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THE STABLE CATEGORY OF GORENSTEIN FLAT SHEAVES

ON A NOETHERIAN SCHEME

LARS WINTHER CHRISTENSEN SERGIO ESTRADA AND PEDER THOMPSON

Abstract For a semi-separated noetherian scheme we show that the cate-

gory of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a nat-ural non-affine analogue of the category of Gorenstein projective modules over

a noetherian ring We show that this coheres perfectly with the work of Mur-

fet and Salarian that identifies the pure derived category of F-totally acycliccomplexes of flat quasi-coherent sheaves as the natural non-affine analogue of

the homotopy category of totally acyclic complexes of projective modules

Introduction

A classic result due to Buchweitz [2] says that the singularity category of a Goren-stein local ring A is equivalent to the homotopy category Ktac(prj(A)) of totallyacyclic complexes of finitely generated projective A-modules The latter category isalso equivalent to the stable category of finitely generated maximal Cohen-MacaulayA-modules or in a different terminology to the stable category StGprj(A) of finitelygenerated Gorenstein projective A-modules This second equivalence extends be-yond the realm of Gorenstein local rings and finitely generated modules For ev-ery ring A the category Ktac(Prj(A)) of totally acyclic complexes of projectiveA-modules is equivalent to the stable category StGPrj(A) of Gorenstein projectiveA-modules We obtain this folklore result as a special case of [5 Corollary 39]What is the analogue in the non-affine setting

Murfet and Salarian [23] offer a non-affine analogue of the category Ktac(Prj(A))over a semi-separated noetherian scheme X in the form of the Verdier quotient

DF-tac(Flat(X)) =KF-tac(Flat(X))

Kpac(Flat(X))

of the homotopy category of F-totally acyclic complexes of flat quasi-coherentsheaves on X by its subcategory of pure-acyclic complexes Indeed for a commu-tative noetherian ring A of finite Krull dimension and X = Spec(A) the categoriesKtac(Prj(A)) and DF-tac(Flat(X)) are equivalent by [23 Lemma 422] What re-mains is to identify an analogue of the category StGPrj(A) in the non-affine settingand that is the goal of this paper

Date 14 July 20202020 Mathematics Subject Classification 14F08 18G35Key words and phrases Cotorsion sheaf Gorenstein flat sheaf noetherian scheme stable

category totally acyclic complexLWC was partly supported by Simons Foundation collaboration grant 428308 SE was

partly supported by grants PRX1800057 MTM2016-77445-P and 19880GERM15 by the Fun-

dacion Seneca-Agencia de Ciencia y Tecnologıa de la Region de Murcia and FEDER funds

2 LW CHRISTENSEN S ESTRADA AND P THOMPSON

The stable category of Gorenstein projective modules is a standard constructionthat applies to any Frobenius category The category of Gorenstein flat modules israrely Frobenius it is essentially only Frobenius if it coincides with the category ofGorenstein projective modules see [5 Theorem 45] The cotorsion Gorenstein flatmodules however do form a Frobenius category and a special case of [5 Corol-lary 59] says that for a commutative noetherian ring A of finite Krull dimensionthe category StGPrj(A) is equivalent to the stable category StGFC(A) of cotorsionGorenstein flat modules This identifies a candidate category and indeed the goalstated above is obtained (in 46) with

Theorem A Let X be a semi-separated noetherian scheme The stable categoryStGFC(X) of cotorsion Gorenstein flat sheaves is equivalent to DF-tac(Flat(X))

In the statement of this theorem and everywhere else in this paper a sheaf meansa quasi-coherent sheaf A sheaf on X is called cotorsion if it is right Ext-orthogonalto flat sheaves on X

A crucial step towards the equivalence in Theorem A is to prove (in 42 and43) that the sheaves that are both cotorsion and Gorenstein flat are precisely thesheaves that arise as cycles in F-totally acyclic complexes of flat-cotorsion sheavesThis is exactly what happens in the affine case and it transpires that the maintake-away from [5] also applies in the non-affine setting One should work withsheaves that are both cotorsion and Gorenstein flat rather than all Gorenstein flatsheaves One manifestation is a result (47) that sharpens [23 Theorem 427]

Theorem B A semi-separated noetherian scheme X is Gorenstein if and only ifevery acyclic complex of flat-cotorsion sheaves on X is F-totally acyclic

A second manifestationmdashactually the result behind Theorem Amdashis that thecategory DF-tac(Flat(X)) considered by Murfet and Salarian is equivalent to thehomotopy category KF-tac(Flat(X) cap Cot(X)) of F-totally acyclic complexes of flatcotorsion sheaves (see 45) That is passing from the affine to the non-affine settingone can replace the homotopy category Ktac(Prj(A)) by another homotopy category

Up to equivalence the category KF-tac(Flat(X) cap Cot(X)) arises in a relatedyet different context The category of Gorenstein flat sheaves on a semi-separatednoetherian scheme X is part of a complete hereditary cotorsion pair (see 22) andone that is comparable to the cotorsion pair of flat sheaves and cotorsion sheaveson X Through work of Hovey [21] and Gillespie [16] these cotorsion pairs inducea model structure on the category of sheaves on X We prove (see 44) that theassociated homotopy category is equivalent to KF-tac(Flat(X) cap Cot(X))

1 Gorenstein flat sheaves

In this paper the symbol X denotes a scheme with structure sheaf OX By asheaf on X we shall always mean a quasi-coherent sheaf and Qcoh(X) denotes thecategory of (quasi-coherent) sheaves on X We frequently add the assumption thatX is semi-separated by which we mean that X has an open affine covering U suchthat UcapV is affine for all U V isin U such a covering is referred to as semi-separatingWe use standard cohomological notation for cochain complexes

In this first section we show that over a semi-separated noetherian scheme onecan equivalently define Gorenstein flatness of sheaves globally locally or stalkwiseLet A be a commutative ring An acyclic complex F of flat A-modules is called

THE STABLE CATEGORY OF GORENSTEIN FLAT SHEAVES 3

F-totally acyclic if the complex I otimesA F is acyclic for every injective A-module IAn A-module M is Gorenstein flat if there exists an F-totally acyclic complex Fwith M = Z0(F ) Denote by GFlat(A) the category of Gorenstein flat A-modules

Remark 11 For an acyclic complex F of flat sheaves on X there is a global alocal and a stalkwise notion of F-total acyclicity

bull For every injective sheaf I on X the complex I otimes F is acyclicbull For every open affine U sube X the OX(U)-complex F (U) is F-totally acyclicbull For every x isin X the OXx-complex Fx is F-totally acyclic

It is proved in [23 Lemmas 44 and 45] that all three notions agree if the schemeX is semi-separated noetherian Christensen Estrada and Iacob [4 Corollary 28]show that the local notion is Zariski-local and by [4 Proposition 210] the localand global notions agree if X is semi-separated and quasi-compact (which is weakerthan noetherian)

Definition 12 Assume that X is semi-separated noetherian An acyclic complexF of flat sheaves on X is called F-totally acyclic if it satisfies the equivalent con-ditions in Remark 11 A sheaf M on X is called Gorenstein flat if there exists anF-totally acyclic complex F of flat sheaves on X with M = Z0(F ) Denote byGFlat(X) the category of Gorenstein flat sheaves on X

Over any scheme Gorenstein flatness can also be defined locally or stalkwiseand we proceed to show that these notions agree with Gorenstein flatness as definedabove if the scheme is semi-separated noetherian

Definition 13 A sheaf M on X is called locally Gorenstein flat if for every openaffine subset U sube X the OX(U)-module M (U) is Gorenstein flat and M is calledstalkwise Gorenstein flat if Mx is a Gorenstein flat OXx-module for every x isin X

Like local F-total acyclicity local Gorenstein flatness is a Zariski-local propertyat least under mild assumptions on the scheme As shown in [4] this follows fromthe next proposition

Proposition 14 Let ϕ Ararr B be a flat homomorphism of commutative rings

(a) If M is a Gorenstein flat A-module then B otimesAM is a Gorenstein flatB-module

(b) Assume that A is coherent and ϕ is faithfully flat An A-module M isGorenstein flat if the B-module B otimesAM is Gorenstein flat

Proof (a) Let F be an F-totally acyclic complex of flatA-modules withM = Z0(F )By [4 Proposition 27(1)] the B-complex B otimesA F is an F-totally acyclic complexof flat B-modules so B otimesAM = Z0(B otimesA F ) is a Gorenstein flat B-module

(b) It follows from work of Saroch and S rsquotovıcek [28 Corollary 412] that thecategory GFlat(A) is closed under extensions so the assertion is immediate from aresult of Christensen Koksal and Liang [6 Theorem 11]

Corollary 15 Assume that X is locally coherent A sheaf M on X is locallyGorenstein flat if there exists an open affine covering U of X such that the OX(U)-module M (U) is Gorenstein flat for every U isin U

Proof Proposition 14 shows that Gorenstein flatness is an ascentndashdescent propertyfor modules over commutative coherent rings Now invoke [4 Lemma 24]


Theorem 16 Assume that X is semi-separated noetherian For a sheaf M onX the following conditions are equivalent

(i) M is Gorenstein flat(ii) M is locally Gorenstein flat

(iii) M is stalkwise Gorenstein flat

Proof The implication (i)rArr (ii) is trivial by the definition of Gorenstein flatnesssee Remark 11

(ii) rArr (iii) Let x isin X and choose an open affine subset U sube X with x isin U Localization is exact and commutes with tensor products so it preserves Gorensteinflatness of modules whence the module Mx

sim= M (U)x is Gorenstein flat over thelocal ring OXx sim= OX(U)x

(iii) rArr (i) This argument is inspired by Yang and Liu [29 Lemmas 38 and39] Let U = U0 Un be a semi-separating open affine covering of X Forevery x isin X there is a short exact sequence of OXx-modules

(1) 0 minusrarrMx minusrarr Fx minusrarr Tx minusrarr 0

where Fx is flat and Tx is Gorenstein flat For x isin X and U isin U consider thecanonical maps

ix Spec(OXx) minusrarr X and iU Spec(OX(U)) minusrarr X

for x isin U the map ix factors through iU The map M rarrprodxisinX(ix)lowast(Mx) is a

monomorphism locally at every y isin X as one hasprodxisinX

(ix)lowast(Mx) sim= (iy)lowast(My)oplusprod

xisinXy

(ix)lowast(Mx)

so it is a monomorphism in Qcoh(X) Now (1) yields a monomorphism

M minusrarrprodxisinX

(ix)lowast(Fx)

so with F 0 =prodxisinX(ix)lowast(Fx) there is an exact sequence in Qcoh(X)

(2) 0 minusrarrM minusrarr F 0 minusrarr K 1 minusrarr 0

The first goal is to show that F 0 is flat For every x isin X choose only one elementUk in U with x isin Uk and for k = 0 n let Ik sube Uk denote the correspondingsubset such that X is the disjoint union

⋃nk=0 Ik Now one has

F 0 =prodxisinX

(ix)lowast(Fx) =

noplusk=0

prodxisinIk

(iUk)lowast(Fx) sim=

(dagger)sim=noplusk=0

(iUk)lowast(prodxisinIk

Fx) sim=noplusk=0


Fx)

where the isomorphism (dagger) holds as (iUk)lowast being a right adjoint functor preservesdirect products Since Fx is a flat OX(Uk)-module and OX(Uk) is noetherian itfollows that

prodxisinIk Fx is a flat OX(Uk)-module Hence F 0 is a flat sheaf


The second goal is to show that K 1 is Gorenstein flat locally at every pointy isin X Consider the commutative diagram of OXy-modules

0 My F 0

y

π

K 1y

$

0

0 My Fy Ty 0

where π is the canonical projection with kernel L this is a flat module as it is thekernel of an epimorphism between flat OXx-modules By the Snake Lemma $ issurjective with kernel L so K 1

y is Gorenstein flat see eg [28 Corollary 412]Let I be an injective sheaf on X we argue that (2) remains exact after tensoring

with I by showing that TorQcoh(X)1 (I K 1) = 0 holds For x isin X let J(x) be the

sheaf on Spec(OXx) associated to the injective hull of the residue field of the localring OXx One has

I sim=oplusxisinX

(ix)lowastJ(x)(Λx)

for some index sets Λx see Hartshorne [19 Proposition II717] Therefore it

suffices to verify that TorQcoh(X)1 ((ix)lowastJ(x)K 1) = 0 holds for every x isin X This

can be verified locally and every localization TorQcoh(X)1 ((ix)lowastJ(x)K 1)xprime is 0 or

isomorphic to TorOXx1 (J(x)K 1

x ) and the latter is also 0 as K 1x is a Gorenstein

flat OXx-moduleRepeating this process one gets an exact sequence of sheaves

(3) 0 minusrarrM minusrarr F 0 minusrarr F 1 minusrarr F 2 minusrarr middot middot middot

which remains exact after tensoring with any injective sheaf on X Since X inparticular is semi-separated quasi-compact every sheaf is a homomorphic imageof a flat sheaf see for example Efimov and Positselski [7 Lemma A1] Thereforethere is an exact sequence

(4) 0 minusrarr K minus1 minusrarr Fminus1 minusrarrM minusrarr 0

with Fminus1 a flat sheaf The class of Gorenstein flat modules is closed under kernelsof epimorphisms see eg [28 Corollary 412] so K minus1

x is a Gorenstein flat OXx-module for every x isin X By the same argument as above the sequence (4) remainsexact after tensoring with any injective sheaf on X Repeating this process oneobtains an exact sequence

(5) middot middot middot minusrarr Fminus3 minusrarr Fminus2 minusrarr Fminus1 minusrarrM minusrarr 0

that remains exact after tensoring with any injective sheaf on X Splicing together(3) and (5) one gets per Definition 12 an F-totally acyclic complex of flat sheavesF = middot middot middot rarr Fminus1 rarr F 0 rarr F 1 rarr middot middot middot Thus M = Z0(F ) is Gorenstein flat

Henceforth we work mainly over semi-separated noetherian schemes In that set-ting we consistently refer to the sheaves described in Theorem 16 by their shortestname Gorenstein flat some proofs though rely crucially on their local properties


2 The Gorenstein flat model structure on Qcoh(X)

Let G be a Grothendieck category that is an abelian category that has colimitsexact direct limits (filtered colimits) and a generator A class C of objects in Gis called resolving if it contains all projective objects and is closed under exten-sions and kernels of epimorphisms To a class C of objects in G one associates theorthogonal classes

Cperp = G isin G | Ext1G(CG) = 0 for all C isin C and

perpC = G isin G | Ext1G(GC) = 0 for all C isin C

Let S sube C be a set The pair (C Cperp) is said to be generated by the set S if an objectG belongs to Cperp if and only if Ext1

G(CG) = 0 holds for all C isin S A pair (F C) of

classes in G with Fperp = C and perpC = F is called a cotorsion pair The intersectionF cap C is called the core of the cotorsion pair

A cotorsion pair (F C) in G is called hereditary if for all F isin F and C isin C onehas ExtiG(FC) = 0 for all i ge 1 Notice that the class F in this case is resolving

A cotorsion pair (F C) in G is called complete provided that for every G isin Gthere are short exact sequences 0rarr C rarr F rarr Grarr 0 and 0rarr Grarr C prime rarr F prime rarr 0with F F prime isin F and CC prime isin C

Abelian model category structures from cotorsion pairs Gillespie [16]shows how to construct a hereditary abelian model structure on G from two com-

parable cotorsion pairs Namely if (Q R) and (QR) are complete hereditary

cotorsion pairs in G with R sube R Q sube Q and Q cap R = Q cap R then there existsa unique thick (ie full closed under direct summands and having the two-out-of-

three property) subcategory W of G such that Q = Q capW and R = R capW Inother words (QWR) is a so-called Hovey triple and from work of Hovey [21] itis known that there is a unique abelian model structure on G in which Q R andW are the classes of cofibrant fibrant and trivial objects respectively refer to [21]for this standard terminology We are now going to apply this machine to cotorsion

pairs with Q and Q the categories of flat and Gorenstein flat sheaves on X

Remark 21 If X is semi-separated quasi-compact then Qcoh(X) is a locallyfinitely presentable Grothendieck category This was proved already in EGA [18I6912] though not using that terminology Being a Grothendieck category Qcoh(X)has a generator and hence by [7 Lemma A1] a flat generator Slavik and S rsquotovıcek[26] have recently proved that if X is quasi-separated and quasi-compact thenQcoh(X) has a flat generator if and only if X is semi-separated

Theorem 22 Assume that X is semi-separated noetherian The pair

(GFlat(X)GFlat(X)perp)

is a complete hereditary cotorsion pair

Proof For an open affine subset U sube X we write GFlat(U) for the categoryGFlat(OX(U)) of Gorenstein flat OX(U)-modules For every open affine subsetU sube X the pair (GFlat(U)GFlat(U)perp) is a complete hereditary cotorsion pair seeEnochs Jenda and Lopez-Ramos [10 Theorems 211 and 212] The proof of [10Theorem 211] shows that the pair is generated by a set SU see also the moreprecise statement in [28 Corollary 412]


A result of Estrada Guil Asensio Prest and Trlifaj [11 Corollary 315] nowshows that (GFlat(X)GFlat(X)perp) is a complete cotorsion pair Indeed the flatgenerator of Qcoh(X) belongs to GFlat(X) As the quiver in [11 Notation 312]one takes the quiver with vertices all open affine subsets of X and the class L in[11 Corollary 315] is in this case

L = L isin Qcoh(X) | L (U) isin SU for every open affine subset U sube X Moreover since (GFlat(U)GFlat(U)perp) is hereditary the class GFlat(U) is resolvingfor every open affine subset U sube X It follows that GFlat(X) is also resolvingwhence (GFlat(X)GFlat(X)perp) is hereditary as GFlat(X) contains a generator seeSaorın and S rsquotovıcek [25 Lemma 425]

Let X be a semi-separated noetherian scheme By Flat(X) we denote the cate-gory of flat sheaves on X The proof of the next result is modeled on an argumentdue to Estrada Iacob and Perez [12 Proposition 41]

Lemma 23 Assume that X is semi-separated noetherian In Qcoh(X) one has

GFlat(X) cap GFlat(X)perp = Flat(X) cap Flat(X)perp

Proof ldquosuberdquo Let M isin GFlat(X) cap GFlat(X)perp The inclusion Flat(X) sube GFlat(X)yields GFlat(X)perp sube Flat(X)perp so it remains to show that M is flat Since M is inGFlat(X) there is an exact sequence in Qcoh(X)

0 minusrarrM minusrarr F minusrarr N minusrarr 0

with F a flat sheaf and N a Gorenstein flat sheaf on X Since M belongs toGFlat(X)perp the sequence splits whence M is flat

ldquosuperdquo Let M isin Flat(X) cap Flat(X)perp As the inclusion Flat(X) sube GFlat(X) holdsit remains to show that M is in GFlat(X)perp Since (GFlat(X)GFlat(X)perp) is acomplete cotorsion pair in Qcoh(X) see Theorem 22 there is an exact sequencein Qcoh(X)

(6) 0 minusrarrM minusrarr G minusrarr N minusrarr 0

with G isin GFlat(X)perp and N isin GFlat(X) Moreover since GFlat(X) is closed underextensions by Theorem 22 also G belongs to GFlat(X) Thus the sheaf G is inGFlat(X)capGFlat(X)perp so by the containment already proved G is flat Since M isalso flat it follows that

flat dimOX(U) N (U) le 1

holds for every open affine subset U sube X Thus N (U) is a Gorenstein flat OX(U)-module of finite flat dimension and therefore flat see [9 Corollary 1034] Itfollows that N is a flat sheaf Since M isin Flat(X)perp by assumption the sequence(6) splits Therefore M is a direct summand of G and thus in GFlat(X)perp

We call sheaves in the subcategory Cot(X) = Flat(X)perp of Qcoh(X) cotorsionSheaves in the intersection Flat(X) cap Cot(X) are called flat-cotorsion

Remark 24 Assume that X is semi-separated quasi-compact In this case thecategory Flat(X) contains a generator for Qcoh(X) so it follows from work ofEnochs and Estrada [8 Corollary 42] that (Flat(X)Cot(X)) is a complete cotorsionpair and since Flat(X) is resolving it follows from [25 Lemma 425] that the pair(Flat(X)Cot(X)) is hereditary This fact can also be deduced from work of Gillespie[14 Proposition 64] and Hovey [21 Corollary 66]


The next theorem establishes what we call the Gorenstein flat model structureon Qcoh(X) it may be regarded as a non-affine version of [17 Theorem 33]

Theorem 25 Assume that X is semi-separated noetherian There exists a uniqueabelian model structure on Qcoh(X) with GFlat(X) the class of cofibrant objects andCot(X) the class of fibrant objects In this structure Flat(X) is the class of triviallycofibrant objects and GFlat(X)perp is the class of trivially fibrant objects

Proof It follows from Theorem 22 and Remark 24 that (GFlat(X)GFlat(X)perp)and (Flat(X)Cot(X)) are complete hereditary cotorsion pairs Every flat sheaf isGorenstein flat and by Lemma 23 the two pairs have the same core so they satisfythe conditions in [16 Theorem 12] Thus the pairs determine a Hovey triple andby [21 Theorem 22] a unique abelian model category structure on Qcoh(X) withfibrant and cofibrant objects as asserted

Corollary 26 Assume that the scheme X is semi-separated noetherian Thecategory Cot(X)capGFlat(X) is Frobenius and the projectivendashinjective objects are theflat-cotorsion sheaves Its associated stable category is equivalent to the homotopycategory of the Gorenstein flat model structure

Proof Applied to the Gorenstein flat model structure from the theorem [15 Propo-sition 52(4)] shows that Cot(X)capGFlat(X) is a Frobenius category with the statedprojectivendashinjective objects The last assertion follows from [15 Corollary 54]

3 Acyclic complexes of cotorsion sheaves

We assume throughout this section that X is semi-separated quasi-compact Thecategory of cochain complexes of sheaves on X is denoted C(Qcoh(X)) The goalis to establish a result Theorem 33 below which in the affine case is proved byBazzoni Cortes Izurdiaga and Estrada [1 Theorem 13] It says in part that everyacyclic complex of cotorsion sheaves has cotorsion cycles Our proof is inspired byarguments of Hosseini [20] and S rsquotovıcek [27]

Let CZac(Flat(X)) denote the full subcategory of C(Qcoh(X)) whose objects are

the acyclic complexes F of flat sheaves with Zn(F ) isin Flat(X) for every n isin Zsimilarly let CZ

ac(Cot(X)) denote the full subcategory whose objects are the acycliccomplexes C of cotorsion sheaves with Zn(C ) isin Cot(X) for every n isin Z FurtherCsemi(Cot(X)) denotes the category of complexes C of cotorsion sheaves with theproperty that the total Hom complex Hom(F C ) of abelian groups is acyclic forevery complex F isin CZ

ac(Flat(X)) In the literature such complexes are referred toas dg- or semi-cotorsion complexes it is part of Theorem 33 that every complex ofcotorsion sheaves on X has this property

Remark 31 The pair (CZac(Flat(X))Csemi(Cot(X))) is by [14 Theorem 67] and

[21 Theorem 22] a complete cotorsion pair in C(Qcoh(X))

For complexes A and B of sheaves on X let Hom(A B) denote the standardtotal Hom complex of abelian groups There is an isomorphism

(7) Ext1C(Qcoh(X))dw(A Σnminus1B) sim= Hn Hom(A B)

where Ext1C(Qcoh(X))dw(A Σnminus1B) is the subgroup of Ext1

C(Qcoh(X))(A Σnminus1B)

consisting of degreewise split short exact sequences see eg [13 Lemma 21] For


a complex F of flat sheaves and a complex C of cotorsion sheaves every extension0rarr C rarrX rarr F rarr 0 is degreewise split so (7) reads

(8) Ext1C(Qcoh(X))(F Σnminus1C ) sim= Hn Hom(F C )

Lemma 32 Let (Fλ)λisinΛ be a direct system of complexes in CZac(Flat(X)) If each

complex Fλ is contractible then

Ext1C(Qcoh(X))(limminusrarr

λisinΛ

FλC ) sim= H1 Hom(limminusrarrλisinΛ

FλC ) = 0

holds for every complex C of cotorsion sheaves on X

Proof The category Qcoh(X) is locally finitely presentable and therefore finitelyaccessible see Remark 21 It follows that the results and arguments in [27] applyThe argument in the proof of [27 Proposition 53] yields an exact sequence

(9) 0 minusrarr K minusrarroplusλisinΛ

Fλ minusrarr limminusrarrλisinΛ

Fλ minusrarr 0

where K is filtered by finite direct sums of complexes Fλ That is there is anordinal number β and a filtration (Kα | α lt β) where K0 = 0 Kβ = K andKα+1Kα

sim=oplus

λisinJα Fλ for α lt β and Jα a finite set

Let C be a complex of cotorsion sheaves on X As CZac(Flat(X)) is closed under

direct limits one has limminusrarrFλ isin CZac(Flat(X)) Thus Ext1

Qcoh(X)((limminusrarrFλ)iC j) = 0holds for all i j isin Z whence there is an exact sequence of complexes of abeliangroups

(10) 0 minusrarr Hom(limminusrarrλisinΛ

FλC ) minusrarr Hom(oplusλisinΛ

FλC ) minusrarr Hom(K C ) minusrarr 0

By (8) it now suffices to show that the left-hand complex in this sequence is acyclicThe middle complex is acyclic because each complex Fλ and therefore the directsum

oplusFλ is contractible Thus it is enough to prove that Hom(K C ) is acyclic

Since Flat(X) is resolving it follows from (9) that K is a complex of flat sheavesAs C is a complex of cotorsion sheaves (8) yields

Hn Hom(K C ) sim= Ext1C(Qcoh(X))(K Σnminus1C )

Hence it suffices to show that Ext1C(Qcoh(X))(K Σnminus1C ) = 0 holds for all n isin Z

Let (Kα | α le λ) be the filtration of K described above For every n isin Z one has

Ext1C(Qcoh(X))(

oplusλisinJα

FλΣnminus1C ) = 0

so Eklofrsquos lemma [27 Proposition 210] yields Ext1C(Qcoh(X))(K Σnminus1C ) = 0

Theorem 33 Assume that X is semi-separated quasi-compact Every complexof cotorsion sheaves on X belongs to Csemi(Cot(X)) and every acyclic complex ofcotorsion sheaves belongs to CZ

ac(Cot(X))

Proof As (CZac(Flat(X))Csemi(Cot(X))) is a cotorsion pair see Remark 31 the

first assertion is that for every complex M of cotorsion sheaves and every F inCZ

ac(Flat(X)) one has Ext1C(Qcoh(X))(F M ) = 0 Fix F isin CZ

ac(Flat(X)) and a

semi-separating open affine covering U = U0 Ud of X Consider the doublecomplex of sheaves obtained by taking the Cech resolutions of each term in F


see Murfet [22 Section 31] The rows of the double complex form a sequence inC(Qcoh(X))

(11) 0 minusrarr F minusrarr C 0(U F ) minusrarr C 1(U F ) minusrarr middot middot middot minusrarr C d(U F ) minusrarr 0

with

C p(U F ) =oplus

j0ltmiddotmiddotmiddotltjp

ilowast( ˜F (Uj0jp))

where j0 jp belong to the set 0 d and i Uj0jp minusrarr X is the inclusionof the open affine subset Uj0jp = Uj0 cap cap Ujp of X For a tuple of indicesj0 lt middot middot middot lt jp the complex F (Uj0jp) is an acyclic complex of flat OX(Uj0jp)-modules whose cycle modules are also flat It follows that the complex F (Uj0jp)is a direct limit

F (Uj0jp) = limminusrarrλisinΛ

PUj0jpλ

of contractible complexes of projective hence flat OX(Uj0jp)-modules see forexample Neeman [24 Theorem 86] The functor ilowast preserves split exact sequences

so ilowast(˜

PUj0jpλ ) is for every λ isin Λ a contractible complex of flat sheaves The

functor also preserves direct limits so C p(U F ) is a finite direct sum of directlimits of contractible complexes in CZ

ac(Flat(X)) hence C p(U F ) is itself a directlimit of contractible complexes in CZ

ac(Flat(X)) For every complex M of cotorsionsheaves and every n isin Z Lemma 32 now yields

Hn Hom(C p(U F )M ) sim= H1 Hom(C p(U F )Σnminus1M ) = 0 for 0 le p le d

That is the complex Hom(C p(U F )M ) is acyclic for every 0 le p le d and everycomplex M of cotorsion sheaves Applying Hom(minusM ) to the exact sequence

0 minusrarr Ldminus1 minusrarr C dminus1(U F ) minusrarr C d(U F ) minusrarr 0

one gets an exact sequence of complexes of abelian groups

0 minusrarr Hom(C d(U F )M ) minusrarr Hom(C dminus1(U F )M ) minusrarr Hom(Ldminus1M ) minusrarr 0

The first two terms are acyclic and hence so is Hom(Ldminus1M ) Repeating thisargument dminus 1 more times one concludes that Hom(F M ) is acyclic whence onehas Ext1

C(Qcoh(X))(F M ) = 0 per (8)

The second assertion now follows from [14 Corollary 39] which applies as(Flat(X)Cot(X)) is a complete hereditary cotorsion pair and Flat(X) contains agenerator for Qcoh(X) see Remark 21

4 The stable category of Gorenstein flat-cotorsion sheaves

In this last section we give a description of the stable category associated to thecotorsion pair of Gorenstein flat sheaves described in Theorem 22 In particularwe prove Theorems A and B from the introduction

Here we use the symbol hom to denote the morphism sets in Qcoh(X) as wellas the induced functor to abelian groups Further the tensor product on Qcoh(X)has a right adjoint functor denoted H omqc see for example [23 21]

We recall from [5 Definition 11 Proposition 13 and Definition 21]


Definition 41 An acyclic complex F of flat-cotorsion sheaves on X is calledtotally acyclic if the complexes hom(C F ) and hom(F C ) are acyclic for everyflat-cotorsion sheaf C on X

A sheaf M on X is called Gorenstein flat-cotorsion if there exists a totally acycliccomplex F of flat-cotorsion sheaves on X with M = Z0(F ) Denote by GFC(X)the category of Gorenstein flat-cotorsion sheaves on X1

We proceed to show that the sheaves defined in 41 are precisely the cotorsionGorenstein flat sheaves ie the sheaves that are both cotorsion and Gorenstein flat

The next result is analogous to [5 Theorem 44]

Proposition 42 Assume that X is semi-separated noetherian An acyclic com-plex of flat-cotorsion sheaves on X is totally acyclic if and only if it is F-totallyacyclic

Proof Let F be a totally acyclic complex of flat-cotorsion sheaves Let I be aninjective sheaf and E an injective cogenerator in Qcoh(X) By [23 Lemma 32]the sheaf H omqc(I E ) is flat-cotorsion The adjunction isomorphism

hom(I otimesF E ) sim= hom(F H omqc(I E ))(12)

along with faithful injectivity of E implies that I otimes F is acyclic hence F isF-totally acyclic

For the converse let F be an F-totally acyclic complex of flat-cotorsion sheavesand C be a flat-cotorsion sheaf Recall from [23 Proposition 33] that C is a directsummand of H omqc(I E ) for some injective sheaf I and injective cogenerator E Thus (12) shows that hom(F C ) is acyclic Moreover it follows from Theorem 33that Zn(F ) is cotorsion for every n isin Z so hom(C F ) is acyclic

Theorem 43 Assume that X is semi-separated noetherian A sheaf on X isGorenstein flat-cotorsion if and only if it is cotorsion and Gorenstein flat that is

GFC(X) = Cot(X) cap GFlat(X)

Proof The containment ldquosuberdquo is immediate by Theorem 33 and Proposition 42For the reverse containment let M be a cotorsion Gorenstein flat sheaf on XThere exists an F-totally acyclic complex F of flat sheaves with M = Z0(F ) As(CZ

ac(Flat(X))Csemi(Cot(X))) is a complete cotorsion pair see Remark 31 thereis an exact sequence in C(Qcoh(X))

0 minusrarr F minusrarr T minusrarrP minusrarr 0

with T isin Csemi(Cot(X)) and P isin CZac(Flat(X)) As F and P are F-totally acyclic

so is T in particular Zn(T ) is cotorsion for every n isin Z see Theorem 33 Theargument in [5 Theorem 52] now applies verbatim to finish the proof

Remark 44 One upshot of Theorem 43 is that the Frobenius category describedin Corollary 26 coincides with the one associated to GFC(X) per [5 Theorem 211]In particular the associated stable categories are equal One of these is equivalentto the homotopy category of the Gorenstein flat model structure and the other isby [5 Corollary 39] and Proposition 42 equivalent to the homotopy category

KF-tac(Flat(X) cap Cot(X))

of F-totally acyclic complexes of flat-cotorsion sheaves on X

1In [5] this category is denoted GorFlatCot(A) in the case of an affine scheme X = SpecA


In [23 25] the pure derived category of flat sheaves on X is the Verdier quotient

D(Flat(X)) =K(Flat(X))

Kpac(Flat(X))

where Kpac(Flat(X)) is the full subcategory of K(Flat(X)) of pure acyclic complexesthat is the objects in Kpac(Flat(X)) are precisely the objects in CZ

ac(Flat(X)) Stillfollowing [23] we denote by DF-tac(Flat(X)) the full subcategory of D(Flat(X))whose objects are F-totally acyclic As the category Kpac(Flat(X)) is containedin KF-tac(Flat(X)) it can be expressed as the Verdier quotient


Kpac(Flat(X))

Theorem 45 Assume that X is semi-separated noetherian The composite ofcanonical functors

KF-tac(Flat(X) cap Cot(X)) minusrarr KF-tac(Flat(X)) minusrarr DF-tac(Flat(X))

is a triangulated equivalence of categories

Proof In view of Theorem 33 and the fact that (CZac(Flat(X))Csemi(Cot(X))) is

a complete cotorsion pair see Remark 31 the proof of [5 Theorem 56] appliesmutatis mutandis

We denote by StGFC(X) the stable category of Gorenstein flat-cotorsion sheavescf Remark 44 Let A be a commutative noetherian ring of finite Krull dimensionFor the affine scheme X = SpecA this category is by [5 Corollary 59] equivalent tothe stable category StGPrj(A) of Gorenstein projective A-modules This togetherwith the next result suggests that StGFC(X) is a natural non-affine analogue ofStGPrj(A) Indeed the category DF-tac(Flat(X)) is Murfet and Salarianrsquos non-affine analogue of the homotopy category of totally acyclic complexes of projectivemodules see [23 Lemma 422]

Corollary 46 There is a triangulated equivalence of categories

StGFC(X) DF-tac(Flat(X))

Proof Combine the equivalence StGFC(X) KF-tac(Flat(X) cap Cot(X)) from Re-mark 44 with Theorem 45

We emphasize that Proposition 42 and Theorem 45 offer another equivalent ofthe category StGFC(X) namely the homotopy category of totally acyclic complexesof flat-cotorsion sheaves

A noetherian scheme X is called Gorenstein if the local ring OXx is Gorensteinfor every x isin X We close this paper with a characterization of Gorenstein schemesin terms of flat-cotorsion sheaves it sharpens [23 Theorem 427] In a paper inprogress [3] we show that Gorensteinness of a scheme X can be characterized by theequivalence of the category StGFC(X) to a naturally defined singularity category

Theorem 47 Assume that X is semi-separated noetherian The following condi-tions are equivalent

(i) X is Gorenstein(ii) Every acyclic complex of flat sheaves on X is F-totally acyclic

(iii) Every acyclic complex of flat-cotorsion sheaves on X is F-totally acyclic(iv) Every acyclic complex of flat-cotorsion sheaves on X is totally acyclic


Proof The equivalence of conditions (i) and (ii) is [23 Theorem 427] and con-ditions (iii) and (iv) are equivalent by Proposition 42 As (ii) evidently implies(iii) it suffices to argue the converse

Assume that every acyclic complex of flat-cotorsion sheaves is F-totally acyclicLet F be an acyclic complex of flat sheaves As (CZ

ac(Flat(X))Csemi(Cot(X))) is acomplete cotorsion pair see Remark 31 there is an exact sequence in C(Qcoh(X))

0 minusrarr F minusrarr C minusrarrP minusrarr 0

with C isin Csemi(Cot(X)) and P isin CZac(Flat(X)) Since F and P are acyclic

the complex C is also acyclic Moreover C is a complex of flat-cotorsion sheavesBy assumption C is F-totally acyclic and so is P whence it follows that F isF-totally acyclic

Acknowledgment

We thank Alexander Slavik for helping us correct a mistake in an earlier version ofthe proof of Theorem 16 We also acknowledge the anonymous refereersquos promptsto improve the presentation

References

[1] Silvana Bazzoni Manuel Cortes Izurdiaga and Sergio Estrada Periodic modules and acyclic

complexes Algebr Represent Theory online 2019

[2] Ragnar-Olaf Buchweitz Maximal CohenndashMacaulay modules and Tate-cohomology over Gorenstein rings University of Hannover 1986 available at

httphdlhandlenet180716682

[3] Lars Winther Christensen Nanqing Ding Sergio Estrada Jiangsheng Hu Huanhuan Li andPeder Thompson Singularity categories of exact categories with applications to the flatndash

cotorsion theory in preparation[4] Lars Winther Christensen Sergio Estrada and Alina Iacob A Zariski-local notion of F-total

acyclicity for complexes of sheaves Quaest Math 40 (2017) no 2 197ndash214 MR3630500

[5] Lars Winther Christensen Sergio Estrada and Peder Thompson Homotopy categories oftotally acyclic complexes with applications to the flatndashcotorsion theory Categorical Homo-

logical and Combinatorial Methods in Algebra Contemp Math vol 751 Amer Math Soc

Providence RI 2020 pp 99ndash118[6] Lars Winther Christensen Fatih Koksal and Li Liang Gorenstein dimensions of unbounded

complexes and change of base (with an appendix by Driss Bennis) Sci China Math 60

(2017) no 3 401ndash420 MR3600932[7] Alexander I Efimov and Leonid Positselski Coherent analogues of matrix factorizations

and relative singularity categories Algebra Number Theory 9 (2015) no 5 1159ndash1292

MR3366002[8] Edgar Enochs and Sergio Estrada Relative homological algebra in the category of quasi-

coherent sheaves Adv Math 194 (2005) no 2 284ndash295 MR2139915[9] Edgar E Enochs and Overtoun M G Jenda Relative homological algebra de Gruyter Ex-

positions in Mathematics vol 30 Walter de Gruyter amp Co Berlin 2000 MR1753146

[10] Edgar E Enochs Overtoun M G Jenda and Juan A Lopez-Ramos The existence of Goren-stein flat covers Math Scand 94 (2004) no 1 46ndash62 MR2032335

[11] Sergio Estrada Pedro A Guil Asensio Mike Prest and Jan Trlifaj Model category structuresarising from Drinfeld vector bundles Adv Math 231 (2012) no 3-4 1417ndash1438 MR2964610

[12] Sergio Estrada Alina Iacob and Marco Perez Model structures and relative Gorenstein flatmodules Categorical Homological and Combinatorial Methods in Algebra Contemp Math

vol 751 Amer Math Soc Providence RI 2020 pp 135ndash176[13] James Gillespie The flat model structure on Ch(R) Trans Amer Math Soc 356 (2004)

no 8 3369ndash3390 MR2052954

[14] James Gillespie Kaplansky classes and derived categories Math Z 257 (2007) no 4 811ndash843 MR2342555


[15] James Gillespie Model structures on exact categories J Pure Appl Algebra 215 (2011)

no 12 2892ndash2902 MR2811572

[16] James Gillespie How to construct a Hovey triple from two cotorsion pairs Fund Math 230(2015) no 3 281ndash289 MR3351474

[17] James Gillespie The flat stable module category of a coherent ring J Pure Appl Algebra

221 (2017) no 8 2025ndash2031 MR3623182[18] A Grothendieck and J A Dieudonne Elements de geometrie algebrique I Grundlehren

der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] vol

166 Springer-Verlag Berlin 1971 MR3075000[19] Robin Hartshorne Residues and duality Lecture notes of a seminar on the work of A

Grothendieck given at Harvard 196364 With an appendix by P Deligne Lecture Notes

in Mathematics vol 20 Springer-Verlag Berlin 1966 MR0222093[20] Esmaeil Hosseini The pure derived category of quasi-coherent sheaves Comm Algebra 47

(2019) no 9 3781ndash3788 MR3969478[21] Mark Hovey Cotorsion pairs model category structures and representation theory Math

Z 241 (2002) no 3 553ndash592 MR1938704

[22] Daniel Murfet The mock homotopy category of projectives and Grothendieck duality PhDthesis Australian National University 2007 x+145 pp Available from wwwtherisingseaorg

[23] Daniel Murfet and Shokrollah Salarian Totally acyclic complexes over Noetherian schemes

Adv Math 226 (2011) no 2 1096ndash1133 MR2737778[24] Amnon Neeman The homotopy category of flat modules and Grothendieck duality Invent

Math 174 (2008) no 2 255ndash308 MR2439608

[25] Manuel Saorın and Jan Stovıcek On exact categories and applications to triangulated adjoints

and model structures Adv Math 228 (2011) no 2 968ndash1007 MR2822215

[26] Alexander Slavik and Jan S rsquotovıcek On the existence of flat generators of quasicoherentsheaves preprint arXiv190205740 [mathAG] 8 pp

[27] Jan S rsquotovıcek On purity and applications to coderived and singularity categories preprintarXiv14121615 [mathCT] 45 pp

[28] Jan Saroch and Jan Stovıcek Singular compactness and definability for Σ-cotorsion andGorenstein modules Selecta Math (NS) 26 (2020) no 2 Paper No 23 40 MR4076700

[29] Gang Yang and Zhongkui Liu Stability of Gorenstein flat categories Glasg Math J 54

(2012) no 1 177ndash191 MR2862396

LWC Texas Tech University Lubbock TX 79409 USAEmail address larswchristensenttuedu

URL httpwwwmathttuedu~lchriste

SE Universidad de Murcia Murcia 30100 Spain

Email address sestradaumes

URL httpswebsumessestrada

PT Norwegian University of Science and Technology 7491 Trondheim NorwayEmail address pederthompsonntnuno

URL httpsfolkntnunopedertho

Introduction



Abelian model category structures from cotorsion pairs



Acknowledgment

References


The stable category of Gorenstein projective modules is a standard constructionthat applies to any Frobenius category The category of Gorenstein flat modules israrely Frobenius it is essentially only Frobenius if it coincides with the category ofGorenstein projective modules see [5 Theorem 45] The cotorsion Gorenstein flatmodules however do form a Frobenius category and a special case of [5 Corol-lary 59] says that for a commutative noetherian ring A of finite Krull dimensionthe category StGPrj(A) is equivalent to the stable category StGFC(A) of cotorsionGorenstein flat modules This identifies a candidate category and indeed the goalstated above is obtained (in 46) with

Theorem A Let X be a semi-separated noetherian scheme The stable categoryStGFC(X) of cotorsion Gorenstein flat sheaves is equivalent to DF-tac(Flat(X))

In the statement of this theorem and everywhere else in this paper a sheaf meansa quasi-coherent sheaf A sheaf on X is called cotorsion if it is right Ext-orthogonalto flat sheaves on X

A crucial step towards the equivalence in Theorem A is to prove (in 42 and43) that the sheaves that are both cotorsion and Gorenstein flat are precisely thesheaves that arise as cycles in F-totally acyclic complexes of flat-cotorsion sheavesThis is exactly what happens in the affine case and it transpires that the maintake-away from [5] also applies in the non-affine setting One should work withsheaves that are both cotorsion and Gorenstein flat rather than all Gorenstein flatsheaves One manifestation is a result (47) that sharpens [23 Theorem 427]

Theorem B A semi-separated noetherian scheme X is Gorenstein if and only ifevery acyclic complex of flat-cotorsion sheaves on X is F-totally acyclic

A second manifestationmdashactually the result behind Theorem Amdashis that thecategory DF-tac(Flat(X)) considered by Murfet and Salarian is equivalent to thehomotopy category KF-tac(Flat(X) cap Cot(X)) of F-totally acyclic complexes of flatcotorsion sheaves (see 45) That is passing from the affine to the non-affine settingone can replace the homotopy category Ktac(Prj(A)) by another homotopy category

Up to equivalence the category KF-tac(Flat(X) cap Cot(X)) arises in a relatedyet different context The category of Gorenstein flat sheaves on a semi-separatednoetherian scheme X is part of a complete hereditary cotorsion pair (see 22) andone that is comparable to the cotorsion pair of flat sheaves and cotorsion sheaveson X Through work of Hovey [21] and Gillespie [16] these cotorsion pairs inducea model structure on the category of sheaves on X We prove (see 44) that theassociated homotopy category is equivalent to KF-tac(Flat(X) cap Cot(X))


In this paper the symbol X denotes a scheme with structure sheaf OX By asheaf on X we shall always mean a quasi-coherent sheaf and Qcoh(X) denotes thecategory of (quasi-coherent) sheaves on X We frequently add the assumption thatX is semi-separated by which we mean that X has an open affine covering U suchthat UcapV is affine for all U V isin U such a covering is referred to as semi-separatingWe use standard cohomological notation for cochain complexes

In this first section we show that over a semi-separated noetherian scheme onecan equivalently define Gorenstein flatness of sheaves globally locally or stalkwiseLet A be a commutative ring An acyclic complex F of flat A-modules is called































xisinXy

(ix)lowast(Mx)



(ix)lowast(Fx)





F 0 =prodxisinX

(ix)lowast(Fx) =

noplusk=0

prodxisinIk




Fx) sim=noplusk=0


Fx)





0 My F 0

y

π

K 1y

$

0

0 My Fy Ty 0





I sim=oplusxisinX

(ix)lowastJ(x)(Λx)












































































λisinΛ


FλC ) = 0





Fλ minusrarr 0


sim=oplus














Ext1C(Qcoh(X))(

oplusλisinJα

FλΣnminus1C ) = 0



ac(Cot(X))




ac(Flat(X)) and a





with

C p(U F ) =oplus





PUj0jpλ


so ilowast(˜











C(Qcoh(X))(F M ) = 0 per (8)






























Kpac(Flat(X))




Kpac(Flat(X))






















Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References































xisinXy

(ix)lowast(Mx)



(ix)lowast(Fx)





F 0 =prodxisinX

(ix)lowast(Fx) =

noplusk=0

prodxisinIk




Fx) sim=noplusk=0


Fx)





0 My F 0

y

π

K 1y

$

0

0 My Fy Ty 0





I sim=oplusxisinX

(ix)lowastJ(x)(Λx)












































































λisinΛ


FλC ) = 0





Fλ minusrarr 0


sim=oplus














Ext1C(Qcoh(X))(

oplusλisinJα

FλΣnminus1C ) = 0



ac(Cot(X))




ac(Flat(X)) and a





with

C p(U F ) =oplus





PUj0jpλ


so ilowast(˜











C(Qcoh(X))(F M ) = 0 per (8)






























Kpac(Flat(X))




Kpac(Flat(X))






















Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References



0 My F 0

y

π

K 1y

$

0

0 My Fy Ty 0





I sim=oplusxisinX

(ix)lowastJ(x)(Λx)












































































λisinΛ


FλC ) = 0





Fλ minusrarr 0


sim=oplus














Ext1C(Qcoh(X))(

oplusλisinJα

FλΣnminus1C ) = 0



ac(Cot(X))




ac(Flat(X)) and a





with

C p(U F ) =oplus





PUj0jpλ


so ilowast(˜











C(Qcoh(X))(F M ) = 0 per (8)






























Kpac(Flat(X))




Kpac(Flat(X))






















Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References






























































λisinΛ


FλC ) = 0





Fλ minusrarr 0


sim=oplus














Ext1C(Qcoh(X))(

oplusλisinJα

FλΣnminus1C ) = 0



ac(Cot(X))




ac(Flat(X)) and a





with

C p(U F ) =oplus





PUj0jpλ


so ilowast(˜











C(Qcoh(X))(F M ) = 0 per (8)






























Kpac(Flat(X))




Kpac(Flat(X))






















Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References




with

C p(U F ) =oplus





PUj0jpλ


so ilowast(˜











C(Qcoh(X))(F M ) = 0 per (8)






























Kpac(Flat(X))




Kpac(Flat(X))






















Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References

























Kpac(Flat(X))




Kpac(Flat(X))






















Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References








Acknowledgment


References





















no 8 3369ndash3390 MR2052954




no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References



no 12 2892ndash2902 MR2811572









Z 241 (2002) no 3 553ndash592 MR1938704




Math 174 (2008) no 2 255ndash308 MR2439608







(2012) no 1 177ndash191 MR2862396








Introduction






Acknowledgment

References

Introduction - math.ttu.edulchriste/download/gflatsheaf.pdf · AF is acyclic for every injective...

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Transcript of Introduction - math.ttu.edulchriste/download/gflatsheaf.pdf · AF is acyclic for every injective...