Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their...
Transcript of Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their...
![Page 1: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/1.jpg)
p-adic Hodge theory
Peter Scholze
Algebraic GeometrySalt Lake City
![Page 2: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/2.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp.
ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).
![Page 3: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/3.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p.
Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).
![Page 4: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/4.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).
![Page 5: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/5.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p,
with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).
![Page 6: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/6.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).
![Page 7: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/7.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)),
where t corresponds to(p, p1/p, . . .).
![Page 8: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/8.jpg)
Fontaine’s construction
Let C be an algebraically closed complete extension of Qp. ThenFrobenius Φ is surjective on OC/p. Have Fontaine’s field
C [ = Frac(lim←−Φ
OC/p) , O[C = lim←−Φ
OC/p .
Then C [ is a complete algebraically closed complete field ofcharacteristic p, with ring of integers O[C .
Example
If C = Qp, then C [ = Fp((t)), where t corresponds to(p, p1/p, . . .).
![Page 9: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/9.jpg)
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim←−x 7→xp
C .
This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p, and
(1 + t)] = limn→∞
(1 + p1/pn)pn.
The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.
![Page 10: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/10.jpg)
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim←−x 7→xp
C .
This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ].
For example, t] = p, and
(1 + t)] = limn→∞
(1 + p1/pn)pn.
The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.
![Page 11: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/11.jpg)
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim←−x 7→xp
C .
This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p,
and
(1 + t)] = limn→∞
(1 + p1/pn)pn.
The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.
![Page 12: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/12.jpg)
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim←−x 7→xp
C .
This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p, and
(1 + t)] = limn→∞
(1 + p1/pn)pn.
The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.
![Page 13: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/13.jpg)
Fontaine’s construction
Fact. There is an identification of multiplicative monoids
C [ = lim←−x 7→xp
C .
This gives rise to a continuous multiplicative mapC [ → C : x 7→ x ]. For example, t] = p, and
(1 + t)] = limn→∞
(1 + p1/pn)pn.
The map C [ → C : x 7→ x ] is analytic, highly non-algebraic.
![Page 14: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/14.jpg)
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:A1C [ ≈ lim←−
T 7→T p
A1C .
On points this is the identification
C [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
![Page 15: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/15.jpg)
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:A1C [ ≈ lim←−
T 7→T p
A1C .
On points this is the identification
C [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
![Page 16: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/16.jpg)
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:A1C [ ≈ lim←−
T 7→T p
A1C .
On points this is the identification
C [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
![Page 17: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/17.jpg)
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:A1C [ ≈ lim←−
T 7→T p
A1C .
On points this is the identification
C [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
![Page 18: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/18.jpg)
Geometry over C vs. geometry over C [
Example (The affine line A1 with coordinate T .)
Claim:A1C [ ≈ lim←−
T 7→T p
A1C .
On points this is the identification
C [ = lim←−x 7→xp
C .
As x 7→ x ] is non-algebraic, need to formalize this in an analyticworld.
![Page 19: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/19.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.
In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 20: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/20.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 21: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/21.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded,
andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 22: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/22.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 23: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/23.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots.
This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 24: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/24.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 25: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/25.jpg)
Perfectoid Algebras
In algebraic geometry, spaces defined by their sheaves of functions.In analytic geometry, these rings are Banach algebras.
DefinitionLet L = C or L = C [ (or any perfectoid field). A perfectoidL-algebra is a Banach L-algebra R such that the subring R ⊂ R ofpowerbounded elements is bounded, andΦ : R/p → R/p : x 7→ xp is surjective.
Example
R = C 〈T 1/p∞〉, convergent power series in variable T and all itsp-power roots. This R gives functions on (an open subset of)lim←−T 7→T p A1
C .
Note: If L = C [ is of characteristic p, last condition is equivalentto requiring R perfect.
![Page 26: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/26.jpg)
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
R[ = lim←−Φ
R/p'← lim←−
x 7→xpR ,
and R[ = R[ ⊗OC[
C [.
Example
If R = C 〈T 1/p∞〉, then R[ = C [〈T 1/p∞〉.
Theorem (S., 2011)
The functor R 7→ R[ is an equivalence between the category ofperfectoid C -algebras and the category of perfectoid C [-algebras.
![Page 27: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/27.jpg)
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
R[ = lim←−Φ
R/p'← lim←−
x 7→xpR ,
and R[ = R[ ⊗OC[
C [.
Example
If R = C 〈T 1/p∞〉, then R[ = C [〈T 1/p∞〉.
Theorem (S., 2011)
The functor R 7→ R[ is an equivalence between the category ofperfectoid C -algebras and the category of perfectoid C [-algebras.
![Page 28: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/28.jpg)
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
R[ = lim←−Φ
R/p'← lim←−
x 7→xpR ,
and R[ = R[ ⊗OC[
C [.
Example
If R = C 〈T 1/p∞〉, then R[ = C [〈T 1/p∞〉.
Theorem (S., 2011)
The functor R 7→ R[ is an equivalence between the category ofperfectoid C -algebras and the category of perfectoid C [-algebras.
![Page 29: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/29.jpg)
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
R[ = lim←−Φ
R/p'← lim←−
x 7→xpR ,
and R[ = R[ ⊗OC[
C [.
Example
If R = C 〈T 1/p∞〉, then R[ = C [〈T 1/p∞〉.
Theorem (S., 2011)
The functor R 7→ R[ is an equivalence between the category ofperfectoid C -algebras and the category of perfectoid C [-algebras.
![Page 30: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/30.jpg)
Tilting for perfectoid algebras
Let R a perfectoid C -algebra. Define
R[ = lim←−Φ
R/p'← lim←−
x 7→xpR ,
and R[ = R[ ⊗OC[
C [.
Example
If R = C 〈T 1/p∞〉, then R[ = C [〈T 1/p∞〉.
Theorem (S., 2011)
The functor R 7→ R[ is an equivalence between the category ofperfectoid C -algebras and the category of perfectoid C [-algebras.
![Page 31: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/31.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 32: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/32.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 33: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/33.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 34: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/34.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 35: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/35.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 36: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/36.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 37: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/37.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 38: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/38.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[).
The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 39: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/39.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic,
OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 40: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/40.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras,
with tilt OX [ .
![Page 41: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/41.jpg)
Perfectoid Spaces
p-adic analytic geometry:
I Tate’s rigid-analytic varieties (late 60’s)
I Berkovich’s analytic spaces (late 80’s)
I Huber’s adic spaces (early 90’s)
To perfectoid C -algebra R (+ R+ ⊂ R) can attach an ’affinoidperfectoid space’ X = Spa(R) of continuous valuations, equippedwith a structure sheaf OX .
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
![Page 42: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/42.jpg)
Perfectoid Spaces
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
Define general perfectoid spaces by gluing affinoid perfectoidspaces.
Corollary
The categories of perfectoid spaces over C and C [ are equivalent.
![Page 43: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/43.jpg)
Perfectoid Spaces
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
Define general perfectoid spaces by gluing affinoid perfectoidspaces.
Corollary
The categories of perfectoid spaces over C and C [ are equivalent.
![Page 44: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/44.jpg)
Perfectoid Spaces
Theorem (S., 2011)
Let X = Spa(R) and X [ = Spa(R[). The underlying topologicalspaces |X | ∼= |X [| are homeomorphic, OX is a sheaf of perfectoidC -algebras, with tilt OX [ .
Define general perfectoid spaces by gluing affinoid perfectoidspaces.
Corollary
The categories of perfectoid spaces over C and C [ are equivalent.
![Page 45: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/45.jpg)
Example
The inverse limit
X = lim←−T 7→T p
A1C has tilt X [ = lim←−
T 7→T p
A1C [ .
|X [|...
T 7→T p∼=
∼= |X |...
T 7→T p
|A1
C [ |
T 7→T p∼=
|A1C |
T 7→T p
|A1
C [ | |A1C |
![Page 46: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/46.jpg)
Example
The inverse limit
X = lim←−T 7→T p
A1C has tilt X [ = lim←−
T 7→T p
A1C [ .
|X [|...
T 7→T p∼=
∼= |X |...
T 7→T p
|A1
C [ |
T 7→T p∼=
|A1C |
T 7→T p
|A1
C [ | |A1C |
![Page 47: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/47.jpg)
Example
The inverse limit
X = lim←−T 7→T p
A1C has tilt X [ = lim←−
T 7→T p
A1C [ .
X [
...
T 7→T p
≈ X...
T 7→T p
A1C [
T 7→T p
A1C
T 7→T p
A1C [ A1
C
![Page 48: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/48.jpg)
Example
The inverse limit
X = lim←−T 7→T p
A1C has tilt X [ = lim←−
T 7→T p
A1C [ .
|X [|...
T 7→T p∼=
∼= |X |...
T 7→T p
|A1
C [ |
T 7→T p∼=
|A1C |
T 7→T p
|A1
C [ | |A1C |
![Page 49: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/49.jpg)
Example
|X [|...
T 7→T p∼=
∼= |X |...
T 7→T p
|A1
C [ |
T 7→T p∼=
|A1C |
T 7→T p
|A1
C [ | |A1C |
Thus, homeomorphism of topological spaces (underlying adicspaces)
|A1C [ | ∼= lim←−
T 7→T p
|A1C | .
char p geometry as infinite covering of char 0 geometry.
![Page 50: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/50.jpg)
Example
|X [|...
T 7→T p∼=
∼= |X |...
T 7→T p
|A1
C [ |
T 7→T p∼=
|A1C |
T 7→T p
|A1
C [ | |A1C |
Thus, homeomorphism of topological spaces (underlying adicspaces)
|A1C [ | ∼= lim←−
T 7→T p
|A1C | .
char p geometry as infinite covering of char 0 geometry.
![Page 51: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/51.jpg)
Example
|X [|...
T 7→T p∼=
∼= |X |...
T 7→T p
|A1
C [ |
T 7→T p∼=
|A1C |
T 7→T p
|A1
C [ | |A1C |
Thus, homeomorphism of topological spaces (underlying adicspaces)
|A1C [ | ∼= lim←−
T 7→T p
|A1C | .
char p geometry as infinite covering of char 0 geometry.
![Page 52: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/52.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C.
There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 53: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/53.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,
such that Xet∼= X [
et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 54: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/54.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 55: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/55.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 56: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/56.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+,
and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 57: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/57.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0,
i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 58: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/58.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 59: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/59.jpg)
The almost purity theorem
TheoremLet X be a perfectoid space over C. There is an etale site Xet,such that Xet
∼= X [et under tilting.
There is the sheaf O+X ⊂ OX of functions bounded by 1.
TheoremThe global sections H0(Xet,O+
X ) = R+, and H i (Xet,O+X ) is almost
zero for i > 0, i.e., killed by p1/n ∈ OC for all n > 0.
This is closely related to Faltings’s celebrated “almost puritytheorem”.
![Page 60: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/60.jpg)
The key computation
Let R = OC 〈T±1〉,
and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 61: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/61.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R).
Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 62: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/62.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R),
where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 63: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/63.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 64: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/64.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 65: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/65.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site,
and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 66: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/66.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn.
One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 67: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/67.jpg)
The key computation
Let R = OC 〈T±1〉, and X = Spa(R[1/p],R). Then X has anaffinoid perfectoid cover by X = Spa(R[1/p], R), where
R = OC 〈T±1/p∞〉 .
Proposition
H0(Xproet, O+X ) = R ,
H1(Xproet, O+X ) = Ω1
R/OC⊕ (p1/(p−1)−torsion) .
Here, Xproet is the pro-etale site, and O+X = lim←−O
+X /pn. One has
H i (Xproet, O+X ) = lim←−H i (Xet,O+
X /pn) .
![Page 68: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/68.jpg)
The key computation
The p1/(p−1)-torsion is sometimes called “junk torsion”.
In the lasttalk, we will see how to get rid of it.
Step 1. The Zp-cover X → X induces a map
H icont(Zp, R)→ H i (Xproet, O+
X ) .
This map is an almost isomorphism as H i (Xproet, O+X ) is almost
zero for i > 0.
Remains to compute
H icont(Zp,OC 〈T±1/p∞〉) =
⊕j∈Z[1/p]
H icont(Zp,OC · T j) .
![Page 69: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/69.jpg)
The key computation
The p1/(p−1)-torsion is sometimes called “junk torsion”. In the lasttalk, we will see how to get rid of it.
Step 1. The Zp-cover X → X induces a map
H icont(Zp, R)→ H i (Xproet, O+
X ) .
This map is an almost isomorphism as H i (Xproet, O+X ) is almost
zero for i > 0.
Remains to compute
H icont(Zp,OC 〈T±1/p∞〉) =
⊕j∈Z[1/p]
H icont(Zp,OC · T j) .
![Page 70: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/70.jpg)
The key computation
The p1/(p−1)-torsion is sometimes called “junk torsion”. In the lasttalk, we will see how to get rid of it.
Step 1. The Zp-cover X → X induces a map
H icont(Zp, R)→ H i (Xproet, O+
X ) .
This map is an almost isomorphism as H i (Xproet, O+X ) is almost
zero for i > 0.
Remains to compute
H icont(Zp,OC 〈T±1/p∞〉) =
⊕j∈Z[1/p]
H icont(Zp,OC · T j) .
![Page 71: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/71.jpg)
The key computation
The p1/(p−1)-torsion is sometimes called “junk torsion”. In the lasttalk, we will see how to get rid of it.
Step 1. The Zp-cover X → X induces a map
H icont(Zp, R)→ H i (Xproet, O+
X ) .
This map is an almost isomorphism as H i (Xproet, O+X ) is almost
zero for i > 0.
Remains to compute
H icont(Zp,OC 〈T±1/p∞〉) =
⊕j∈Z[1/p]
H icont(Zp,OC · T j) .
![Page 72: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/72.jpg)
The key computation
The p1/(p−1)-torsion is sometimes called “junk torsion”. In the lasttalk, we will see how to get rid of it.
Step 1. The Zp-cover X → X induces a map
H icont(Zp, R)→ H i (Xproet, O+
X ) .
This map is an almost isomorphism as H i (Xproet, O+X ) is almost
zero for i > 0.
Remains to compute
H icont(Zp,OC 〈T±1/p∞〉) =
⊕j∈Z[1/p]
H icont(Zp,OC · T j) .
![Page 73: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/73.jpg)
The key computation
Step 2. Computation of
H icont(Zp,OC · T j) .
Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via
γ · T j = ζnpmT j , j = n/pm , n,m ∈ Z .
Then H icont(Zp,OC · T j) computed by the complex
(OC · T j γ−1−→ OC · T j) ∼= (OC
ζnpm−1−→ OC ) .
![Page 74: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/74.jpg)
The key computation
Step 2. Computation of
H icont(Zp,OC · T j) .
Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC .
Then the generator γ = 1 ∈ Zp acts via
γ · T j = ζnpmT j , j = n/pm , n,m ∈ Z .
Then H icont(Zp,OC · T j) computed by the complex
(OC · T j γ−1−→ OC · T j) ∼= (OC
ζnpm−1−→ OC ) .
![Page 75: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/75.jpg)
The key computation
Step 2. Computation of
H icont(Zp,OC · T j) .
Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via
γ · T j = ζnpmT j , j = n/pm , n,m ∈ Z .
Then H icont(Zp,OC · T j) computed by the complex
(OC · T j γ−1−→ OC · T j) ∼= (OC
ζnpm−1−→ OC ) .
![Page 76: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/76.jpg)
The key computation
Step 2. Computation of
H icont(Zp,OC · T j) .
Fix a compatible system of p-power roots of unityζp, ζp2 , . . . ∈ OC . Then the generator γ = 1 ∈ Zp acts via
γ · T j = ζnpmT j , j = n/pm , n,m ∈ Z .
Then H icont(Zp,OC · T j) computed by the complex
(OC · T j γ−1−→ OC · T j) ∼= (OC
ζnpm−1−→ OC ) .
![Page 77: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/77.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 78: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/78.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0.
Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 79: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/79.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1.
This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 80: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/80.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.
Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 81: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/81.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z.
Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 82: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/82.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.
End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 83: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/83.jpg)
The key computation
OC
ζnpm−1−→ OC .
Case 1. j ∈ Z, i.e., m = 0. Then ζnpm − 1 = 0, and so one gets OC
in degrees 0, 1. This contributes OC · T j in degrees i = 0, 1 to thefinal result.Case 2. j 6∈ Z. Then ζnpm − 1 divides p1/(p−1), and one gets no
H0, and p1/(p−1)-torsion in H1.End result.
H0(Xproet, O+X ) =
⊕j∈ZOCT j = R ,
H1(Xproet, O+X ) =
⊕j∈ZOCT j ⊕
⊕j=n/pm∈Z[1/p]\Z
(OC/(ζnpm − 1))T j
= Ω1R/OC
⊕ (p1/(p−1)−torsion) .
![Page 84: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/84.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra.
An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 85: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/85.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated
if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 86: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/86.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M
such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 87: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/87.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 88: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/88.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R).
Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 89: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/89.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0,
is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 90: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/90.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d,
and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 91: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/91.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 92: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/92.jpg)
Almost finite generation: Local case
DefinitionLet R be an OC -algebra. An R-module M is called almost finitelygenerated if for any n ≥ 1, there is a finitely generated submoduleMn ⊂ M such that M/Mn is killed by p1/n.
Corollary
Let R = OC 〈T±11 , . . . ,T±1
d 〉, X = Spa(R[1/p],R). Then
H i (Xproet, O+X ) is an almost finitely generated R-module for all
i ≥ 0, is almost zero for i > d, and
H i (Xproet, O+X ) = Ωi
R/OC⊕ (pi/(p−1)−torsion) .
For the proof, redo the computation in any dimension, or use theKunneth formula.
![Page 93: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/93.jpg)
Almost finite generation: Global case
TheoremLet X be a proper smooth rigid-analytic variety over C.
ThenH i (Xproet, O+
X ) is an almost finitely generated OC -module for alli ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument: Take two(nice) affinoid covers X =
⋃i∈I Ui =
⋃i∈I Vi such that Ui is
strictly contained in Vi .
The key point is that the transition maps
H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+
X /p)
have almost finitely generated image (over OC ).
![Page 94: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/94.jpg)
Almost finite generation: Global case
TheoremLet X be a proper smooth rigid-analytic variety over C. ThenH i (Xproet, O+
X ) is an almost finitely generated OC -module for alli ≥ 0,
and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument: Take two(nice) affinoid covers X =
⋃i∈I Ui =
⋃i∈I Vi such that Ui is
strictly contained in Vi .
The key point is that the transition maps
H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+
X /p)
have almost finitely generated image (over OC ).
![Page 95: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/95.jpg)
Almost finite generation: Global case
TheoremLet X be a proper smooth rigid-analytic variety over C. ThenH i (Xproet, O+
X ) is an almost finitely generated OC -module for alli ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument: Take two(nice) affinoid covers X =
⋃i∈I Ui =
⋃i∈I Vi such that Ui is
strictly contained in Vi .
The key point is that the transition maps
H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+
X /p)
have almost finitely generated image (over OC ).
![Page 96: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/96.jpg)
Almost finite generation: Global case
TheoremLet X be a proper smooth rigid-analytic variety over C. ThenH i (Xproet, O+
X ) is an almost finitely generated OC -module for alli ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument:
Take two(nice) affinoid covers X =
⋃i∈I Ui =
⋃i∈I Vi such that Ui is
strictly contained in Vi .
The key point is that the transition maps
H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+
X /p)
have almost finitely generated image (over OC ).
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Almost finite generation: Global case
TheoremLet X be a proper smooth rigid-analytic variety over C. ThenH i (Xproet, O+
X ) is an almost finitely generated OC -module for alli ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument: Take two(nice) affinoid covers X =
⋃i∈I Ui =
⋃i∈I Vi such that Ui is
strictly contained in Vi .
The key point is that the transition maps
H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+
X /p)
have almost finitely generated image (over OC ).
![Page 98: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/98.jpg)
Almost finite generation: Global case
TheoremLet X be a proper smooth rigid-analytic variety over C. ThenH i (Xproet, O+
X ) is an almost finitely generated OC -module for alli ≥ 0, and almost zero for i > 2 dim X .
The proof is a version of the Cartan–Serre argument: Take two(nice) affinoid covers X =
⋃i∈I Ui =
⋃i∈I Vi such that Ui is
strictly contained in Vi .
The key point is that the transition maps
H j(Vi ,et,O+X /p)→ H j(Ui ,et,O+
X /p)
have almost finitely generated image (over OC ).
![Page 99: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/99.jpg)
Finiteness of Zp-cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C.
Thenatural map
H i (Xet,Zp)⊗Zp OC → H i (Xproet, O+X )
is an almost isomorphism. In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xet,Fp)⊗Fp OC/p → H i (Xet,O+X /p) .
There, use Artin–Schreier sequence
0→ Fp → O+X /p → O+
X /p → 0 .
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Finiteness of Zp-cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C. Thenatural map
H i (Xet,Zp)⊗Zp OC → H i (Xproet, O+X )
is an almost isomorphism.
In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xet,Fp)⊗Fp OC/p → H i (Xet,O+X /p) .
There, use Artin–Schreier sequence
0→ Fp → O+X /p → O+
X /p → 0 .
![Page 101: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/101.jpg)
Finiteness of Zp-cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C. Thenatural map
H i (Xet,Zp)⊗Zp OC → H i (Xproet, O+X )
is an almost isomorphism. In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xet,Fp)⊗Fp OC/p → H i (Xet,O+X /p) .
There, use Artin–Schreier sequence
0→ Fp → O+X /p → O+
X /p → 0 .
![Page 102: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/102.jpg)
Finiteness of Zp-cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C. Thenatural map
H i (Xet,Zp)⊗Zp OC → H i (Xproet, O+X )
is an almost isomorphism. In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xet,Fp)⊗Fp OC/p → H i (Xet,O+X /p) .
There, use Artin–Schreier sequence
0→ Fp → O+X /p → O+
X /p → 0 .
![Page 103: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/103.jpg)
Finiteness of Zp-cohomology
Corollary
Let X be a proper smooth rigid-analytic variety over C. Thenatural map
H i (Xet,Zp)⊗Zp OC → H i (Xproet, O+X )
is an almost isomorphism. In particular, H i (Xet,Zp) is finitelygenerated, and vanishes for i > 2 dim X .
Enough to prove similar result for
H i (Xet,Fp)⊗Fp OC/p → H i (Xet,O+X /p) .
There, use Artin–Schreier sequence
0→ Fp → O+X /p → O+
X /p → 0 .
![Page 104: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/104.jpg)
The Hodge–Tate decomposition
Recall that we want to prove the Hodge–Tate decomposition, for aproper smooth rigid-analytic variety X0 over K with X = X0 ⊗K C ,
H i (Xet,Zp)⊗Zp C ∼=i⊕
j=0
H i−j(X ,ΩjX )(−j) .
At this point, we have an isomorphism
H i (Xet,Zp)⊗Zp C ∼= H i (Xproet, OX ) ,
where OX = O+X [1/p].
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The Hodge–Tate decomposition
Recall that we want to prove the Hodge–Tate decomposition, for aproper smooth rigid-analytic variety X0 over K with X = X0 ⊗K C ,
H i (Xet,Zp)⊗Zp C ∼=i⊕
j=0
H i−j(X ,ΩjX )(−j) .
At this point, we have an isomorphism
H i (Xet,Zp)⊗Zp C ∼= H i (Xproet, OX ) ,
where OX = O+X [1/p].
![Page 106: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/106.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.
![Page 107: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/107.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant;
itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.
![Page 108: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/108.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)
If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.
![Page 109: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/109.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0.
Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.
![Page 110: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/110.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.
This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.
![Page 111: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/111.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.
Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.
![Page 112: Peter Scholze Algebraic Geometry Salt Lake City · In algebraic geometry, spaces de ned by their sheaves of functions. In analytic geometry, these rings are Banach algebras. De nition](https://reader033.fdocuments.net/reader033/viewer/2022042622/5f93effc53977e71a323474a/html5/thumbnails/112.jpg)
The Hodge–Tate decomposition
The local computation of the cohomology of OX in terms ofdifferentials gives the Hodge–Tate spectral sequence
E ij2 = H i (X ,Ωj
X )(−j)⇒ H i+j(Xproet, OX ) .
(The Tate twist is needed to make things Galois equivariant; itappears because we had to choose roots of unity in thecomputation.)If X = X0 ⊗K C , then Galois equivariance forces all differentials inthe spectral sequence to be 0. Moreover, there is a uniqueGalois-equivariant splitting of the resulting abutment filtration.This proves the Hodge–Tate decomposition.Remark. One can show that the Hodge–Tate spectral sequencedegenerates always, for a proper smooth rigid-analytic variety Xover C . However, it does not canonically split.