +

s∑

j=1

∂f

∂wjdwj .

Let I be an ideal of B, and let C = B/I. Let N be the submodule of ΩB/A

generated by elements of the form df , with f ∈ I. The universal continuousderivation on B induces a continuous derivation d : C → (ΩB/A/N)⊗AC of Cover A which satisfies the universal property. This leads to the fundamentalexact sequence of C-modules:

(3.3.5.1) I/I2 → ΩB/A ⊗B C → ΩC/A → 0.

The corresponding complex

(3.3.5.2) 0 → I/I2 → ΩB/A ⊗B C → ΩC/A → 0

will be called the fundamental complex associated with the pair (B, I).

(3.3.6). — This definition of continuous differentials can be globalized todefine the coherent sheaf of (continuous) differentials ΩY/X for an arbitrarymorphism Y → X between locally noetherian schemes which is locally for-mally of finite type. In particular, if Y = Spf(B) → X = Spf(A) is amorphism of finite type of noetherian affine formal schemes, then the sheafof differentials ΩY/X is the coherent sheaf on Y associated with the B-moduleof finite type ΩB/A.

These sheaves of differentials satisfy the usual calculus. Let f : Y → Xand g : Z → Y be morphisms locally formally of finite type between locallynoetherian formal schemes.

488 APPENDIX

a) There is a canonical exact sequence of OZ-modules:(3.3.6.1) g∗ΩY/X → ΩZ/X → ΩZ/Y → 0.

b) Assume moreover that g is a closed immersion, defined by a coherentideal sheaf J on Y, then there is a canonical fundamental exact sequence ofcoherent OZ-modules:(3.3.6.2) J /J 2 → g∗ΩY/X → ΩZ/X → 0,

and a canonical fundamental complex of coherent OZ-modules associates withthe immersion g:(3.3.6.3) 0 → J /J 2 → g∗ΩY/X → ΩZ/X → 0,

which generalize the exact sequences (3.3.5.1) and (3.3.5.2) in the affine case.(3.3.7). — One has the expected relations between infinitesimal lifting prop-erties and modules of differentials; proofs can be found in section 4 of Tarríoet al. (2007).

Let f : Y → X and g : Z → Y be morphisms locally formally of finite typebetween locally noetherian formal schemes.

Then f is unramified if and only if ΩY/X = 0.If g is étale, then the map g∗ΩY/X → ΩZ/X is an isomorphism.If f is smooth, then f is flat, and ΩY/X is locally free.Moreover, if f is smooth, then g is smooth if and only if g ◦ f is smooth

and the sequence of OZ-modules0 → g∗ΩY/X → ΩZ/X → ΩZ/Y → 0

is exact and locally split.Finally, there is also a version of Zariski’s Jacobian criterion in this setting:

if f is smooth and g is a closed immersion defined by a coherent ideal sheafJ on Y, then g ◦ f is smooth if and only if the sequence of OZ-modules

0 → J /J 2 → g∗ΩY/X → ΩZ/X → 0is exact and locally split.

3.4. Formal Schemes over a Complete Discrete ValuationRing

(3.4.1). — For our purposes, the most important class of formal schemesis the following. Let R be a complete discrete valuation ring with maximalideal m. We endow R with its m-adic topology; then R becomes an adictopological ring.

A formal R-scheme of finite type (resp. formally of finite type) is a formalscheme X over Spf(R) such that the structural morphism X → Spf(R) is offinite type (resp. formally of finite type).

Formal schemes formally of finite type over R are excellent; this is Propo-sition 7 of Valabrega (1975) when R has equal characteristic and Theorem 9of Valabrega (1976) in the mixed characteristic case.


(3.4.2). — For every integer n � 0, we set Rn = R/mn+1 and

Xn = X ⊗R Rn.

If X is formally of finite type over R, then Xn is a formal Rn-scheme formallyof finite type, for every n. The formal k-scheme X0 is called the special fiberof X. If X is of finite type over R, then Xn is a scheme of finite type over Rn

for every n.Conversely, if (Xn)n>0 is a direct system of R-schemes of finite type such

that mn+1 = 0 on Xn for every n � 0 and such that the transition morphismfmn : Xm → Xn induces an isomorphism of Rm-schemes Xm

∼= Xn ⊗RnRm

for all 0 � m � n, then the direct limit

X = lim−→

n>0Xn

in the category of topologically ringed spaces is a formal R-scheme of finitetype, and the natural morphism Xn → X induces an isomorphism Xn → Xn

for every n � 0.(3.4.3). — In the same way, giving a morphism f : Y → X of formal R-schemes of finite type amounts to giving a family of morphisms (fn : Yn →Xn)n�0 such that fn is a morphism of Rn-schemes and all the squares

Ym Xm

Yn Xn

fm

fn

are Cartesian.(3.4.4). — Let X be a formal scheme formally of finite type over R.

We define the relative dimension of X to be the dimension of its specialfiber X0. More generally, for every point x ∈ X(k), the dimension of thespecial fiber X0 at the point x is called the relative dimension of X at x.

We say that X is flat over R if the structural morphism X → Spf(R) isa flat morphism of locally ringed spaces; this is equivalent to saying thatfor every affine open formal subscheme U of X, the R-algebra OX(U) has nom-torsion.(3.4.5). — An immersion of formal R-schemes f : X → Y is called regularat x ∈ X if the kernel of the local homomorphism OY,f(x) → OX,x can begenerated by the elements of a regular sequence in OY,f(x).

This property is stable under base change to arbitrary extensions of R,because the regularity of a sequence of elements in a ring is preserved underflat morphisms (Liu 2002, 6.3.10).

We say that a formal R-scheme of finite type X is a local complete inter-section at a point x ∈ X if x has an open neighborhood U in X such thatthere exist a smooth formal R-scheme Y and an immersion X → Y which is

490 APPENDIX

regular at x. We say that X is a local complete intersection if it is so at everypoint.

Lemma 3.4.6. — Let X be a formal R-scheme of finite type, and let x ∈X(k); let d be the relative dimension of X at x over R.

If X is a local complete intersection at x, then there exist an integer � 0,a regular sequence (f1, . . . , f�) in R[[z1, . . . , z�+d]], and an isomorphism ofcompleted local rings

R[[z1, . . . , z�+d]]/(f1, . . . , f�)�−→ OX,x.

Proof. — Up to replacing X by a formal affine open neighborhood of x, wemay assume that there exists a regular immersion f : X → Y, where Y is asmooth formal R-scheme. Let y = f(x) and let r be the relative dimensionof Y at y. The completed local ring of Y is isomorphic to R[[z1, . . . , zr]]. Byflatness of the completion morphism OX,x → OX,x, the immersion f inducesa surjective morphism

R[[z1, . . . , zr]]→ OX,x,

whose kernel I is generated by a regular sequence (f1, . . . , f�). It then followsfrom (ÉGA IV2, 7.1.4) that

dim(OX,x) = dim(OX,x) = dim(R[[z1, . . . , zr]]) − = r − .

This concludes the proof.

(3.4.7) Rig-Irreducible Components of Formal Schemes. — Let X be a for-mal R-scheme formally of finite type. We want to define a notion of irre-ducible components of X. This is not obvious, because the topology of Xreflects its geometry rather poorly. For instance, if X is the formal spectrumof a complete local ring, then its underlying topological space consists of asingle point, but its ring of regular functions may have several minimal primeideals. In order to solve this issue, we adopt a similar strategy as the one ofConrad (1999).

Let h : X → X be the normalization morphism constructed in Conrad(1999, 2.1). If X is affine and N is the nilradical of O(X), then X is the formalspectrum of the integral closure of O(X)/N in its total ring of fractions. Thegeneral construction is then carried out by gluing. Since X is excellent, thenormalization morphism h : X → X is finite.

The rig-irreducible components of X are the closed formal subschemes of Xdefined by the coherent ideal sheaves of the form ker(OX → h∗OC) where C

is a connected component of X.If X is affine, then its rig-irreducible components are simply the closed

formal subschemes defined by the minimal prime ideals in OX(X).We say that X is rig-irreducible if it is nonempty and has a unique rig-

irreducible component.


Remark 3.4.8. — The rig-irreducible components of X may display someunexpected behavior: the following example shows that a nonempty opensubscheme of a rig-irreducible formal scheme is not necessarily rig-irreducible!

Let π be a uniformizer in R and letX = Spf(R{x, y}/(π − xy)).

Then X is normal (even regular) and connected. However, if we remove thepoint defined by the ideal (π, x, y), then we find an open formal subschemeof X which is disconnected and has two rig-irreducible components (definedby the equations x = 0 and y = 0, respectively).

Proposition 3.4.9. — Let X be a formal R-scheme formally of finite typeand let d be a nonnegative integer. Then the following are equivalent:

a) Every rig-irreducible component of X has dimension d;b) The local ring at each closed point of X is equidimensional of dimen-

sion d.

If X satisfies the equivalent conditions of Proposition 3.4.9, we say that Xhas pure dimension d.

Proof. — Since normalization commutes with base change to an open formalsubscheme, we may assume that X is affine, say, X = Spf(A). Then therig-irreducible components of X are the closed formal subschemes defined bythe minimal prime ideals of A. It follows from (ÉGA IV2, 7.1.2) that thelocalization of A at an open prime ideal p is equidimensional if and only ifthe local ring of X at the point corresponding to p is equidimensional. Thuswe may assume that A is integral, and it is enough to show that the Krulldimension of A is equal to the height of each maximal ideal. Every maximalideal of A contains m, because A is m-adically complete. Thus, in order toprove the desired property, we may replace A by A ⊗R k, since this eitherreduces the dimension of A and the height of each maximal ideal by 1 (ifm is nonzero on A) or leaves them invariant (if m = 0 on A). The ringA ⊗R k is an adic completion of a finitely generated k-algebra, so that theresult follows from the analogous property for finitely generated k-algebras(ÉGA IV2, 5.2.1).

3.5. Non-Archimedean Analytic Spaces

(3.5.1). — Let R be a complete discrete valuation ring with residue field kand quotient field K. The valuation vK on K gives rise to an absolute valueby setting |x| = exp(−vK(x)) for every element x of K×. Non-Archimedeananalytic geometry is a theory of analytic spaces over the non-Archimedeanfield K.

Naïvely mimicking the definitions and constructions over the complexnumbers gives rise to the theory discussed in §1/1. Despite its interest, itis not satisfactory for the purpose of algebraic geometry, because the metrictopology on a non-Archimedean field is totally disconnected. Historically,

492 APPENDIX

various theories have been proposed to solve this issue: rigid analytic spaces(Tate 1971), formal geometry up to admissible blowing-ups (Raynaud 1974;Abbes 2010; Bosch 2014), and, more recently, the theory of analytic spacesdeveloped by Berkovich (1990, 1993) and the theory of adic spaces (Huber1994). For the spaces that we will consider, all of these theories give rise toessentially equivalent categories.

For practical reasons, and because they are closer to the standard intuition,we use in this book Berkovich’s theory (1).

We will give a concise overview of some important features of the theory,with an emphasis on the theory of formal models, which is a fundamentalviewpoint for the applications in this volume.(3.5.2) Affinoid Algebras. — The basic building blocks of the theory arespectra of affinoid algebras. For every integer r > 0, we consider the Tatealgebra Tr of convergent power series with coefficients in K, defined by

Tr = K{z1, . . . , zr}= {f =

∑

ν∈Nr

aνzν ∈ K[[z1, . . . , zr]] | |aν | → 0 as ‖ν‖ → ∞},

in which we use the standard multi-index notation zν = zν11 · . . . · zνr

r andset ‖ν‖ =

∑ri=1 νi. The convergence condition in the definition implies in

particular that the coefficient aν belongs to R when ‖ν‖ is sufficiently large.More precisely, the canonical morphism of K-algebras

R{z1, . . . , zr} ⊗R K → K{z1, . . . , zr}is an isomorphism.

The algebra Tr is a Banach algebra over K with respect to the Gaussnorm

f‖ = maxν

|aν |.One can show that Tr is noetherian and that every ideal I of Tr is closedwith respect to this Gauss norm, so that we can consider the residue normon the quotient Tr/I.

A Banach algebra A over K is called (strictly) affinoid if there exists asurjective morphism of K-algebras ϕ : Tr → A such that the norm on A isequivalent to the residue norm on Tr/ ker(ϕ).(3.5.3) Berkovich Spectrum. — Let A be Banach algebra. The Berkovichspectrum of A is the set M (A) of all bounded multiplicative seminorms onA. In other words, it is the set of maps x : A → R�0 such that

– x(0) = 0 and x(a + b) � x(a) + x(b) for all elements a, b in A;– x(1) = 1 and x(ab) = x(a)x(b) for all elements a, b in A;– x(a) = |a| for every a ∈ K;– There exists a constant C > 0 such that x(a) � C ‖a‖ for every a in A.

(1)We will only consider strictly K-analytic and Hausdorff analytic spaces.


We endow M (A) with the topology of pointwise convergence, that is, theweakest topology such that the evaluation map x �→ x(a) from M (A) to R�0is continuous, for every a in A.

One also defines a Grothendieck topology on M (A), called the G-topology,and a sheaf of analytic functions with respect to the G-topology.(3.5.4) K-analytic Spaces. — K-analytic spaces are defined by consideringtopological spaces endowed with a suitable notion of affinoid atlas, using thelanguage of nets—see Berkovich (1993, §1). They are endowed both with aGrothendieck topology and with a usual topology.

Every G-open of an analytic space inherits a natural structure of analyticspace, and such a subspace is called an analytic domain. The embeddingof an analytic domain in a K-analytic space is called an analytic domainimmersion .

A K-analytic space is called good if every point has an affinoid neighbor-hood. This is the category of spaces that was originally defined in Berkovich(1990), but non-good analytic spaces arise naturally, for instance, by consid-ering generic fibers of certain formal R-schemes. See, for instance, Exam-ple 4.2.1.4 in Temkin (2015).

To every separated K-scheme X of finite type, one can associate a K-analytic space Xan by a process of analytification. The space Xan is good,and it is compact if and only if X is proper over K.(3.5.5) Analytic Spaces Versus Rigid Varieties. — Historically, the first fullydeveloped theory of non-Archimedean analytic spaces was Tate’s theory ofrigid varieties. The category of Hausdorff strictly K-analytic spaces ad-mits a fully faithful embedding into the category of quasi-separated rigidK-varieties, and this embedding restricts to an equivalence between the cat-egory of paracompact Hausdorff strictly K-analytic spaces and the categoryof quasi-separated rigid K-varieties that have an admissible affinoid coveringof finite type (Berkovich 1993, 1.6.1). In particular, we have an equivalencebetween the category of compact strictly K-analytic spaces and the categoryof quasi-compact quasi-separated rigid K-varieties.

Under this embedding, strict affinoid domains in K-analytic spaces cor-respond to affinoid open subvarieties in rigid K-varieties, and strict com-pact analytic domains correspond to quasi-compact open rigid subvarieties(Berkovich 1993, 1.6.2).(3.5.6). — A little extra care is needed in dealing with the notion of smooth-ness: if f is a morphism of Hausdorff strictly K-analytic spaces, then f isquasi-smooth (or rig-smooth) if and only if the associated morphism of rigidK-varieties is smooth. This notion will be more useful for us than Berkovich’sdefinition of smooth morphisms from Berkovich (1993). A nice reference forthe theory of quasi-smooth morphisms is chapter 5 of (Ducros 2018).

If the source and target of f are good K-analytic spaces, the morphism fis smooth if and only if it is quasi-smooth and has no boundary. A typical ex-ample of a morphism that is quasi-smooth (even quasi-étale) but not smooth

494 APPENDIX

is the embedding of the closed unit disk in the analytification of the affineline over K; the existence of a boundary point prevents this embedding frombeing smooth.(3.5.7) Formal Models. — The comparison results for the categories of rigidK-varieties and K-analytic spaces allow us to apply results from the literatureon rigid analytic geometry to K-analytic spaces, in particular, Raynaud’sdescription of the category of rigid K-varieties in terms of their formal models(Bosch and Lütkebohmert 1993).

Let us briefly recall the main ingredients of this theory, translated intothe language of K-analytic spaces and applying our convention that all K-analytic spaces are assumed to be strictly K-analytic and Hausdorff.

There is a generic fiber functor X �→ Xη from the category of formal R-schemes of finite type to the category of compact K-analytic spaces. If Xis affine, say X = Spf(A), then Xη = M (A ⊗R K). If Y → X is an open(resp. closed) immersion of formal R-schemes of finite type, then Yη → Xη

is an analytic domain (resp. closed) immersion of K-analytic spaces. If Xis a flat formal R-scheme of finite type whose generic fiber Xη is good andquasi-smooth, then Xη is smooth if and only if X is proper over R.(3.5.8). — We call a formal R-scheme admissible if it is flat and of finitetype (beware that, in Bosch and Lütkebohmert (1993), admissible means flatand locally of finite type).

Let X be an admissible formal R-scheme with ideal of definition I . LetA be an open coherent sheaf of ideals on X, and let Y = V (A ) be the formalsubscheme of X that it defines. For every n ∈ N, let Xn = X ⊗R Rn andYn = Y ⊗R Rn. The formal R-scheme

X′ = lim−→n

BlYn(Xn)

is an admissible formal R-scheme, called the formal blow-up of X along Y;the canonical morphism X′ → X is called a formal blowing-up, and Y is calledits center.

The generic fiber functor maps admissible blowing-ups of admissible for-mal R-schemes to isomorphisms of K-analytic spaces. Thus, it induces afunctor from the category of admissible formal R-schemes modulo admissibleblowing-ups to the category of compact K-analytic spaces. The key result inRaynaud’s theory states that this is an equivalence of categories (Bosch andLütkebohmert 1993, 4.1).

In more explicit terms, every compact K-analytic space is isomorphic tothe generic fiber of an admissible formal R-scheme (called an admissible for-mal model of the analytic space), and two morphisms of admissible formalR-schemes X → Y coincide if and only if the induced morphisms Xη → Yη

are equal. Moreover, for every pair of admissible formal R- schemes X, Y andevery morphism of K-analytic spaces f : Yη → Xη, there exists an admissibleblowing-up Y′ → Y such that f extends to a morphism of formal R-schemes


Y′ → X. In particular, the natural map X(R′) → Xη(K ′) is a bijection, forevery finite extension R′ of R with quotient field K ′.

We will also use that, for every admissible formal R-scheme X and everycompact analytic domain U in Xη, there exist an admissible blowing-up X′ →X and an open immersion U → X such that U is the image of Uη → Xη (Boschand Lütkebohmert 1993, 4.4).

(3.5.9). — Let X be an R-scheme of finite type and denote by X its formalm-adic completion. There exists a natural morphism of K-analytic spacesXη → (XK)an. This is an analytic domain immersion if X is separated overR, and it is an isomorphism if X is proper over R.(3.5.10) The Generic Fiber of a Formal Scheme Formally of Finite Type. —In Berthelot (1996), the construction of the generic fiber functor for formalR-schemes of finite type is extended to formal R-schemes that are formallyof finite type. A summary together with some additional results can also befound in de Jong (1995, §7). An equivalent construction is given in Berkovich(1996a) in the framework of K-analytic spaces.

For our purposes, it is sufficient to recall the following properties. Let Ybe a formal R-scheme formally of finite type, and let Yη be its generic fiber.This is a paracompact Hausdorff strictly K-analytic space. It is compact ifX is of finite type over R, but not in general.

For every finite extension R′ of R with quotient field K ′, there is a naturalbijection Y(R′) → Yη(K ′). Moreover, if X → Y is the dilatation of Y(defined in section 7/5.1.1), then the induced morphism Xη → Yη is ananalytic domain immersion, and it induces a bijection Xη(K ′) → Yη(K ′)for every finite unramified extension K ′ of K (the latter property followsfrom the fact that X(R′) → Y(R′) is a bijection for every finite unramifiedextension R′ of R).

If X is a K-analytic space, then a formal R-model of X is a formal R-scheme X formally of finite type endowed with an isomorphism of K-analyticspaces Xη → X.

If X and X′ are formal R-models of X, then a morphism of formal R-models X′ → X is a morphism of formal R-schemes whose restriction to thegeneric fibers commutes with the isomorphisms to X. If such a morphismexists, we say that X′ dominates X. If X′ is flat over R, then there exists atmost one morphism of formal R-models X′ → X.

Example 3.5.11. — Assume that Y is affine, say

Y = Spf(R{z1, . . . , zr}[[w1, . . . , ws]]/(f1, . . . , f�)).

The generic fiber Yη can be explicitly described as follows.Let B = M (K{z}) be the closed unit disk over K, and let E be the open

unit disk over K. Then each power series fi ∈ O(Y) defines an analyticfunction on Br ×K Es, and Yη is the closed analytic subspace of Br ×K Es

defined by the equations f1 = . . . = f� = 0.

496 APPENDIX

(3.5.12) The Specialization Map. — For every formal R-scheme formally offinite type X, there exists a canonical specialization map

spX : Xη → X.

This is a morphism of locally ringed spaces with respect to the G-topologyon Xη, but the map on underlying topological spaces is anti-continuous withrespect to the Berkovich topology on Xη: the inverse image of a closed set inX is open in Xη.

If R′ is a finite extension of R with quotient field K ′ and x is a K ′-pointon Xη, then spX(x) is the image of the unique extension of x to a morphismSpf(R′) → X.

If Y is a locally closed subset of X0, then sp−1X (Y ) is an analytic domain

in Xη that is canonically isomorphic to the generic fiber of the completion ofX along Y .(3.5.13) Dimension and Irreducible Components. — The irreducible com-ponents of rigid K-varieties have been defined by Conrad (1999) via a processof normalization similar to the one we have used to define the rig-irreduciblecomponents of formal schemes in §3.4.7. A different and more general def-inition for analytic spaces was given by Ducros (2009), and he showed thathis definition is equivalent to the one of Conrad for rigid varieties. In orderto preserve the parallels with the definition for formal schemes, and since weonly need to deal with Hausdorff strictly K-analytic varieties, we will followConrad’s definition, translated to the language of Berkovich spaces by meansof the results in Ducros (2009).

Let X be a K-analytic space. The normalization h : X → X is definedin Ducros (2009, 5.6, 5.10); it is a finite surjective morphism of K-analyticspaces (Ducros 2009, 5.11.1).

We define the irreducible components of X to be the closed analytic sub-spaces defined by a coherent ideal sheaf of the form ker(OX → h∗OC) whereC is a connected component of X. This is equivalent to Ducros’ definition,by (Ducros 2009, 5.15).

Proposition 3.5.14. — Let X be a flat formal R-scheme formally of finitetype. Then the generic fiber functor (·)η induces a bijection between the setof rig-irreducible components of X and the set of irreducible components ofXη.

Proof. — This property was proven in Conrad (1999, 2.3.1) for rigid varieties,so that it suffices to show that the notion of irreducible component behaveswell under the comparison functors between the categories of rigid varietiesand analytic spaces in §3.5.5. For every K-analytic space X, we will denotethe associated rigid variety by Xrig. Then we must show that the functor (·)riginduces a bijection between the set of irreducible components of X and the setof irreducible components of Xrig in the sense of Conrad (1999). This followsfrom the fact that (·)rig preserves connected components and commutes withnormalization (since the constructions are the same for affinoid spaces).


(3.5.15). — We refer to Ducros (2007, §1) and (Ducros 2009, 0.26) for thedimension theory of analytic spaces. If A is a K-affinoid algebra, then thedimension of M (A) is the Krull dimension of A. The dimension of a K-analytic space X is the supremum of the dimensions of its affinoid domains.If X is irreducible, then all of its nonempty affinoid domains have the samedimension as X. We say that a K-analytic space X has pure dimension dif each of its irreducible components has dimension d; this is equivalent tosaying that every nonempty affinoid domain in X has pure dimension d.

Proposition 3.5.16. — Let X be a formal R-scheme formally of finite type.Then the dimension of Xη is at most the dimension of X0. If X is flat over R,we have dim(X0) = dim(Xη) = dim(X) − 1.

In other words, the dimension of the generic fiber of the formal R-scheme Xis at most the relative dimension of X, and both dimensions coincide whenX is flat over R. We will say that X has pure relative dimension d if both Xη

and X0 have pure dimension d.By Proposition 3.5.17 below, this happens, in particular, when X is flat

over R and has pure dimension d + 1.

Proof. — Replacing X by its maximal R-flat closed formal subscheme, itsuffices to consider the case where X is flat. We may assume that X is affine,say X = Spf(A). Let d be the Krull dimension of A. Since X is catenary,the special fiber X0 has dimension d − 1 by flatness of X and the KrullHauptidealsatz. We must show that the dimension of Xη also equals d − 1.Note that d − 1 is precisely the Krull dimension of A ⊗R K. By de Jong(1995, 7.1.9), there exists a bijective correspondence between the rigid pointsx of Xη (i.e., the points of (Xη)rig) and the maximal ideals m of A ⊗R K.Moreover, the completion of the localization of A ⊗R K at m is isomorphicto the completion of the G-local ring of Xη at x. Since the completion of anoetherian local ring preserves its dimension, it now suffices to show that thedimension of Xη equals the supremum of the Krull dimensions of the G-localrings at the rigid points of Xη. This follows from Ducros (2007, 0.26.10).

Proposition 3.5.17. — Let X be a flat formal R-scheme formally of finitetype, and let d be a nonnegative integer. Then the following are equivalent:

a) The formal scheme X has pure dimension d + 1;b) The generic fiber Xη has pure dimension d;c) The special fiber X0 has pure dimension d.

Proof. — Since X is catenary, the equivalence of a) and c) is an easy conse-quence of Proposition 3.4.9, flatness of X and the Krull Hauptidealsatz. Solet us prove the equivalence of a) and b). The rig-irreducible components ofX are still flat over R, because the normalization of X is flat over R. Thusthe result follows from Propositions 3.5.14 and 3.5.16.

BIBLIOGRAPHY

In this book, Grothendieck’s Éléments de géométrie algébrique and the Sémi-naires de géométrie algébrique he conducted are referred to as ÉGA and SGArespectively. The precise correspondence is as follows: Éléments de géométriealgébrique:

– ÉGA I = (Grothendieck and Dieudonné 1960);– ÉGA Isv = (Grothendieck and Dieudonné 1971);– ÉGA II = (Grothendieck and Dieudonné 1961a);– ÉGA III2 = (Grothendieck and Dieudonné 1963);– ÉGA III1 = (Grothendieck and Dieudonné 1961b);– ÉGA IV1 = (Grothendieck and Dieudonné 1964);– ÉGA IV2 = (Grothendieck and Dieudonné 1965);– ÉGA IV3 = (Grothendieck and Dieudonné 1966);– ÉGA IV4 = (Grothendieck and Dieudonné 1967).

Séminaire de géométrie algébrique:– SGA I = (Grothendieck 1971),– SGA IV = (Grothendieck et al. 1972–1973a),– SGA IV3 = (Grothendieck et al. 1973a),– SGA V = (Grothendieck 1972–1973),– SGA VII1 = (Grothendieck et al. 1972),– SGA VII2 = (Grothendieck et al. 1973b).

Finally, we refer to (Deligne 1977) as SGA IV12 .

A. Abbes (2010), Éléments de géométrie rigide, volume I. Construction etétude géométrique des espaces rigides. Progress in Mathematics, vol. 286(Birkhäuser/Springer Basel AG, Basel)

S. Abhyankar (1956), On the valuations centered in a local domain. Am. J.Math. 78, 321–348


499

https://doi.org/10.1007/978-1-4939-7887-8

500 BIBLIOGRAPHY

S.S. Abhyankar (1998), Resolution of Singularities of Embedded AlgebraicSurfaces, 2nd edn. Springer Monographs in Mathematics (Springer,Berlin)

D. Abramovich, K. Karu, K. Matsuki, J. Włodarczyk (2002), Torificationand factorization of birational maps. J. Am. Math. Soc. 15(3), 531–572

N. A’Campo (1975), La fonction zêta d’une monodromie. Comment. Math.Helv. 50, 233–248

F. Ambro (1999), On minimal log discrepancies. Math. Res. Lett. 6(5–6),573–580

Y. André (2004), Une introduction aux motifs (motifs purs, motifs mixtes,périodes). Panoramas et Synthèses 17 (Soc. Math. France)

D. Arapura, S.-J. Kang (2006), Coniveau and the Grothendieck group ofvarieties. Mich. Math. J. 54(3), 611–622

E. Artal Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle Hernández (2002a),The Denef–Loeser zeta function is not a topological invariant. J. Lond.Math. Soc. (2), 65, 45–54

E. Artal Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle Hernéndez (2002b),Monodromy conjecture for some surface singularities. Ann. Sci. ÉcoleNorm. Sup. (4) 35(4), 605–640

E. Artal Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle Hernéndez (2005),Quasi-ordinary power series and their zeta functions. Mem. Am. Math.Soc. 178(841), vi+85

M. Artin (1969), Algebraic approximation of structures over complete localrings. Inst. Hautes Études Sci. Publ. Math. 36, 23–58

M. Artin (1986), Néron models, in Arithmetic Geometry (Storrs, Connecti-cut, 1984) (Springer, New York), pp. 213–230

M.F. Atiyah (1970), Resolution of singularities and division of distributions.Commun. Pure Appl. Math. 23, 145–150

J. Ayoub (2007a/2008), Les six opérations de Grothendieck et le formal-isme des cycles évanescents dans le monde motivique. I. Astérisque 314,x+466 pp.

J. Ayoub (2007b/2008), Les six opérations de Grothendieck et le formal-isme des cycles évanescents dans le monde motivique. II. Astérisque 315,vi+364 pp.

J. Ayoub (2015), Motifs des variétés analytiques rigides. Mém. Soc. Math.Fr. (N.S.) 140–141, vi+386

J. Ayoub, F. Ivorra, J. Sebag (2017), Motives of rigid analytic tubes andnearby motivic sheaves. Ann. Sci. École Norm. Sup. 50(6), 1335–1382

V.V. Batyrev (1999a), Birational Calabi-Yau n-folds have equal Betti num-bers, in New Trends in Algebraic Geometry (Warwick, 1996). LondonMathematical Society, Lecture Note Series, vol. 264 (Cambridge Univer-sity Press, Cambridge), pp. 1–11

V.V. Batyrev (1999b), Non-Archimedean integrals and stringy Euler num-bers of log-terminal pairs. J. Eur. Math. Soc. 1(1), 5–33

BIBLIOGRAPHY 501

V.V. Batyrev, D.I. Dais (1996), Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry. Topology 35(4), 901–929

A. Beauville (1983), Variétés Kähleriennes dont la première classe de Chernest nulle. J. Differ. Geom. 18(4), 755–782

A. Beauville, J.-L. Colliot-Thélène, J.-J. Sansuc, P. Swinnerton-Dyer (1985),Variétés stablement rationnelles non rationnelles. Ann. Math. (2),121(2), 283–318

A.A. Beılinson (1987), On the derived category of perverse sheaves, in K-Theory, Arithmetic and Geometry (Moscow, 1984–1986). Lecture Notesin Mathematics, vol. 1289 (Springer, Berlin), pp. 27–41

A.A. Beılinson, J. Bernstein, P. Deligne (1982), Faisceaux pervers. Analysisand topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100(Soc. Math.), pp. 5–171

V.G. Berkovich (1990), Spectral Theory and Analytic Geometry Over Non-Archimedean Fields. Mathematical Surveys and Monographs, vol. 33(American Mathematical Society, Providence)

V.G. Berkovich (1993), Étale cohomology for non-Archimedean analyticspaces. Publ. Math. Inst. Hautes Études Sci. 78, 5–161

V.G. Berkovich (1996a), Vanishing cycles for formal schemes. II. Invent.Math. 125(2), 367–390

V.G. Berkovich (1996b), Vanishing cycles for non-Archimedean analyticspaces. J. Am. Math. Soc. 9(4), 1187–1209

J.N. Bernstein (1973), Analytic continuation of generalized functions withrespect to a parameter. Funct. Anal. Appl. 6, 273–285

J.N. Bernstein, S.I. Gel’fand (1969), Meromorphy of the function P λ. Funct.Anal. Appl. 3, 68–69

P. Berthelot (1974), Cohomologie cristalline des schémas de caractéristiquep > 0. Lecture Notes in Mathematics, vol. 407 (Springer, Berlin)

P. Berthelot (1986), Géométrie rigide et cohomologie des variétés algébriquesde caractéristique p. Mém. Soc. Math. France, vol. 23, pp. 7–32. Intro-ductions aux cohomologies p-adiques (Luminy, 1984)

P. Berthelot (1996), Cohomologie rigide et cohomologie rigide à supportspropres. Première partie. Prépublication, IRMAR, Université Rennes 1

P. Berthelot, A. Ogus (1983), F -isocrystals and de Rham cohomology. I.Invent. Math. 72(2), 159–199

B. Bhatt (2016), Algebraization and Tannaka duality. Camb. J. Math. 4(4),403–461. http://dx.doi.org/10.4310/CJM.2016.v4.n4.a1

B. Bhatt, P. Scholze (2015), The pro-étale topology for schemes, Dela géométrie algébrique aux formes automorphes, I : Une collectiond’articles en l’honneur du soixantième anniversaire de Gérard Laumon,ed. by J.-B. Bost, P. Boyer, A. Genestier, L. Lafforgue, S. Lysenko,S. Morel, B.C. Ngô, vol. 369 (American Mathematical Society, Provi-dence), pp. 99–201

http://dx.doi.org/10.4310/CJM.2016.v4.n4.a1

502 BIBLIOGRAPHY

E. Bilgin (2014), On the classes of hypersurfaces of low degree in theGrothendieck ring of varieties. Int. Math. Res. Not. 16, 4534–4546.http://dx.doi.org/10.1093/ imrn/rnt089 . arXiv:1112.2131

C. Birkar, P. Cascini, C.D. Hacon, J. McKernan (2010), Existence of min-imal models for varieties of log general type. J. Am. Math. Soc. 23(2),405–468

C. Birkenhake, H. Lange (2004), Complex Abelian Varieties, 2nd edn.Grundlehren der Mathematischen Wissenschaften [Fundamental Prin-ciples of Mathematical Sciences], vol. 302 (Springer, Berlin)

F. Bittner (2004), The universal Euler characteristic for varieties of charac-teristic zero. Compos. Math. 140(4), 1011–1032

M.V. Bondarko (2009), Differential graded motives: weight complex, weightfiltrations and spectral sequences for realizations; Voevodsky versusHanamura. J. Inst. Math. Jussieu 8(1), 39–97

A. Borel (1991), Linear Algebraic Groups, 2nd edn. (Springer, New York)Z.I. Borevich, I.R. Shafarevich (1966), Number Theory (Academic, New

York)B. Bories, W. Veys (2016), Igusa’s p-adic local zeta function and the mon-

odromy conjecture for non-degenerate surface singularities. Mem. Am.Math. Soc. 242(1145), vii+131

L. Borisov (2014), Class of the affine line is a zero divisor in the grothendieckring. arXiv:1412.6194

L. Borisov, A. Căldăraru (2009), The Pfaffian-Grassmannian derived equiv-alence. J. Algebraic Geom. 18(2), 201–222

S. Bosch (2014), Lectures on Formal and Rigid Geometry. Lecture Notes inMathematics, vol. 2105 (Springer, Berlin)

S. Bosch, W. Lütkebohmert (1993), Formal and rigid geometry. I. Rigidspaces. Math. Ann. 295(2), 291–317

S. Bosch, K. Schlöter (1995), Néron models in the setting of formal and rigidgeometry. Math. Ann. 301(2), 339–362

S. Bosch, W. Lütkebohmert, M. Raynaud, Néron Models (1990), Ergebnisseder Mathematik und ihrer Grenzgebiete, vol. 21 (Springer, Berlin)

S. Bosch, W. Lütkebohmert, M. Raynaud (1995), Formal and rigid geometry.III. The relative maximum principle. Math. Ann. 302(1), 1–29

N. Bourbaki (1963), Éléments de mathématique. Fascicule XXIX. Livre VI:Intégration. Chapitre 7: Mesure de Haar. Chapitre 8: Convolution etreprésentations. Actualités Scientifiques et Industrielles, No. 1306 (Her-mann, Paris)

N. Bourbaki (1985), Éléments de mathématique, Masson, Paris. Algèbrecommutative. Chapitres 5 à 7. [Commutative algebra. Chapters 5–7].Reprint

N. Bourbaki (2006), Éléments de mathématique. Algèbre commutative.Chapitres 8 et 9 (Springer, Berlin). Reprint of the 1983 original

D. Bourqui, J. Sebag (2017a), The Drinfeld–Grinberg–Kazhdan theorem forformal schemes and singularity theory. Confluentes Math. 9(1), 29–64

http://dx.doi.org/10.1093/imrn/rnt089

BIBLIOGRAPHY 503

D. Bourqui, J. Sebag (2017b), The Drinfeld–Grinberg–Kazhdan theorem isfalse for singular arcs. J. Inst. Math. Jussieu 16(4), 879–885

D. Bourqui, J. Sebag (2017c), Smooth arcs on algebraic varieties. J. Singul.16, 130–140

A. Bouthier, B.C. Ngô, Y. Sakellaridis (2016), On the formal arc space of areductive monoid. Am. J. Math. 138(1), 81–108

M. Brandenburg, A. Chirvasitu (2014), Tensor functors between categoriesof quasi-coherent sheaves. J. Algebra 399, 675–692. arXiv:1202.5147

N. Budur, M. Mustaţă, Z. Teitler (2011), The monodromy conjecture forhyperplane arrangements. Geom. Dedicata 153, 131–137

E. Bultot, J. Nicaise (2016), Computing motivic zeta functions on logsmooth models. arXiv:1610.00742

J. Burillo (1990), El polinomio de Poincaré-Hodge de un producto simétricode variedades kählerianas compactas. Collect. Math. 41(1), 59–69

J.W.S. Cassels, A. Fröhlich (eds.) (1986), Algebraic Number Theory (Aca-demic) [Harcourt Brace Jovanovich Publishers, London]. Reprint of the1967 original

W. Chen, Y. Ruan (2004), A new cohomology theory of orbifold. Commun.Math. Phys. 248(1), 1–31

B. Chiarellotto (1998), Weights in rigid cohomology. applications to unipo-tent F -isocrystals. Ann. Sci. École Norm. Sup. 31(5), 683–715

B. Chiarellotto, B. Le Stum (1999), Pentes en cohomologie rigide et F -isocristaux unipotents. Manuscripta Math. 100(4), 455–468

B. Chiarellotto, B. Le Stum (2002), A comparison theorem for weights. J.Reine Angew. Math. 546, 159–176

R. Cluckers, F. Loeser (2008), Constructible motivic functions and motivicintegration. Invent. Math. 173(1), 23–121

P. Colmez, J.-P. Serre (eds.) (2001), Correspondance Grothendieck-Serre.Documents Mathématiques (Paris) [Mathematical Documents (Paris)],2 (Société Mathématique de France, Paris)

B. Conrad (1999), Irreducible components of rigid spaces. Ann. Inst. Fourier(Grenoble) 49(2), 473–541

V. Cossart, O. Piltant (2014), Resolution of singularities of arithmeticalthreefolds, II. arXiv:1412.0868

O. Debarre, A. Laface, R. Xavier (2017), Lines on cubic hypersurfaces overfinite fields, in Geometry over Nonclosed Fields (Simons Publications,New York). arXiv:1510.05803

T. de Fernex (2013), Three-dimensional counter-examples to the Nash prob-lem. Compos. Math. 149(9), 1519–1534. http://dx.doi.org/10.1112/S0010437X13007252 . arXiv:1205.0603

T. de Fernex, R. Docampo (2016), Terminal valuations and the Nash prob-lem. Invent. Math. 203(1), 303–331

A.J. de Jong (1995/1996), Crystalline Dieudonné module theory via formaland rigid geometry. Inst. Hautes Études Sci. Publ. Math. 82, 5–96

http://dx.doi.org/10.1112/S0010437X13007252

http://dx.doi.org/10.1112/S0010437X13007252

504 BIBLIOGRAPHY

S. del Baño Rollin, V. Navarro Aznar (1998), On the motive of a quotientvariety. Collect. Math. 49(2–3), 203–226. Dedicated to the memory ofFernando Serrano

P. Deligne (1971a), Théorie de Hodge. I. Actes du Congrès Internationaldes Mathématiciens (Nice, 1970), Tome 1 (Gauthier-Villars, Paris), pp.425–430

P. Deligne (1971b), Théorie de Hodge II. Publ. Math. Inst. Hautes ÉtudesSci. 40, 5–57

P. Deligne (1974a), La conjecture de Weil. I. Inst. Hautes Études Sci. Publ.Math., 43, 273–307

P. Deligne (1974b), Théorie de Hodge. III. Inst. Hautes Études Sci. Publ.Math. 44, 5–77

P. Deligne (1977), Cohomologie étale — SGA IV 12 . Lecture Notes in Mathe-

matics, vol. 569 (Springer, Berlin). Avec la collaboration de J.-F. Boutot,A. Grothendieck, L. Illusie et J.-L. Verdier

P. Deligne (1980), La conjecture de Weil, II. Publ. Math. Inst. Hautes ÉtudesSci. 52, 137–252

J.-P. Demailly, J. Kollár (2001), Semi-continuity of complex singularity ex-ponents and Kähler-Einstein metrics on Fano orbifolds. Ann. Sci. ÉcoleNorm. Sup. (4) 34(4), 525–556

M. Demazure (1969–1970), Motifs des variétés algébriques. Séminaire Bour-baki 12, 19–38

M. Demazure, P. Gabriel (1970), Groupes algébriques. Tome I : Géométriealgébrique, généralités, groupes commutatifs (North-Holland PublishingCompany, Amsterdam)

J. Denef (1984), The rationality of the Poincaré series associated to thep-adic points on a variety. Invent. Math. 77(1), 1–23

J. Denef (1987), On the degree of Igusa’s local zeta function. Am. J. Math.109, 991–1008

J. Denef (1991), Report on Igusa’s Local Zeta Function. Séminaire Bourbaki,1990/1991, 201–203 (Soc. Math., France), pp. 359–386

J. Denef, F. Loeser (1992), Caractéristiques d’Euler-Poincaré, fonctions zêtalocales et modifications analytiques. J. Am. Math. Soc. 5(4), 705–720

J. Denef, F. Loeser (1998), Motivic Igusa zeta functions. J. Algebraic Geom.7(3), 505–537

J. Denef, F. Loeser (1999), Germs of arcs on singular algebraic varieties andmotivic integration. Invent. Math. 135(1), 201–232

J. Denef, F. Loeser (2001), Geometry on arc spaces of algebraic varieties,in European Congress of Mathematics, Volume I (Barcelona, 2000).Progress in Mathematics, vol. 201 (Birkhäuser, Basel), pp. 327–348

J. Denef, F. Loeser (2002a), Lefschetz numbers of iterates of the monodromyand truncated arcs. Topology 41(5), 1031–1040

J. Denef, F. Loeser (2002b), Lefschetz numbers of iterates of the monodromyand truncated arcs. Topology 41(5), 1031–1040

BIBLIOGRAPHY 505

J. Denef, F. Loeser (2002c), Motivic integration, quotient singularities andthe McKay correspondence. Compos. Math. 131(3), 267–290

J. Denef, F. Loeser (2004), On some rational generating series occurring inarithmetic geometry, in Geometric Aspects of Dwork Theory, vols. I, II(Walter de Gruyter, Berlin), pp. 509–526

A. Dimca (1992), Singularities and Topology of Hypersurfaces. Universitext(Springer, New York)

V. Drinfeld (2002), On the Grinberg–Kazhdan formal arc theorem.arXiv:math/0203263

A. Ducros (2007), Variation de la dimension relative en géométrie analytiquep-adique. Compos. Math. 143(6), 1511–1532

A. Ducros (2009), Les espaces de Berkovich sont excellents. Ann. Inst.Fourier (Grenoble) 59(4), 1443–1552

A. Ducros (2018), Families of Berkovich spaces. arXiv:1107.4259v5B. Dwork (1960), On the rationality of the zeta function of an algebraic

variety. Am. J. Math. 82, 631–648L. Ein, M. Mustaţa (2004), Inversion of adjunction for local complete inter-

section varieties. Am. J. Math. 126(6), 1355–1365L. Ein, M. Mustaţă, T. Yasuda (2003), Jet schemes, log discrepancies and

inversion of adjunction. Invent. Math. 153(3), 519–535L. Ein, R. Lazarsfeld, M. Mustaţa (2004), Contact loci in arc spaces. Com-

pos. Math. 140(5), 1229–1244D. Eisenbud (1995), Commutative Algebra with a View Towards Algebraic

Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, Berlin)T. Ekedahl (1983), Sur le groupe fondamental d’une variété unirationnelle.

C. R. Acad. Sci. Paris Sér. I Math. 297(12), 627–629T. Ekedahl (1990), On the adic formalism, in The Grothendieck Festschrift,

Volume II. Progress in Mathematics, vol. 87 (Birkhäuser, Boston), pp.197–218

T. Ekedahl (2009), The Grothendieck group of algebraic stacks.arXiv:0903.3143

T. Ekedahl (2010), Is the Grothendieck ring of varieties re-duced? MathOverflow. http://mathoverflow.net/questions/37737/is-the-grothendieck-ring-of-varieties-reduced

R. Elkik (1973/1974), Solutions d’équations à coefficients dans un anneauhensélien. Ann. Sci. École Norm. Sup. 6, 553–603

H. Esnault (2003), Varieties over a finite field with trivial Chow groupof 0-cycles have a rational point. Invent. Math. 151(1), 187–191.arXiv:math.AG/0207022

H. Esnault, J. Nicaise (2011), Finite group actions, rational fixed points andweak Néron models. Pure Appl. Math. Q. 7(4), 1209–1240. Special Issue:In memory of Eckart Viehweg

J.-Y. Étesse, B. Le Stum (1993), Fonctions L associées aux F -isocristaux sur-convergents. I. Interprétation cohomologique. Math. Ann. 296(3), 557–576

http://mathoverflow.net/questions/37737/is-the-grothendieck-ring-of-varieties-reduced

http://mathoverflow.net/questions/37737/is-the-grothendieck-ring-of-varieties-reduced

506 BIBLIOGRAPHY

G. Faltings (1988), p-adic Hodge theory. J. Am. Math. Soc. 1(1), 255–299B. Fantechi, L. Göttsche, L. Illusie, S.L. Kleiman, N. Nitsure, A. Vistoli

(2005), Fundamental Algebraic Geometry — Grothendieck’s FGA Ex-plained. Mathematical Surveys and Monographs, vol. 123 (AmericanMathematical Society, Providence)

J. Fernández de Bobadilla (2012), Nash problem for surface singularities isa topological problem. Adv. Math. 230(1), 131–176

J. Fernández de Bobadilla, M. Pe Pereira (2012), The Nash problem forsurfaces. Ann. Math. (2) 176(3), 2003–2029

J. Fogarty (1968), Algebraic families on an algebraic surface. Am. J. Math.90, 511–521. http://dx.doi.org/10.2307/2373541

J.-M. Fontaine, W. Messing (1985/1987), p-adic Periods and p-Adic étaleCohomology. Contemporary Mathematics, vol. 67 (American Mathemat-ical Society, Arcata), pp. 179–207

E. Freitag, R. Kiehl (1988), Étale cohomology and the Weil Conjecture.Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Math-ematics and Related Areas (3)], vol. 13 (Springer, Berlin). Translatedfrom the German by Betty S. Waterhouse and William C. Waterhouse,With an historical introduction by J.A. Dieudonné

W. Fulton (1998), Intersection Theory, 2nd edn. (Springer, Berlin)P. Gabriel, M. Zisman (1967), Calculus of Fractions and Homotopy Theory.

Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35 (Springer,New York)

S. Galkin, E. Shinder (2014), The Fano variety of lines and rationality prob-lem for a cubic hypersurface. arXiv:1405.5154

H. Gillet, C. Soulé (1996), Descent, motives and K-theory. J. Reine Angew.Math. 478, 127–176

H. Gillet, C. Soulé (2009), Motivic weight complexes for arithmetic varieties.J. Algebra 322(9), 3088–3141

L. Göttsche (2001), On the motive of the Hilbert scheme of points on asurface. Math. Res. Lett. 8(5–6), 613–627

M.J. Greenberg (1961), Schemata over local rings. Ann. Math. (2) 73, 624–648

M.J. Greenberg (1966), Rational points in Henselian discrete valuation rings.Inst. Hautes Études Sci. Publ. Math. 31, 59–64

M. Grinberg, D. Kazhdan (2000), Versal deformations of formal arcs. Geom.Funct. Anal. 10(3), 543–555

M. Gromov (1999), Endomorphisms of symbolic algebraic varieties. J. Eur.Math. Soc. 1(2), 109–197

A. Grothendieck (1965), Formule de Lefschetz et rationalité des fonctionsL, Séminaire Bourbaki, Volume 9, Année 1964–1965 (Société Mathéma-tique de France, Paris). Reprint 1995

A. Grothendieck (1968), Crystals and the de Rham cohomology of schemes,in Dix exposés sur la cohomologie des schémas. Advanced Studies in PureMathematics (North-Holland, Amsterdam), pp. 306–358

http://dx.doi.org/10.2307/2373541

BIBLIOGRAPHY 507

A. Grothendieck (1971), Revêtements étales et groupe fondamental —SGA I. Lecture Notes in Mathematics, vol. 224 (Springer, Berlin).Quoted as (SGA I)

A. Grothendieck, J. Dieudonné (1960), Éléments de géométrie algébrique.I. Le langage des schémas. Publ. Math. Inst. Hautes Études Sci. 4, 228.Quoted as (ÉGA I)

A. Grothendieck, J. Dieudonné (1961a), Éléments de géométrie algébrique.II. Étude globale élémentaire de quelques classes de morphismes. Publ.Math. Inst. Hautes Études Sci. 8, 5–222. Quoted as (ÉGA II)

A. Grothendieck, J. Dieudonné (1961b), Éléments de géométrie algébrique.III. Étude cohomologique des faisceaux cohérents. I. Publ. Math. Inst.Hautes Études Sci., 11, 5–167. Quoted as (ÉGA III1)

A. Grothendieck, J. Dieudonné (1963), Éléments de géométrie algébrique.III. Étude cohomologique des faisceaux cohérents. II. Publ. Math. Inst.Hautes Études Sci. 17, 91. Quoted as (ÉGA III2)

A. Grothendieck, J. Dieudonné (1964), Éléments de géométrie algébrique.IV. Étude locale des schémas et des morphismes de schémas. I. Publ.Math. Inst. Hautes Études Sci. 20, 5–259. Quoted as (ÉGA IV1)

A. Grothendieck, J. Dieudonné (1965), Éléments de géométrie algébrique.IV. Étude locale des schémas et des morphismes de schémas. II. Publ.Math. Inst. Hautes Études Sci. 24, 5–231. Quoted as (ÉGA IV2)

A. Grothendieck, J. Dieudonné (1966), Éléments de géométrie algébrique.IV. Étude locale des schémas et des morphismes de schémas. III. Inst.Hautes Études Sci. Publ. Math. 28, 5–255. Quoted as (ÉGA IV3)

A. Grothendieck, J. Dieudonné (1967), Éléments de géométrie algébrique.IV. Étude locale des schémas et des morphismes de schémas IV. Publ.Math. Inst. Hautes Études Sci. 32, 5–361. Quoted as (ÉGA IV4)

A. Grothendieck, J.-A. Dieudonné (1971), Éléments de géométrie algébrique,vol. 1. Grundlehren der Mathematischen Wissenschaften, vol. 166(Springer, Berlin). Quoted as (ÉGA Isv)

A. Grothendieck, P. Deligne, N.M. Katz (1972), Groupes de monodromieen géométrie algébrique, I — SGA VII1. Lecture Notes in Mathematics,vol. 288 (Springer, Berlin). Quoted as (SGA VII1)

A. Grothendieck (1972–1973), Cohomologie -adique et fonctions L. LectureNotes in Mathematics, vol. 589 (Springer, Berlin). Quoted as (SGA V)

A. Grothendieck, M. Artin, J.-L. Verdier (1972–1973a), Théorie des topos etcohomologie étale des schémas. Lecture Notes in Mathematics, vols. 269–270–305 (Springer, Berlin). Quoted as (SGA IV)

A. Grothendieck, P. Deligne, N.M. Katz (1972–1973b), Groupes de mon-odromie en géométrie algébrique — SGA VII. Lecture Notes in Mathe-matics, vols. 288–340 (Springer, Berlin). Quoted as ((alias?))

A. Grothendieck, M. Artin, J.-L. Verdier (1973a), Théorie des topos et coho-mologie étale des schémas, III. Lecture Notes in Mathematics, vol. 305(Springer, Berlin). Quoted as (SGA IV3)

508 BIBLIOGRAPHY

A. Grothendieck, P. Deligne, N.M. Katz (1973b), Groupes de monodromieen géométrie algébrique, II — SGA VII2. Lecture Notes in Mathematics,vol. 340 (Springer, Berlin). Quoted as (SGA VII2)

F. Guillén, V. Navarro Aznar (2002), Un critère d’extension des foncteursdéfinis sur les schémas lisses. Publ. Math. Inst. Hautes Études Sci. 95,1–91

V. Guletskiı, C. Pedrini (2002), The Chow motive of the Godeaux surface,in Algebraic Geometry (de Gruyter, Berlin), pp. 179–195

L.H. Halle, J. Nicaise (2011), Motivic zeta functions of abelian varieties, andthe monodromy conjecture. Adv. Math. 227(1), 610–653

L.H. Halle, J. Nicaise (2016), Néron Models and Base Change. Lecture Notesin Mathematics, vol. 2156 (Springer, Cham)

L.H. Halle, J. Nicaise (2017), Motivic zeta functions of degenerating Calabi-Yau varieties. Math. Ann., arXiv:1701.09155

E. Hamann (1975), On power-invariance. Pac. J. Math. 61(1), 153–159A. Hartmann (2015), Equivariant motivic integration on formal schemes and

the motivic zeta function. arXiv:1511.08656T. Hausel, F. Rodriguez-Villegas (2008), Mixed Hodge polynomials of char-

acter varieties. Invent. Math. 174(3), 555–624. With an appendix byNicholas M. Katz

O. Haution (2017), On rational fixed points of finite group actions on theaffine space. Trans. Am. Math. Soc. 369(11), 8277–8290. http://dx.doi.org/10.1090/ tran/7184

F. Heinloth (2007), A note on functional equations for zeta functions withvalues in Chow motives. Ann. Inst. Fourier (Grenoble) 57(6), 1927–1945

H. Hironaka (1964), Resolution of singularities of an algebraic variety overa field of characteristic zero. I, II. Ann. Math. (2) 79, 109–203, 205–326

R. Hotta, K. Takeuchi, T. Tanisaki (2008), D-Modules, PerverseSheaves, and Representation Theory. Progress in Mathematics, vol. 236(Birkhäuser Boston, Inc., Boston) Translated from the 1995 Japaneseedition by Takeuchi

E. Hrushovski, D. Kazhdan (2006), Integration in valued fields, in Alge-braic Geometry and Number Theory. Progress in Mathematics, vol. 253(Birkhäuser Boston, Boston), pp. 261–405

E. Hrushovski, F. Loeser (2015), Monodromy and the Lefschetz fixed pointformula. Ann. Sci. Éc. Norm. Supér. (4) 48(2), 313–349

A. Huber (1995), Mixed Motives and Their Realization in Derived Cate-gories. Lecture Notes in Mathematics, vol. 1604 (Springer, Berlin)

A. Huber (1997), Mixed perverse sheaves for schemes over number fields.Compos. Math. 108(1), 107–121

R. Huber (1994), A generalization of formal schemes and rigid ana-lytic varieties. Math. Z. 217(4), 513–551. http://dx.doi.org/10.1007/BF02571959

J.-I. Igusa (1975), Complex powers and asymptotic expansions. II. Asymp-totic expansions. J. Reine Angew. Math. 278/279, 307–321

http://dx.doi.org/10.1090/tran/7184

http://dx.doi.org/10.1090/tran/7184

http://dx.doi.org/10.1007/BF02571959

http://dx.doi.org/10.1007/BF02571959

BIBLIOGRAPHY 509

J.-I. Igusa (1978), Forms of Higher Degree, Tata Institute of FundamentalResearch Lectures on Mathematics and Physics, vol. 59 (Tata Instituteof Fundamental Research, Bombay)

J.-I. Igusa (1989), b-functions and p-adic integrals, in Algebraic Analysis.Papers Dedicated to Professor Mikio Sato on the Occasion of His SixtiethBirthday, vol. 1 (Academic, New York), pp. 231–241

J.-I. Igusa (2000), An Introduction to the Theory of Local Zeta Functions.AMS/IP Studies in Advanced Mathematics, vol. 14 (American Mathe-matical Society, Providence)

L. Illusie (1981), Théorie de Brauer et caractéristique d’Euler-Poincaré(d’après P. Deligne). The Euler-Poincaré characteristic (French),Astérisque 82, pp. 161–172, Soc. Math. France, Paris

L. Illusie, Y. Laszlo, F. Orgogozo (eds.) (2014), Travaux de Gabbersur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents, Astérisque, vols. 363–364, Société Mathématique de France,Paris. Séminaire à l’École polytechnique 2006–2008. With the collabo-ration of Frédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Ray-naud, Joël Riou, Benoît Stroh, Michael Temkin and Weizhe Zheng.

S. Ishii (2004), Extremal functions and prime blow-ups. Commun. Algebra32(3), 819–827

S. Ishii (2008), Maximal divisorial sets in arc spaces, in Algebraic Geome-try in East Asia—Hanoi 2005. Advanced Studies in Pure Mathematics,vol. 50 (Mathematical Society of Japan, Tokyo), pp. 237–249

S. Ishii, J. Kollár (2003), The Nash problem on arc families of singularities.Duke Math. J. 120(3), 601–620

S. Ishii, A.J. Reguera (2013), Singularities with the highest Mather minimallog discrepancy. Math. Z. 275(3–4), 1255–1274

A. Ito, M. Miura, S. Okawa, K. Ueda (2016), The class of the affineline is a zero divisor in the grothendieck ring: via g2-Grassmannians.arXiv:1606.04210

T. Ito (2003), Birational smooth minimal models have equal Hodge numbersin all dimensions, Calabi-Yau varieties and mirror symmetry (Toronto,ON, 2001). Fields Institute Communication, vol. 38 (American Mathe-matical Society, Providence), pp. 183–194

T. Ito (2004), Stringy Hodge numbers and p-adic Hodge theory. Compos.Math. 140(6), 1499–1517

F. Ivorra (2014), Finite dimension motives and applications (following S-I.Kimura, P. O’Sullivan and others). Autour des motifs, II. Asian-Frenchsummer school on algebraic geometry and number theory. Panoramas etsynthèses, vol. 38, Soc. Math. France

F. Ivorra, J. Sebag (2012), Géométrie algébrique par morceaux, K-équivalence et motifs. Enseign. Math. (2), 58, 375–403

F. Ivorra, J. Sebag (2013), Nearby motives and motivic nearby cycles. Se-lecta Math. (N.S.) 19(4), 879–902

510 BIBLIOGRAPHY

U. Jannsen (1992), Motives, numerical equivalence, and semi-simplicity. In-vent. Math. 107, 447–452

J.M. Johnson, J. Kollár (2013), Arc spaces of cA-type singularities. J. Singul.7, 238–252

B. Kahn (2009), Zeta functions and motives. Pure Appl. Math. Q. 5(1),507–570

M. Kapranov (2000), The elliptic curve in the S-duality theory and Eisen-stein series for Kac-Moody groups. arXiv:math/0001005

M. Kashiwara (1976/1977), B-functions and holonomic systems. Rationalityof roots of B-functions. Invent. Math. 38(1), 33–53

N.M. Katz (1972–1973), Le niveau de la cohomologie des intersectionscomplètes, in Groupes de monodromie en géométrie algébrique, I —SGA VII1, ed. by A. Grothendieck, P. Deligne, N.M. Katz. LectureNotes in Mathematics, vol. 288 (Springer, Berlin), pp. 363–399. Quotedas ((alias?))

N.M. Katz (1979), Slope filtration of F-crystals. Journées de Géométrie al-gébrique de Rennes Astérisque 63, pp. 113–164, Soc. Math. France

N.M. Katz, W. Messing (1974), Some consequences of the Riemann hypoth-esis for varieties over finite fields. Invent. Math. 23, 73–77

S. Kawaguchi, J.H. Silverman (2009), Nonarchimedean Green functions anddynamics on projective space. Math. Z. 262(1), 173–197

S.-I. Kimura (2005), Chow groups are finite dimensional, in some sense.Math. Ann. 331(1), 173–201

T. Kimura (1982), The b-functions and holonomy diagrams of irreducibleregular prehomogeneous vector spaces. Nagoya Math. J. 85, 1–80

T. Kimura (2003), Introduction to Prehomogeneous Vector Spaces. Trans-lations of Mathematical Monographs, vol. 215 (American Mathemati-cal Society, Providence). Translated from the 1998 Japanese original byMakoto Nagura and Tsuyoshi Niitani and revised by the author

T. Kimura, F. Sato, X.-W. Zhu (1990), On the poles of p-adic complexpowers and the b-functions of prehomogeneous vector spaces. Am. J.Math. 112(3), 423–437

S.L. Kleiman (1968), Algebraic cycles and the Weil conjectures. Dix exposéssur la cohomologie des schémas. Advanced Studies in Pure Mathemat-ics, vol. 3 (North-Holland, Amsterdam), pp. 359–386

E.R. Kolchin (1973), Differential Algebra and Algebraic Groups. Pure andApplied Mathematics, vol. 54 (Academic, New York)

J. Kollár (1997), Quotient spaces modulo algebraic groups. Ann. Math.145(1), 33–79

J. Kollár (1989), Flops. Nagoya Math. J. 113, 15–36J. Kollár (ed.) (1992), Flips and abundance for algebraic threefolds, Société

Mathématique de France, Paris. Papers from the Second Summer Sem-inar on Algebraic Geometry held at the University of Utah, Salt LakeCity, Utah, August 1991, Astérisque No. 211 (1992) (1992)

BIBLIOGRAPHY 511

J. Kollár (1996), Rational Curves on Algebraic Varieties. Ergebnisse derMathematik und ihrer Grenzgebiete, vol. 32 (Springer, Berlin)

J. Kollár (2007), Lectures on Resolution of Singularities. Annals of Mathe-matics Studies, vol. 166 (Princeton University Press, Princeton)

J. Kollár, S. Mori (1998), Birational Geometry of Algebraic Varieties. Cam-bridge Tracts in Mathematics, vol. 134 (Cambridge University Press,Cambridge). With the collaboration of C. H. Clemens and A. Corti,Translated from the 1998 Japanese original

M. Kontsevich (1995), Motivic integration. Lecture at Orsay. http://www.lama.univ-savoie.fr/~raibaut/Kontsevich-MotIntNotes.pdf

K. Kpognon, J. Sebag (2017), Nilpotency in arc scheme of planecurves. Commun. Algebra 45(5), 2195–2221. http://dx.doi.org/10.1080/00927872.2016.1233187

J. Krajíček, T. Scanlon (2000), Combinatorics with definable sets: Eulercharacteristics and Grothendieck rings. Bull. Symb. Log. 6(3), 311–330

M. Larsen, V.A. Lunts (2003), Motivic measures and stable birational ge-ometry. Mosc. Math. J. 3(1), 85–95, 259. arXiv:math.AG/0110255

M. Larsen, V.A. Lunts (2004), Rationality criteria for motivic zeta functions.Compos. Math. 140(6), 1537–1560

G. Laumon (1981), Comparaison de caractéristiques d’Euler-Poincaré encohomologie l-adique. C. R. Acad. Sci. Paris Sér. I Math. 292(3), 209–212

D.T. Lê (1977), Some remarks on relative monodromy, in Real and complexsingularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math.,Oslo, 1976) (Sijthoff and Noordhoff, Alphen aan den Rijn), pp. 397–403

B. Le Stum (2007), Rigid Cohomology. Cambridge Tracts in Mathemat-ics, vol. 172 (Cambridge University Press, Cambridge)

M. Lejeune-Jalabert, A.J. Reguera (2012), Exceptional divisors that are notuniruled belong to the image of the Nash map. J. Inst. Math. Jussieu11(2), 273–287

A. Lemahieu, L. Van Proeyen (2011), Monodromy conjecture for nondegen-erate surface singularities. Trans. Am. Math. Soc. 363(9), 4801–4829

J. Lipman (1976), The Picard group of a scheme over an Artin ring. Inst.Hautes Études Sci. Publ. Math. 46, 15–86

J. Lipman (1978), Desingularization of two-dimensional schemes. Ann.Math. (2) 107(1), 151–207

D. Litt (2015), Zeta functions of curves with no rational points. MichiganMath. J. 64(2), 383–395. arXiv:1405.7380

Q. Liu (2002), Algebraic Geometry and Arithmetic Curves. Oxford Grad-uate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford).Translated from the French by Reinie Erné, Oxford Science Publications

Q. Liu, J. Sebag (2010), The Grothendieck ring of varieties and piecewiseisomorphisms. Math. Z. 265(2), 321–342

F. Loeser (1988), Fonctions d’Igusa p-adiques et polynômes de Bernstein.Am. J. Math. 110(1), 1–21

http://www.lama.univ-savoie.fr/~raibaut/Kontsevich-MotIntNotes.pdf

http://www.lama.univ-savoie.fr/~raibaut/Kontsevich-MotIntNotes.pdf

http://dx.doi.org/10.1080/00927872.2016.1233187

http://dx.doi.org/10.1080/00927872.2016.1233187

512 BIBLIOGRAPHY

F. Loeser (1990), Fonctions d’Igusa p-adiques, polynômes de Bernstein, etpolyèdres de Newton. J. Reine Angew. Math. 412, 75–96

F. Loeser, J. Sebag (2003), Motivic integration on smooth rigid varietiesand invariants of degenerations. Duke Math. J. 119(2), 315–344

E. Looijenga (2002), Motivic measures. Astérisque, 276, 267–297. SéminaireBourbaki, vols. 1999/2000

S. Mac Lane (1998), Categories for the Working Mathematician, 2nd edn.Graduate Texts in Mathematics, vol. 5 (Springer, New York)

I.G. Macdonald (1962), The Poincaré polynomial of a symmetric product.Proc. Camb. Philos. Soc. 58, 563–568

A. MacIntyre (1976), On definable subsets of p-adic fields. J. Symb. Log.41(3), 605–610

B. Malgrange (1974/1975), Intégrales asymptotiques et monodromie. Ann.Sci. École Norm. Sup. (4) 7, 405–430

J.I. Manin (1968), Correspondences, motifs and monoidal transformations.Mat. Sb. (N.S.) 77(119), 475–507

N. Martin (2016), The class of the affine line is a zero divisor in theGrothendieck ring: an improvement. C. R. Math. Acad. Sci. Paris354(9), 936–939

D. Meuser (1981), On the rationality of certain generating functions. Math.Ann. 256(3), 303–310

J.S. Milne (1980), Étale Cohomology. Mathematical Notes, vol. 33 (Prince-ton University Press, Princeton)

J. Milnor (1968), Singular points of complex hypersurfaces, Annals of Math-ematics Studies, vol. 61 (Princeton University Press, Princeton)

P. Monsky, G. Washnitzer (1968), Formal cohomology I. Ann. Math. 88,181–217

S. Morel (2012), Complextes mixtes sur un schéma de type fini sur Q.https://web.math.princeton.edu/~smorel/ sur_Q.pdf

D. Mumford (1974), Abelian Varieties, Tata Institute of Fundamental Re-search Studies in Mathematics, vol. 5. Published for the Tata Instituteof Fundamental Research, Bombay/Oxford University Press, London)

J.P. Murre, J. Nagel, C.A.M. Peters (2013), Lectures on the Theory of PureMotives. University Lecture Series, vol. 61 (American Mathematical So-ciety, Providence)

M. Mustaţă (2001), Jet schemes of locally complete intersection canonicalsingularities. Invent. Math. 145(3), 397–424. With an appendix by DavidEisenbud and Edward Frenkel

M. Mustaţă (2002), Singularities of pairs via jet schemes. J. Am. Math. Soc.15(3), 599–615 [electronic]

C. Nakayama (1998), Nearby cycles for log smooth families. Compos. Math.112(1), 45–75

Y. Nakkajima (2012), Weight filtration and slope filtration on the rigidcohomology of a variety in characteristic p > 0, in Mém. Soc. Math.France, vols. 130–131 (American Mathematical Society, Providence)

https://web.math.princeton.edu/~smorel/sur_Q.pdf

BIBLIOGRAPHY 513

J.F. Nash Jr. (1995/1996), Arc structure of singularities. Duke Math. J.81(1), 31–38. A celebration of John F. Nash, Jr.

N. Naumann (2007), Algebraic independence in the Grothendieck ring ofvarieties. Trans. Am. Math. Soc. 359(4), 1653–1683 [electronic]

A. Neeman (2001), Triangulated Categories. Annals of Mathematics Studies,vol. 148 (Princeton University Press, Princeton)

J. Nicaise (2009), A trace formula for rigid varieties, and motivic Weil gen-erating series for formal schemes. Math. Ann. 343(2), 285–349

J. Nicaise (2011a), Motivic invariants of algebraic tori. Proc. Am. Math.Soc. 139(4), 1163–1174

J. Nicaise (2011b), A trace formula for varieties over a discretely valuedfield. J. Reine Angew. Math. 650, 193–238

J. Nicaise (2013), Geometric criteria for tame ramification. Math. Z. 273(3–4), 839–868

J. Nicaise, J. Sebag (2007a), Motivic Serre invariants of curves. ManuscriptaMath. 123(2), 105–132

J. Nicaise, J. Sebag (2007b), Motivic Serre invariants, ramification, and theanalytic Milnor fiber. Invent. Math. 168(1), 133–173

J. Nicaise, J. Sebag (2011), The Grothendieck ring of varieties, in Motivic In-tegration and its Interactions with Model Theory and Non-ArchimedeanGeometry, Volume I. London Mathematical Society. Lecture Note Se-ries, vol. 383 (Cambridge University Press, Cambridge), pp. 145–188

J. Nicaise, C. Xu (2016), Poles of maximal order of motivic zeta functions.Duke Math. J. 165(2), 217–243

M.V. Nori (2002), Constructible sheaves, in Algebra, Arithmetic and Ge-ometry, Parts I, II (Mumbai, 2000). Tata Institute of Fundamental Re-search Studies in Mathematics, vol. 16 (Tata Institute of FundamentalResearch, Bombay), pp. 471–491

A. Néron (1964), Modèles minimaux des variétés abéliennes sur les corpslocaux et globaux. Inst. Hautes Études Sci. Publ. Math. No. 21, 128

J. Oesterlé (1982), Réduction modulo pn des sous-ensembles analytiquesfermés de ZN

p . Invent. Math. 66(2), 325–341J. Oesterlé (1983), Image modulo pn of a closed analytic subset of ZN

p . (Im-age modulo pn d’un sous-ensemble analytique fermé de Z

Np .). Théorie

des Nombres, Sémin. Delange-Pisot-Poitou, Paris 1981/1982, Prog.Math. 38, 219–224

J.-P. Olivier (1968a), Anneaux absolument plats universels et épimorphismesà buts réduits. Séminaire Samuel. Algèbre commutative (1967/1968),Exposé No. 6. http://www.numdam.org/ item?id=SAC_1967-1968__2__A6_0

J.-P. Olivier (1968b), Le foncteur T −∞. globalisation du foncteur t. Sémi-naire Samuel. Algèbre commutative (1967/1968), Exposé No. 9. http://www.numdam.org/ item?id=SAC_1967-1968__2__A9_0

F. Orgogozo (2014), Exposé XIII. Le théorème de finitude, in Travaux deGabber sur l’uniformisation locale et la cohomologie étale des schémas

http://www.numdam.org/item?id=SAC_1967-1968__2__A6_0




514 BIBLIOGRAPHY

quasi-excellents, ed. by L. Illusie, Y. Laszlo, F. Orgogozo. Astérisque,vols. 363–364 (Société Mathématique de France, Paris), pp. 261–275.Séminaire à l’École polytechnique 2006–2008. With the collaboration ofFrédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, JoëlRiou, Benoît Stroh, Michael Temkin and Weizhe Zheng

P. O’Sullivan (2005), The structure of certain rigid tensor categories. C. R.Math. Acad. Sci. Paris 340(8), 557–562

J. Pas (1989), Uniform p-adic cell decomposition and local zeta functions.J. Reine Angew. Math. 399, 137–172

C.A.M. Peters, J.H.M. Steenbrink (2008), Mixed Hodge structures, in Ergeb-nisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Mod-ern Surveys in Mathematics [Results in Mathematics and Related Ar-eas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 52(Springer, Berlin)

B. Poonen (2002), The Grothendieck ring of varieties is not a domain. Math.Res. Lett. 9(4), 493–497

D. Popescu (1986), General Néron desingularization and approximation.Nagoya Math. J. 104, 85–115

D. Popescu (2000), Artin Approximation. Handbook of Algebra, vol. 2(North-Holland, Amsterdam), pp. 321–356

M. Presburger (1930), Über die Vollständigkeit eines gewissen Systems derArithmetik ganzer Zahlen, in welchem die Addition als einzige Operationhervortritt. Sprawozdanie z 1 Kongresu Matematyków Krajow Slowiańs-kich, pp. 92–101, 395, Warsaw

M. Raynaud (1974), Géométrie analytique rigide d’après Tate, Kiehl, . . . .Bull. Soc. Math. Fr. 39–40, 319–327

M. Raynaud, Y. Laszlo (2014), Exposé I. Anneaux excellents, in Travaux deGabber sur l’uniformisation locale et la cohomologie étale des schémasquasi-excellents, ed. by L. Illusie, Y. Laszlo, F. Orgogozo Astérisque,vols. 363–364. Société Mathématique de France, Paris, pp. 1–19 Sémi-naire à l’École polytechnique 2006–2008. With the collaboration ofFrédéric Déglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, JoëlRiou, Benoît Stroh, Michael Temkin and Weizhe Zheng

A.J. Reguera (2006), A curve selection lemma in spaces of arcs and theimage of the Nash map. Compos. Math. 142(1), 119–130

Z. Reichstein, B. Youssin (2000), Essential dimensions of algebraic groupsand a resolution theorem for G-varieties. Can. J. Math. 52(5), 1018–1056. With an appendix by János Kollár and Endre Szabó

J. Riou (2014), Exposé XVII. Dualité, in Travaux de Gabber surl’uniformisation locale et la cohomologie étale des schémas quasi-excellents, ed. by L. Illusie, Y. Laszlo, F. Orgogozo Astérisque, vols. 363–364, Société Mathématique de France, Paris, pp. 351–453. Séminaireà l’École polytechnique 2006–2008. With the collaboration of FrédéricDéglise, Alban Moreau, Vincent Pilloni, Michel Raynaud, Joël Riou,Benoît Stroh, Michael Temkin and Weizhe Zheng

BIBLIOGRAPHY 515

E.A. Rødland (2000), The Pfaffian Calabi-Yau, its mirror, and their link tothe Grassmannian G(2, 7). Compos. Math. 122(2), 135–149

B. Rodrigues (2004), On the monodromy conjecture for curves on normalsurfaces. Math. Proc. Camb. Philos. Soc. 136(2), 313–324

B. Rodrigues, W. Veys (2001), Holomorphy of Igusa’s and topological zetafunctions for homogeneous polynomials. Pac. J. Math. 201(2), 429–440

B. Rodrigues, W. Veys (2003), Poles of zeta functions on normal surfaces.Proc. Lond. Math. Soc. (3) 87(1), 164–196

M. Saito (1988/1989), Modules de Hodge polarisables. Publ. Res. Inst. Math.Sci. 24(6), 849–995

M. Saito (1990), Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26(2),221–333

M. Sato, T. Kimura (1977), A classification of irreducible prehomogeneousvector spaces and their relative invariants. Nagoya Math. J. 65, 1–155

J. Schepers (2006), Stringy E-functions of varieties with A-D-E singulari-ties. Manuscripta Math. 119(2), 129–157

J. Schepers, W. Veys (2007), Stringy Hodge numbers for a class of isolatedsingularities and for threefolds. Int. Math. Res. Not. 2007(2), Article IDrnm016, 14

J. Schepers, W. Veys (2009), Stringy E-functions of hypersurfaces and ofBrieskorn singularities. Adv. Geom. 9(2), 199–217

J. Schürmann (2011), Characteristic classes of mixed Hodge modules, inTopology of Stratified Spaces. Mathematical Sciences Research InstitutePublications, vol. 58 (Cambridge University Press, Cambridge), pp. 419–470

J. Sebag (2004a), Intégration motivique sur les schémas formels. Bull. Soc.Math. Fr. 132(1), 1–54

J. Sebag (2004b), Rationalité des séries de Poincaré et des fonctions zêtamotiviques. Manuscripta Math. 115(2), 125–162

J. Sebag (2010a), Variations on a question of Larsen and Lunts. Proc. Am.Math. Soc. 138(4), 1231–1242

J. Sebag (2010b), Variétés K-équivalentes et géométrie par morceaux.Arch. Math. (Basel) 94(3), 207–217. http://dx.doi.org/10.1007/s00013-009-0095-3

J. Sebag (2011), Arcs schemes, derivations and Lipman’s theorem. J. Algebra347, 173–183

J. Sebag (2017), A remark on Berger’s conjecture, Kolchin’s theorem andarc schemes. Arch. Math. (Basel) 108(2), 145–150

B. Segre (1950), Sur un problème de M. Zariski. Algèbre et Théorie desNombres, Colloques Internationaux du Centre National de la RechercheScientifique, vol. 24, pp. 135–138, Centre National de la Recherche Sci-entifique, Paris

J.-P. Serre (1965), Classification des variétés analytiques p-adiques com-pactes. Topology 3, 409–412

J.-P. Serre (1968), Corps locaux (Hermann, Paris)

http://dx.doi.org/10.1007/s00013-009-0095-3

http://dx.doi.org/10.1007/s00013-009-0095-3

516 BIBLIOGRAPHY

J.-P. Serre (2012), Lectures on NX(p). Chapman & Hall/CRC ResearchNotes in Mathematics, vol. 11 (CRC Press, Boca Raton)

T. Shioda (1977), Some remarks on Abelian varieties. J. Fac. Sci. Univ.Tokyo Sect. IA Math. 24(1), 11–21

A. Smeets (2017), Logarithmic good reduction, monodromy and the rationalvolume. Algebra & Number Theory 11(1), 213–233

M. Spivakovsky (1999), A new proof of D. Popescu’s theorem on smoothingof ring homomorphisms. J. Am. Math. Soc. 12(2), 381–444

Stacks Project (2017). http:// stacks.math.columbia.eduR.G. Swan (1998), Néron-Popescu desingularization, in Algebra and Geome-

try (Taipei, 1995). Lectures on Algebraic Geometry, vol. 2 (InternationalPress, Cambridge), pp. 135–192

L.A. Tarrío, A.J. López, M.P. Rodríguez (2007), Infinitesimal lifting andJacobi criterion for smoothness on formal schemes. Commun. Algebra35(4), 1341–1367

J. Tate (1971), Rigid analytic spaces. Invent. Math. 12, 257–289B. Teissier (1995), Résultats récents sur l’approximation des morphismes en

algèbre commutative (d’après André, Artin, Popescu et Spivakovsky).Astérisque, 227, pp. Exp. No. 784, 4, 259–282. Séminaire Bourbaki,vols. 1993/1994

M. Temkin (2008), Desingularization of quasi-excellent schemes in charac-teristic zero. Adv. Math. 219(2), 488–522

M. Temkin (2009), Functorial desingularization over q: boundaries and theembedded case. Arxiv:0912.2570

M. Temkin (2012), Functorial desingularization of quasi-excellent schemesin characteristic zero: the nonembedded case. Duke Math. J. 161(11),2207–2254

M. Temkin (2015), Introduction to Berkovich analytic spaces, in BerkovichSpaces and Applications. Lecture Notes in Mathematics, vol. 2119(Springer, Cham), pp. 3–66

P. Valabrega (1975), On the excellent property for power series rings overpolynomial rings. J. Math. Kyoto Univ. 15(2), 387–395

P. Valabrega (1976), A few theorems on completion of excellent rings.Nagoya Math. J. 61, 127–133

A.N. Varchenko (1982), The complex singularity index does not change alongthe stratum μ = const. Funktsional. Anal. i Prilozhen. 16(1), 1–12, 96

J.-L. Verdier (1996/1997), Des catégories dérivées des catégories abéliennes.Astérisque 239, xii+253 pp. With a preface by Luc Illusie, Edited andwith a note by Georges Maltsiniotis

W. Veys (1993), Poles of Igusa’s local zeta function and monodromy. Bull.Soc. Math. Fr. 121, 545–598

W. Veys (2004), Stringy invariants of normal surfaces. J. Algebraic Geom.13(1), 115–141

http://stacks.math.columbia.edu

BIBLIOGRAPHY 517

W. Veys (2006), Vanishing of principal value integrals on surfaces. J. ReineAngew. Math. 598, 139–158

C.-L. Wang (1998), On the topology of birational minimal models. J. Differ.Geom. 50(1), 129–146

C.-L. Wang (2002), Cohomology theory in birational geometry. J. Differ.Geom. 60(2), 345–354

C.A. Weibel (2013), The K-Book: An Introduction to Algebraic K-Theory.Graduate Studies in Mathematics, vol. 145 (American Mathematical So-ciety, Providence)

A. Weil (1982), Adeles and Algebraic Groups. Progress in Mathematics,vol. 23 (Birkhäuser, Boston)

J. Włodarczyk (2003), Toroidal varieties and the weak factorization theo-rem. Invent. Math. 154(2), 223–331

T. Yasuda (2004), Twisted jets, motivic measures and orbifold cohomology.Compos. Math. 140(2), 396–422

O. Zariski (1939), The reduction of the singularities of an algebraic surface.Ann. Math. (2) 40, 639–689

Z. Zhu (2013), Log canonical thresholds in positive characteristic.arXiv:1308.5445

INDEX

Aabsolute ramification index, 218absolute value, 1

p-adic —, 2trivial —, 2

A’Campo’s formula, 47ACFk, first order theory of

algebraically closed fields,117

additive invariant, 56, 77affine bundle, 278

free —, 278affinoid algebra, 492analytic domain, 493analytic domain immersion, 251,

493analytic function, 7analytic manifold, 8analytic Milnor fiber, 450analytic space

formal model of, 494, 495good, 493irreducible component of, 496normalization of, 496

analytic topology, 19analytification, 493angular component, 34, 345

reduced —, 34

arc on a variety, 169constant, 180

Artin approximation theorem,274

Bbase-point

of a jet, 165of an arc, 170

Batyrev’s theorem, 25Berkovich spectrum of a Banach

algebra, 492Bernstein–Sato polynomial, 37Betti numbers, 25, 27

virtual — of a variety, 102Bhatt’s theorem, 174Bittner’s theorem, 121blow-up, 469

admissible, 494formal, 494

blow-up relations, 71, 121blowing-up, 469

formal, 494Borisov’s theorem, 139bounded differential form, 440

CCalabi–Yau manifold, 25Calabi–Yau variety, 475


519

https://doi.org/10.1007/978-1-4939-7887-8

520 INDEX

cancellable local k-algebra, 208canonical divisor, 472

relative —, 473canonical line bundle, 15canonical singularities, 474category

abelian, 82additive, 81derived — of an abelian

category, 84exact, 82semisimple abelian, 83triangulated, 84

C-constructible approximation,331

center of a blowing-up, 469center of a valuation, 386change of variables formula

for motivic integrals, 341for smooth formal schemes, 309in p-adic integrals, 12

chart, 8Chevalley’s theorem, 59, 346, 467C-measurable set, 331codimension of a constructible

subset, 298coherent sheaf on a formal

scheme, 482constructible, 59constructible approximation, 321constructible function, 334constructible sheaf, 90constructible subset

of an arc space, 195constructible subset of a

topological space, 465globally —, 465

constructible topology on ascheme, 467

constructible varieties (categoryof), 67

ConsX , Boolean algebra of allconstructible subsets of X,59

convergent power series, 483, 492

covering family of opensubfunctors, 156

crepant birational morphism, 473CSh(S), category of constructible

sheaves of Q-vector spaceson a variety, 90

CVark, category of constructiblek-varieties, 67

cyclotomic character, 94cylinder, 195

DDenef’s formula, 41Denef’s rationality theorem

for the Poincaré series, 53differential form, 13dilatation, 428dimension

of a formal scheme, 481dimensional filtration, 109discrepancy

— of a valuation, 387discrete valuation ring

absolutely ramified —, 218absolutely unramified —, 218

disk, 2distinguished triangle, 84divisorial valuation, 386domination of formal models, 495

EEisenstein polynomial, 220elimination of quantifiers, 346,

347Elkik-Jacobi constant, 284essential valuation, 400étale cohomology

of a variety, 94Eu(X), Euler characteristic of X,

95e+(X/S), 57Euler characteristic

A-valued —, 56motivic, 131

INDEX 521

Euler–Poincaré polynomial, 101of a variety, 102

excellent formal scheme, 488excellent scheme, 79exceptional divisor in a blow-up,

469exceptional locus, 470extension of complete discrete

valuation rings, 219ramification index of —, 219totally ramified —, 219unramified —, 219

Ffat subset, 257field of p-adic numbers, 3field of representatives, 221finite dimensional model, 197finite dimensional motive, 132finite dimensionality conjecture of

Kimura and O’Sullivan, 133,385

Fitting ideal, 264formal completion, 482formal model, 494formal scheme, 480

adic, 480admissible, 494dimension of, 481flat, 489formally of finite type, 488generic fiber of, 495locally noetherian, 480morphism of, 480noetherian, 481of finite type, 488pure dimensional, 491pure relative dimension, 497reduction of, 481rig-irreducible, 490separated, 480

formal spectrum of an admissibletopological ring, 479

formal subschemeclosed, 483open, 480

formally principal homogeneousspace, 277

trivial —, 277Frobenius, 217functor of arcs

on a variety, 169

GGabber’s cancellation theorem,

208gauge form, 15Gauss norm, 492generic fiber of a formal scheme,

495geometric monodromy group, 460geometrical branch, 179Gorenstein measure, 415Gorenstein variety, 473Gorenstein volume, 415Gr-bijective map, 293Gr-injective map, 293Greenberg approximation

theorem, 271Greenberg function, 275Greenberg functor, 248Greenberg scheme, 248Grinberg–Kazhdan–Drinfeld’s

theorem, 197Gromov’s question, 142Grothendieck group

of a symmetric monoidalcategory, 81

of a triangulated category, 84of an exact category, 82of varieties, 57

Grothendieck monoidof varieties, 57

Grothendieck ringof varieties, 69of varieties modulo universal

homeomorphisms, 115

522 INDEX

Grothendieck semiring ofS-varieties

up to universalhomeomorphisms, 115

HHaar measure, 5Hasse–Weil zeta function, 26Hensel’s lemma, 10henselian ring, 9Hironaka’s theorem, 470Hodge filtration, 85Hodge module, 93

mixed, 91Hodge numbers, 25

of K-equivalent varieties, 26of a mixed Hodge structure, 85virtual — of a variety, 88, 89

Hodge structuremixed, 85mixed — on the cohomology of

a variety, 87polarizable, 86pure, 85, 86Tate, 86

Hodge–Deligne polynomialof a mixed Hodge structure, 85of a variety, 57, 88

homogeneous subset of an arcspace, 300

HS, category of pure Hodgestructures, 86

hX , presheaf associated with X,154

Iideal of definition, 479, 481Igusa’s local zeta function, 33Igusa’s monodromy conjecture, 48Igusa’s rationality theorem

for the local zeta function, 39for the Poincaré series, 51

immersionanalytic domain —, 251of formal schemes, 483

regular — of formal schemes,489

indconstructible subset of ascheme, 467

index of a divisorial valuation,386

integrable function, 334inversion of adjunction, 397irreducible components of an

analytic space, 496isocrystal, 104

JJacobian

order of the, 289Jacobian criterion for

smoothness, 203Jacobian ideal

of a formal scheme, 266of a morphism, 289

Jacobian matrix, 7Jannsen’s theorem, 130, 132jet on a variety, 162

KK0(A), Grothendieck group of

the exact category A, 82K0(VarS), Grothendieck ring of

S-varieties, 57, 69K+

0 (VarS), Grothendieck monoidof S-varieties, 57

K+,uh0 (VarS), Grothendieck

semiring of S-varietiesmodulo universalhomeomorphisms, 115

Ksplit0 (A), Grothendieck group of

the symmetric monoidalcategory A, 81

K0(A), Grothendieck group ofthe triangulated category A,84

Kuh0 (VarS), Grothendieck ring of

S-varieties modulo universalhomeomorphisms, 115

INDEX 523

K-equivalence of varieties, 26,417, 474

Kimura–O’Sullivan’s conjecture,133, 385

Kolchin’s irreducibility theorem,192

Kontsevich’s theorem, 25, 418

LLarsen–Lunts’s theorem, 134LS , class of the affine line in

K0(VarS), 68Lefschetz number, 425Lefschetz trace formula, 96, 106linear topology, 478Ln(X/S), functor/scheme of jets

of level n on X/S, 163local complete intersection, 489local field, 3local inversion theorem, 8local ring, 4local zeta function, 33, 413log canonical threshold, 391

along a subscheme, 392semicontinuity of —, 396

log discrepancy of a valuation,391

log resolution, 470of a formal scheme, 441

log terminal singularities, 474L∞(X/S), functor of arcs on X,

169

Mmeasurable function, 334measurable subset, 321measure, 317metric on a line bundle, 14MHM(X), category of mixed

Hodge modules on X, 91mHS, category of mixed Hodge

structures, 86Milnor fiber, 47Milnor fibration, 46

minimal log discrepancy, 391along a subscheme, 392

minimal variety, 475mixed �-adic sheaf, 100model of a scheme, 22module of continuous

differentials, 487modulus of a local field, 5monodromy conjecture, 48monodromy transformation, 47monodromy zeta function, 425morphism of analytic spaces

quasi-smooth, 493rig-smooth, 493

morphism of formal schemes, 480étale, 486adic, 482formally étale, 486formally of finite type, 485formally smooth, 486formally unramified, 486immersion, 483locally formally of finite type,

485locally of finite type, 484of finite type, 484separated, 480smooth, 486unramified, 486

motive, 129motivic Euler characteristic, 131motivic homotopy of schemes, 108motivic integral

of a volume form on analgebraic variety, 456

of a volume form on ananalytic space, 432

motivic integral on a smoothformal scheme, 309

motivic measure, 67, 78separated, 112

motivic Milnor fiber, 424motivic nearby fiber, 423

with support, 424

524 INDEX

motivic Serre invariantof an algebraic variety, 456of an analytic space, 434

motivic volumeof a constructible set, 317of a measurable set, 324on smooth formal schemes, 308

motivic zeta functionKapranov’s —, 374local — of f , 420of a formal scheme, 439of a motive, 384of an analytic space, 439of f , 420

M∼,R(X), motive of thek-variety X for the adequateequivalence relation ∼, 130

NNash map, 401Nash space, 400negligible subset, 330Néron model (weak), 251Néron smoothening, 251, 429néronian analytic space, 252Newton polygon of an isocrystal,

105Nishimura’s lemma, 137non-Archimedean valued field, 2nondegenerate points (of a

Greenberg scheme), 298

OΩB/A, module of continuous

differentials, 487order function, 260order of the Jacobian, 270ΩY/X, module of continuous

differentials, 487

Pp-adic Hodge theory, 29Pas’s theorem, 347perverse sheaf, 91, 100

pHS, category of polarizablepure Hodge structures, 87

piecewise isomorphism, 64elementary, 63

piecewise morphism, 64piecewise trivial fibration, 69piecewise varieties (category of),

64pMHS, category of polarizable

mixed Hodge structures, 87Poincaré series, 51polydisk, 6Poonen’s theorem, 146power series

convergent —, 6Presburger set, 346Presburger’s theorem, 347presheaf on a category, 154principal homogeneous space, 277proconstructible subset of a

scheme, 467pure motives, 130

category of, 131category of effective, 130

Qquantifier elimination for

semi-algebraic sets, 347quasi-character, 32quasi-excellent scheme, 79

Rrational map, 471

birational, 471dominant, 471proper, 471

rational volume, 461regular immersion, 489relative canonical divisor, 473renormalization of an arc, 181RepGk

Q�, abelian category ofcontinuous representationsof Gk in finite dimensionalQ�-vector spaces, 94

representability criterion, 156

INDEX 525

representable functor, 155representable sheaf, 155resolution of singularities, 470,

471restricted power series, see also

convergent power seriesretrocompact subset of a

topological space, 465rig-irreducible components of

formal schemes, 490rigid cohomology, 105RS′/S(X), Weil restriction of X

with respect to S′/S, 157

SSchwartz–Bruhat function, 33scissor relations, 57semi-algebraic

condition, 345family, 348map, 346set, 346subset of a Greenberg scheme,

347Serre invariant, 18Serre’s theorem, 17fpqc-sheaf, 228simple function, 349singular locus

of a formal scheme, 266slope of an isocrystal, 104sn,X , constant jets of level n on

X, 165s∞,X , constant arcs on X, 180special fiber of a formal scheme,

489specialization map, 496Spf(A), formal spectrum of the

admissible topologicalring A, 479

spHS, set of isomorphism classesof simple polarizable pureHodge structures, 87

split-additive invariant, 81spreading-out, 74

stably birational varieties, 133,476

SBk, set of equivalence classes ofintegral k-varieties for thestably birational relation, 134

strict normal crossings divisor,470

strict transform, 470strongly measurable subset, 321strongly negligible subset, 330subfunctor

closed —, 155open —, 155

sum of a summable family, 319summable family, 319symmetric monoidal category, 80

Ttame formal scheme, 314tame inertia group, 460tame locus, 336tame monodromy operator, 460tame morphism of formal

schemes, 336Tate algebra, 492Tate Hodge structure, 86Teichmüller map, 221Teichmüller representative, 221thin subset, 257

algebraically —, 258θm

n,X , truncation morphismbetween jet schemes of X,165

θ∞n,X , truncation morphism from

arcs to jets on X, 170topological algebra, 478

formally of finite type, 484topologically of finite type, 484

topological ring, 478adic, 479admissible, 479pre-adic, 479pre-admissible, 478

topological zeta function, 43, 414totally discontinuous topology, 2

526 INDEX

trace formulaalgebraic case, 461analytic case, 437

truncation functorfor jets, 165from arcs to jets, 170

Uultrametric inequality, 2uniformizer, 218universal homeomorphism, 114

Vvaluation ring, 4valued field, 2

complete —, 3VarS , category of S-varieties, 55vector bundle, 278vector scheme, 278

sub—, 278Verschiebung, 216virtual étale �-adic realization, 94virtual Hodge realization, 88

Wweak factorization theorem, 472weak Néron model, 251wedge, 171Weierstrass division theorem, 200Weierstrass preparation theorem,

201weight filtration, 85, 100

Weil conjectures, 27Weil restriction, 157wild locus, 336wild morphism of formal schemes,

336Witt polynomials, 212Witt vectors, 214

YYasuda’s theorem, 417Yoneda lemma, 155

ZZariski sheaf, 154zeta function

Hasse–Weil —, 26Igusa’s local zeta function, 33,

413of the monodromy, 47topological —, 43

Zf(T)Zf (T ), motivic zeta function

of f , 420Zf,W(T)

Zf,W (T ), local motivic zetafunction of f with supporton W , 420

Zf,x(T)Zf,x(T ), local motivic zeta

function of f at x, 420ZX(t), motivic zeta function of a

k-variety, 374

link.springer.com978-1-4939-7887-8/1.pdf · APPENDIX § 1. CONSTRUCTIBILITY IN ALGEBRAIC GEOMETRY...

Documents

Transcript of link.springer.com978-1-4939-7887-8/1.pdf · APPENDIX § 1. CONSTRUCTIBILITY IN ALGEBRAIC GEOMETRY...