Quillen equivalent models for the derived category of flats and the resolution property
aa r X i v : . [ m a t h . A T ] M a r QUILLEN EQUIVALENT MODELS FOR THE DERIVEDCATEGORY OF FLATS AND THE RESOLUTION PROPERTY
SERGIO ESTRADA AND ALEXANDER SL ´AVIK
Abstract.
We investigate under which assumptions a subclass of flat quasi-coherent sheaves on a quasi-compact and semi-separated scheme allows to“mock” the homotopy category of projective modules. Our methods are basedon module theoretic properties of the subclass of flat modules involved as wellas their behaviour with respect to Zariski localizations. As a consequence weget that, for such schemes, the derived category of flats is equivalent to thederived category of very flats. If, in addition, the scheme satisfies the resolutionproperty then both derived categories are equivalent to the derived categoryof infinite-dimensional vector bundles. The equivalences are inferred from aQuillen equivalence between the corresponding models. Introduction
Throughout the paper R will denote a commutative ring. In [20] Neeman gives anew description of the homotopy category K (Proj( R )) as a quotient of K (Flat( R )).The main advantage of the new description is that it does not involve projectiveobjects, so it can be generalized to non-affine schemes (see [19, Remark 3.4]). So, inhis thesis [16], Murfet mocks the homotopy category of projectives on a non-affinescheme, by considering the category D (Flat( X )) defined as the Verdier quotient D (Flat( X )) := K (Flat( X )) g Flat K ( X ) , where g Flat K ( X ) denotes the class of acyclic complexes in K (Flat( X )) with flat cy-cles. In the language of model categories, Gillespie showed in [10] that D (Flat( X ))can be realized as the homotopy category of a Quillen model structure on the cat-egory Ch( Qcoh ( X )) of unbounded chain complexes of quasi-coherent sheaves on aquasi compact and semi-separated scheme, and that, in fact, in case X = Spec( R )is affine, both homotopy categories D (Flat( X )) and K (Proj( R )) are triangle equiv-alent, coming from a Quillen equivalence between the corresponding models.However, from an homological point of view, flat modules are much more com-plicated than projective modules. For instance, for a general commutative ring,the exact category of flat modules has infinite homological dimension. In order topartially remedy these complications, recently Positselski in [21] has introduced arefinement of the class of flat quasi-coherent sheaves, the so-called very flat quasi-coherent sheaves (see Section 3 for the definition and main properties) and showedthat this class shares many nice properties with the class of flat sheaves, but it has Mathematics Subject Classification.
Key words and phrases.
Resolution property, very flat sheaf, model category, Quillen equiva-lence, homotopy category, restricted flat Mittag-Leffler.The first author is supported by the grant MTM2016-77445-P and FEDER funds and thegrant 19880/GERM/15 by the Fundaci´on S´eneca-Agencia de Ciencia y Tecnolog´ıa de la Regi´onde Murcia. The original terminology in [16] for D (Flat( X )) was K m (Proj( X )). This is referred in [18]as the pure derived category of flat sheaves on X and denoted by D (Flat( X )). potentially several advantages with respect to it, for instance, it can be applied tomatrix factorizations (see the introduction of the recent preprint [23] for a nice anddetailed treatment on the goodness of the very flat sheaves).Moreover, in the affine case X = Spec( R ), the exact category of very flat moduleshas finite homological dimension (every very flat module has projective dimension ≤ D ( VF ( R )) and K (Proj( R )) (here VF ( R ) denotes the class of very flat R -modules).In particular it is much less involved than the aforementioned triangulated equiva-lence between D (Flat( R )) and K (Proj( R )) ([20, Theorem 1.2]).So, if we denote by VF ( X ) the class of very flat quasi-coherent sheaves, onecan also think in “mocking” the homotopy category of projectives over a non-affinescheme by defining the Verdier quotient D ( VF ( X )) := K ( VF ( X )) f VF K ( X ) . It is then natural to wonder whether or not the (indirect) triangulated equivalencebetween D (Flat( R )) and D ( VF ( R )) still holds over a non-affine scheme. This wasalready proved to be the case for a semi-separated Noetherian scheme of finiteKrull dimension in [21, Corollary 5.4.3]. As a first consequence of the results inthis paper, we extend in Corollary 6.1 this result for arbitrary (quasi-compact andsemi-separated) schemes. Corollary 1.
For any scheme X , the categories D (Flat( X )) and D ( VF ( X )) aretriangle equivalent.Recall from Totaro [28] (see Gross [12] for the general notion) that a scheme X satisfies the resolution property provided that X has enough locally frees, that is, forevery quasi-coherent sheaf M there exists an exact map ⊕ i V i → M →
0, for somefamily { V i : i ∈ I } of vector bundles. In this case the class of infinite-dimensionalvector bundles (in the sense of Drinfeld [4]) constitutes the natural extension of theclass of projective modules for non-affine schemes. And one can define the derivedcategory of infinite-dimensional vector bundles again as the Verdier quotient D (Vect( X )) := K (Vect( X )) g Vect K ( X ) . This definition trivially agrees with K (Proj( R )) in case X = Spec( R ) is affine.By using the class of very flat sheaves we obtain in Corollary 6.2 the followingmeaningful consequence, which does not seem clearly to admit a direct proof (i.e.a proof without using very flat sheaves). Corollary 2.
Let X be a quasi-compact and semi-separated scheme satisfying theresolution property (for instance if X is divisorial [17, Proposition 6(a)]). Mur-fet’s and Neeman’s derived category of flats, D (Flat( X )), is triangle equivalent to D (Vect( X )), the derived category of infinite-dimensional vector bundles.Indeed the methods developed in this paper go beyond the class of very flatquasi-coherent sheaves. More precisely, we investigate which are the conditionsthat a subclass A qc of flat quasi-coherent sheaves has to fulfil in order to get atriangle equivalent category to D (Flat( X )). In fact, we show that the triangulatedequivalence comes from a Quillen equivalence between the corresponding models.We point out that there are well-known examples of non Quillen equivalent modelswith equivalent homotopy categories. The precise statement of our main result isin Theorem 6.1 (see the setup in Section 6 for unexplained terminology). UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 3
Theorem.
Let X be a quasi-compact and semi-separated scheme and let P be aproperty of modules and A its associated class of modules. Assume that A ⊆
Flat,and that the following conditions hold:(1) The class A is Zariski-local.(2) For each R = O X ( U ), U ∈ U , the pair ( A R , B R ) is a hereditary cotorsionpair generated by a set.(3) For each R = O X ( U ), U ∈ U , every flat A R -periodic module is trivial.(4) j ∗ ( A qc( U α ) ) ⊆ A qc( X ) , for each α ⊆ { , . . . , m } .Then the class A qc defines an abelian model category structure in Ch( Qcoh ( X ))whose homotopy category D ( A qc ) is triangle equivalent to D (Flat( X )), induced bya Quillen equivalence between the corresponding model categories.It is interesting to observe that conditions (1), (2) and (3) in the previous theoremonly involve properties of modules. Thus we find useful and of independent interestto explicitly state in Theorem 5.1 the affine version of the previous theorem (andgive an easy proof). Section 4 is meant to make abundantly clear the variety ofexamples of classes of modules that fit into those conditions. Of particular interestis the class A ( κ ) of restricted flat Mittag-Leffler modules considered in Theorem 4.5which has been widely studied in the literature in the recent years (see, for instance,[6, 7, 11, 14, 25]). So regarding this class, we obtain the following meaningfulconsequences: Corollary.
Let κ be an infinite cardinal and A ( κ ) be the class of κ -restricted flatMittag-Leffler modules (notice that A ( κ ) = Proj( R ) in case κ = ℵ ).(1) Every pure acyclic complex with components in A ( κ ) has cycles in A ( κ ).(2) The categories D ( A ( κ )) and K (Proj( R )) are triangle equivalent.The proof of (1) can be found in Theorem 4.5 whereas the proof of (2) is aparticular instance of Theorem 5.1 with A = A ( κ ). In the special case κ = ℵ ,the statement (1) recovers a well-known result due to Benson and Goodearl ([2,Theorem 1.1]). 2. Preliminaries
Zariski local classes of modules. Let P be a property of modules and let A be the corresponding class of modules satisfying P , i.e. for any ring R , the class A R consists of M ∈ R -Mod such that M satisfies P R . We define the class A qc( X ) in Qcoh ( X ) (or just A qc if the scheme is understood) as the class of all quasi-coherent sheaves M such that, for each open affine U ⊆ X , the module of sections M ( U ) ∈ A O X ( U ) . We will be only interested in those properties of modules P such that the property of being in A qc( X ) can be tested on an open affine coveringof X . In this case we will say that the class A of modules (associated to P ) is Zariski-local .The following is a specialization of the ascent-descent conditions ([7, Definition3.4]) that suffices to prove Zariski locality (see Vakil [29, Lemma 5.3.2] and also[15, § Lemma 2.1.
The class of modules A associated to the property of modules P isZariski-local if and only if satisfies the following: (1) If an R -module M ∈ A R , then M f ∈ A R f for all f ∈ R . (2) If ( f , . . . , f n ) = R , and M f i = R f i ⊗ R M ∈ A R fi , for all i ∈ { , . . . , n } ,then M ∈ A R . It is easy to see that the class Flat of flat modules is Zariski-local. A module M is Mittag-Leffler provided that the canonical map M ⊗ R Q i ∈ I M i → Q i ∈ I M ⊗ R M i SERGIO ESTRADA AND ALEXANDER SL´AVIK is monic for each family of left R -modules ( M i | i ∈ I ). The classes FlatML (of flatMittag-Leffler modules) and Proj (of projective modules) are also Zariski-local by3.1.4.(3) and 2.5.2 in [24, Seconde partie, 2.5.2]. The class rFlatML of restricted flat Mittag-Leffler modules (in the sense of [7, Example 2.1(3)]) is also Zariski-localby [7, Theorem 4.2]. Precovers, envelopes and complete cotorsion pairs. Throughout this sectionthe symbol G will denote an abelian category. Let C be a class of objects in G . Amorphism C φ → M in G is called a C -precover if C is in C and Hom G ( C ′ , C ) → Hom G ( C ′ , M ) → C ′ ∈ C . If every object in G has a C -precover,then the class C is called precovering . The dual notions are preenvelope and preen-veloping class.A pair ( A , B ) of classes of objects in G is a cotorsion pair if A ⊥ = B and A = ⊥ B ,where, given a class C of objects in A , the right orthogonal C ⊥ is defined to be theclass of all Y ∈ G such that Ext G ( C, Y ) = 0 for all C ∈ C . The left orthogonal ⊥ C is defined similarly. A cotorsion pair ( A , B ) is called hereditary if Ext i G ( A, B ) = 0for all A ∈ A , B ∈ B , and i >
1. A cotorsion pair ( A , B ) is complete if it has enough projectives and enough injectives , i.e. for each D ∈ G there exist short exactsequences 0 −→ B −→ A −→ D −→ −→ D −→ B ′ −→ A ′ −→ A, A ′ ∈ A and B, B ′ ∈ B . It is then easy to observe that A −→ D is an A -precover of D (such precovers are called special ). Analogously, D −→ B ′ is a special B -preenvelope of D . A cotorsion pair ( A , B ) is generated bya set provided that there exists a set S ⊆ A such that S ⊥ = B . In case G is, inaddition Grothendieck, it is known that a cotorsion pair generated by a set S whichcontains a generating set of G is automatically complete. Exact model categories and Hovey triples. In [13] Hovey relates complete co-torsion pairs with abelian (or exact) model category structures.An abelian model structure on G , that is, a model structure on G which is com-patible with the abelian structure in the sense of [13, Definition 2.1], correspondsby [13, Theorem 2.2] to a triple ( C , W , F ) of classes of objects in A for which W is thick and ( C ∩ W , F ) and ( C , W ∩ F ) are complete cotorsion pairs in G . Inthe model structure on G determined by such a triple, C is precisely the class ofcofibrant objects, F is precisely the class of fibrant objects, and W is precisely theclass of trivial objects (that is, objects weakly equivalent to zero). Such triple isoften referred as a Hovey triple .Gillespie extends in [9, Theorem 3.3] Hovey’s correspondance, mentioned above,from the realm of abelian categories to the realm of weakly idempotent completeexact categories ([9, Definition 2.2]). More precisely, if G is a weakly idempotentcomplete exact categories (not necessarily abelian), then an exact model structure on G (i.e. a model structure on G which is compatible with the exact structure inthe sense of [9, Definition 3.1]) corresponds to a Hovey triple ( C , W , F ) in G . Deconstructible classes. A well ordered direct system, ( M α : α ≤ λ ), ofobjects in G is called continuous if M = 0 and, for each limit ordinal β ≤ λ , wehave M β = lim −→ α<β M α . If all morphisms in the system are monomorphisms, thenthe system is called a continuous directed union .Let S be a class of objects in G . An object M in G is called S -filtered if there is acontinuous directed union ( M α : α ≤ λ ) of subobjects of M such that M = M λ andfor every α < λ the quotient M α +1 /M α is isomorphic to an object in S . We denote Recall that a class W in an abelian (or, more generally, in an exact) category G is thick if itis closed under direct summands and satisfies that whenever two out of three of the terms in ashort exact sequence are in W , then so is the third. UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 5 by Filt ( S ) the class of all S -filtered objects in G . A class C is called deconstructible provided that there exists a set S such that C = Filt ( S ) (see [27, Definition 1.4]). Itis then known by [27, Theorem pg.195] that any deconstructible class is precovering. Chain complexes of modules. We denote by Ch( G ) the category of unboundedchain complexes of objects in G , i.e. complexes G • of the form · · · → G n +1 d Gn +1 −−−→ G n d Gn −−→ G n − → · · · . We will denote by Z n G • the n -cycle of G , i.e. Z n G = Ker( d Gn ). Given a chaincomplex G the n th -suspension of G , Σ n G , is the complex defined as (Σ n G ) k = G k − n and d Σ n Gk = ( − n d k − n . And for a given object A ∈ G , the n -disk complex D n ( A )is the complex with the object A in the components n and n − d n as the identitymap, and 0 elsewhere.We denote by K ( G ) the homotopy category of G , i.e. K ( G ) has the same ob-jects as Ch( G ) and the morphisms are the homotopy classes of morphisms of chaincomplexes.In case G = R -Mod, we will denote Ch( G ) (resp. K ( G )) simply by Ch( R )(resp. K ( R )). Given a class C in G , we shall consider the following classes of chaincomplexes: • Ch( C ) (resp. K ( C )) is the full subcategory of Ch( G ) (resp. of K ( C )) of allcomplexes C • ∈ Ch( G ) such that C n ∈ C . • Ch ac ( C ) (resp. K ac ( C )) is the class of all acyclic complexes in Ch( C ) (resp.in K ( C )). • e C (resp. e C K ) is the class class of all complexes C • ∈ Ch ac ( C ) (resp. C • ∈ K ac ( C )) with the cycles Z n C • in C for all n ∈ Z . A complex in e C is calleda C complex . • If ( A , B ) is a cotorsion pair in G , then dg ( A ) is the class of all complexes A • ∈ Ch( A ) such that every morphism f : A • → B • , with B • a B complex,is null homotopic. Since Ext G ( A n , B n ) = 0 for every n ∈ Z , a standardformula allows to infer that dg ( A ) = ⊥ e B . Analogously, dg ( B ) is the classof all complexes B • ∈ Ch( B ) such that every morphism f : A • → B • , with A • an A complex, is null homotopic. Hence dg ( B ) = e A ⊥ .3. Very flat modules and sheaves
One of the main application of the results in this paper concerns the classes ofvery flat modules and very flat quasi-coherent sheaves, as defined by Positselski in[21]. In the present section we summarize all relevant definitions and propertiesregarding this class and that will be relevant in the sequel.
Very flat and contraadjusted modules. Let us consider the set S = { R [ r − ] : r ∈ R } and let ( VF ( R ) , CA ( R )) the complete cotorsion pair generated by S . Themodules in the class VF ( R ) are called very flat and the modules in the class CA ( R )are called contraadjusted . It is then clear that every projective module is veryflat, and that every very flat module is, in particular, flat. In fact it is easy toobserve that every very flat module has finite projective dimension ≤
1. Thus,the complete cotorsion pair ( VF , CA ) is automatically hereditary and CA is closedunder quotients. We finally notice that L is very flat in any short exact sequence0 → L → V → M → V is very flat and pd R ( M ) ≤ R ( M ) isthe projective dimension of M ). Proposition 3.1 (Positselski) . The class of very flat modules is Zariski-local.Proof.
Condition (1) of Lemma 2.1 holds by [21, Lemma 1.2.2(b)].Condition (2) of Lemma 2.1 follows from [21, Lemma 1.2.6(a)]. (cid:3)
SERGIO ESTRADA AND ALEXANDER SL´AVIK
Very flat and contraadjusted quasi-coherent sheaves. Let X be any scheme.A quasi-coherent sheaf M is very flat if there exists an open affine covering U of X such that M ( U ) is a very flat O X ( U )-module for each U ∈ U . By the previousproposition, the definition of very flat quasi-coherent sheaf is independent of thechoice of the open affine covering. A quasi-coherent sheaf N is contraadjusted if Ext n ( M , N ) = 0 for each very flat quasi-coherent sheaf M and every integer n ≥ Very flat generators in
Qcoh ( X ). Let X be a quasi-compact and semi-separatedscheme, with U = { U , · · · , U d } a semi-separated finite affine covering of X . Let U = U i ∩· · ·∩ U i p be any intersection of open sets in the cover U and let j : U ֒ → X be the inclusion of U in X . The inverse image functor j ∗ is just the restriction,so it is exact and preserves quasi-coherence. The direct image functor j ∗ is ex-act and preserves quasi-coherence because j : U ֒ → X is an affine morphism,due to the semi-separated assumption. Thus we have an adjunction ( j ∗ , j ∗ ) with j ∗ : Qcoh ( U ) → Qcoh ( X ) and j ∗ : Qcoh ( X ) → Qcoh ( U ).The proof of the next proposition is implicit in [1, Proposition 1.1] (see alsoMurfet [16, Proposition 3.29] for a very detailed treatment) by noticing that thedirect image functor j ∗ preserves not just flatness but in fact very flatness (by [21,Corollary 1.2.5(b)]). The reader can find a short and direct proof in [21, Lemma4.1.1]. Proposition 3.2.
Let X be a quasi-compact and semi-separated scheme. Everyquasi-coherent sheaf is a quotient of a very flat quasi-coherent sheaf. Therefore Qcoh ( X ) possesses a family of very flat generators. The very flat cotorsion pair in
Qcoh ( X ). For any scheme X , the class VF ( X )of very flat quasi-coherent sheaves is deconstructible (by [6, Corollary 3.14]). There-fore the class of very flat quasi-coherent sheaves is a precovering class (see 2.4). If,in addition, the scheme X is quasi-compact and semi-separated we infer from [6,Corollary 3.15] and [21, Corollary 4.1.2] that the pair ( VF ( X ) , CA ( X )) is a com-plete hereditary cotorsion pair in Qcoh ( X ) (where CA ( X ) denotes the class of allcontraadjusted quasi-coherent sheaves on X ).By [21, Lemma 1.2.2(d)] the class of very flat modules (and hence the class ofvery flat quasi-coherent sheaves) is closed under tensor products. Thus, in case X isquasi-compact and semi-separated, [6, Theorem 4.5] yields a cofibrantly generatedand monoidal model category structure in Ch( Qcoh ( X )) where the weak equiva-lences are the homology isomorphisms. The cofibrations (resp. trivial cofibrations)are monomorphisms whose cokernels are dg-very flat complexes (resp. very flatcomplexes). The fibrations (resp. trivial fibrations) are epimorphisms whose ker-nels are dg-contraadjusted complexes (resp. contraadjusted complexes). Thereforethe corresponding triple is( dg ( VF ( X )) , Ch ac ( Qcoh ( X )) , dg ( CA ( X ))) . The property of modules involved. Examples
As we will see in the next sections, we are mainly concerned in deconstructibleclasses of modules that are closed under certain periodic modules. We start byrecalling the notion of C -periodic module with respect to a class C of modules. Definition 4.1.
Let C be a class of modules. A module M is called C -periodic ifthere exists a short exact sequence 0 → M → C → M →
0, with C ∈ C . UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 7
The following proposition relating flat periodic A -modules and acyclic complexeswith components in A is standard, but relevant for our purposes. The reader canfind a proof in [5, Proposition 1 and Proposition 2]. Proposition 4.2.
Let A be a class of modules closed under direct sums and directsummands. The following are equivalent: (1) Every cycle of an acyclic complex with flat cycles and with components in A belongs to A . (2) Every flat A -periodic module belongs to A . We are interested in deconstructible classes of modules A satisfying condition(2) in the previous proposition. Of course the first trivial example is the classFlat( R ) of flat modules itself. Since the class of all flat Mittag-Leffler modules isclosed under pure submodules, this class also trivially yields an example of a class A satisfying that every flat A -periodic module is in A . However this class has animportant drawback: it is only deconstructible in the trivial case of a perfect ring(see Herbera and Trlifaj [14, Corollary 7.3]). This setback can be remedied byconsidering the restricted flat Mittag-Leffler modules, in the sense of [7, Example2.1(3)], as we will show in Theorem 4.5 below.Now we will provide with other interesting non-trivial examples of such classes A satisfying condition (2) above, and that will be relevant in the applications ofour main results in the next sections.The first example is the class A = Proj( R ) of projective R -modules and goesback to Benson and Goodearl [2, Theorem 1.1]. Proposition 4.3.
Let
Proj( R ) be the class of all projective R -modules. Every flat Proj( R ) -periodic module is projective. As a consequence every pure acyclic complexof projectives is contractible (i.e. has projective cycles). The second application is the class A = VF ( R ) of very flat modules (this is dueto ˇSˇt’ov´ıˇcek, personal communication). Proposition 4.4.
Every flat VF ( R ) -periodic module is very flat. As a consequenceevery pure acyclic complex of very flat modules has very flat cycles.Proof. Let 0 → F → G → F → F flat and G very flat.Let 0 → F → P → F → P projective; then F is flat.An application of the horseshoe lemma gives the following commutative diagram0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / F (cid:15) (cid:15) / / Q / / (cid:15) (cid:15) F (cid:15) (cid:15) / / / / P (cid:15) (cid:15) / / P ⊕ P / / (cid:15) (cid:15) P (cid:15) (cid:15) / / / / F (cid:15) (cid:15) / / G / / (cid:15) (cid:15) F (cid:15) (cid:15) / /
00 0 0 . where Q is projective, since pd R ( G ) ≤
1. Thus, by Proposition 4.3, F is projectiveand therefore pd R ( F ) ≤
1. Let C ∈ CA ( R ). Then applying Hom R ( − , C ) to theshort exact sequence yields 0 = Ext R ( G, C ) → Ext R ( F, C ) → Ext R ( F, C ) = 0,
SERGIO ESTRADA AND ALEXANDER SL´AVIK hence F ∈ VF ( R ). Finally, the consequence follows from Proposition 4.2(1) (with A = VF ( R )). (cid:3) The last example is the announced deconstructible class of restricted flat Mittag-Leffler modules as defined in [7, Example 2.1(3)].
Theorem 4.5. let κ be an infinite cardinal and A ( κ ) be the class of κ -restrictedflat Mittag-Leffler modules. Every flat A ( κ ) -periodic module is in A ( κ ) . As aconsequence every pure acyclic complex with components in A ( κ ) has cycles in A ( κ ) .Proof. The proof mostly follows the pattern outlined in [2]; the main difference isthat instead of direct sum decomposition, we work with filtrations and Hill Lemma(cf. [11, Theorem 7.10]). Given a short exact sequence(1) 0 → F → G f → F → F flat and G ∈ A ( κ ), we fix a Hill family H for G . The goal is to picka filtration ( G α | α ≤ σ ) from H such that for each α < σ , f ( G α ) = F ∩ G α , f ( G α ) ⊆ ∗ F , and G α +1 /G α is ≤ κ -presented flat Mittag-Leffler.Once this is achieved, we obtain a filtration of the whole short exact sequence(1) by short exact sequences of the form0 → F α +1 /F α → G α +1 /G α → F α +1 /F α → F α = f ( G α ) = F ∩ G α ). Since the property of being flat Mittag-Lefflerpasses to pure submodules (cf. [11, Corollary 3.20]), this would make F α +1 /F α an ≤ κ -presented flat Mittag-Leffler module and hence imply F ∈ A ( κ ). (Note that by[6, Lemma 2.7 (1)], each ≤ κ -generated flat Mittag-Leffler module is (even strongly) ≤ κ -presented.)Put G = 0. For limit ordinals α , it suffices to take unions of already constructedsubmodules G β , β < α ; note that by property (H2) in Hill Lemma, G α ∈ H then.Having constructed modules up to G α (and assuming G α = G ), we construct G α +1 as follows: We pass to the quotient short exact sequence0 → F/F α → G/G α f → F/F α → , which, by assumption, satisfy that F/F α is flat and G/G α ∈ A ( κ ). Note that F/F α ,being (identified with) a pure submodule of G/G α , is flat Mittag-Leffler. The Hillfamily H gives rise to family H ′ for G/G α , which consists of factors of modulesfrom H (containing G α ) by G α .Let us first show that any ≤ κ -generated submodule Y of G/G α can be enlargedto ≤ κ -generated G ∈ H ′ with the property that f ( G ) ⊆ ∗ F/F α and G ∩ F/F α is ≤ κ -generated. To this end, we construct inductively a chain of submodules G n ∈ H ′ with union G (utilizing property (H2)). Let G be an arbitrary ≤ κ -generated module G ∈ H ′ containing Y (obtained via (H4)). Assuming we haveconstructed G n , we get G n +1 by taking these steps:(1) Enlarge f ( G n ) to a ≤ κ -generated pure submodule X n of F/F α ; this ispossible by [6, Lemma 2.7 (2)] once we notice that F/F α , being a puresubmodule of G/G α , is flat Mittag-Leffler.(2) Take ≤ κ -generated G n +1 ∈ H ′ such that X ⊆ f ( G ′ n ); this is again possibleby property (H4) of the Hill Lemma.We have f ( G ) = S n ∈ N f ( G ) = S n ∈ N X n ⊆ ∗ F/F α . This also shows that f ( G ) isflat Mittag-Leffler, hence ≤ κ -presented. The short exact sequence0 → G ∩ ( F/F α ) → G → f ( G ) → G ∩ ( F/F α ) is indeed ≤ κ -generated. UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 9
Now iterate the claim as follows: Start with arbitrary ≤ κ -generated non-zero Y ⊆ G/G α and obtain G from the claim. Enlarge it to G ∈ H ′ satisfying G ∩ ( F/F α ) ⊆ f ( G ) (which we may do using (H4), since G ∩ ( F/F α ) is ≤ κ -generated). Taking Y = G + f ( G ) and applying the claim, we get G etc. Thisway we obtain a chain G ∩ ( F/F α ) ⊆ f ( G ) ⊆ G ∩ ( F/F α ) ⊆ f ( G ) ⊆ . . . , so for G = S n ∈ N G n ∈ H ′ we have G ∩ ( F/F α ) = f ( G ). Also the purity of f ( G ) in F/F α and being ≤ κ -generated is ensured.The desired module G α +1 is now the one satisfying G α +1 /G α = G . (cid:3) Note that in the case κ = ℵ , A ( κ ) is just the class of projective modules by [24,Seconde partie, Section 2.2], so this also covers the case of [2].5. Quillen equivalent models for K (Proj( R ))It is known (see Bravo, Gillespie and Hovey [3, Corollary 6.4]) that the homotopycategory of projectives K (Proj( R )) can be realized as the homotopy category ofthe model M proj = (Ch(Proj( R )) , Ch(Proj( R )) ⊥ , Ch( R )) in Ch( R ). Now, by [10,Remark 4.2], the class Ch(Flat( R )) induces model category in Ch( R ) given by thetriple, (Ch(Flat( R )) , Ch(Proj( R )) ⊥ , dg (Cot( R ))) . The last model is thus Quillen equivalent to M proj . Therefore, its homotopycategory, the derived category of flats D (Flat( R )), is triangulated equivalent to K (Proj( R )). The next theorem gives sufficient conditions on a class of modules A to get D ( A ) and D (Flat( R )) to be triangulated equivalent. For concrete examplesof such classes the reader should have in mind the classes of modules considered inSection 4. Theorem 5.1.
Let
A ⊆
Flat( R ) be a class of modules such that: (1) The pair ( A , B ) is a hereditary cotorsion pair generated by a set. (2) Every flat A -periodic module is trivial.Then there is an abelian model category structure M = (Ch( A ) , Ch(Proj( R )) ⊥ , dg e B ) in Ch( R ) . If we denote by D ( A ) the homotopy category of M , then D (Flat( R )) , D ( A ) and K (Proj( R )) are triangulated equivalent, induced by a Quillen equivalencebetween the corresponding model categories.Proof. Let M = (Ch( A ) , W , dg e B ) be the model associated to the complete hered-itary cotorsion pairs (Ch( A ) , Ch( A ) ⊥ ) and ( e A , dg e B ) in Ch( R ). To get the claimit suffices to show that W = Ch(Proj( R )) ⊥ . To this aim we will use [10, Lemma4.3(1)], i.e. we need to prove:(i) e A = Ch( A ) ∩ Ch(Proj( R )) ⊥ .(ii) Ch( A ) ⊥ ⊆ Ch(Proj( R )) ⊥ .Condition (ii) is clear because Proj( R ) ⊆ A . Now, by Neeman [20, Theorem 8.6],Ch( A ) ∩ Ch(Proj( R )) ⊥ = g Flat( R ) ∩ Ch( A ). But, by the assumption (2), we followthat g Flat( R ) ∩ Ch( A ) = e A . (cid:3) Remark 5.1.
Starting with a class A in the assumptions of Theorem 5.1, we mayconstruct, for each integer n ≥
0, the class A ≤ n of modules M possessing an exactsequence 0 → A n → A n − → . . . → A → M → A i ∈ A , i = 1 , . . . , n . The derived categories D ( A ≤ n ) and D ( A ) are triangu-lated equivalent (see Positselski [21, Proposition A.5.6]). In particular we can inferfrom this a triangulated equivalence between K (Proj( R )) and D ( VF ( R )). By using a standard argument of totalization one can also check that D ( A ≤ n ) and D ( A ) canbe realized as the homotopy categories of two models M and M and that thesemodels are Quillen equivalent without using Neeman [20, Theorem 8.6]. From thispoint of view it seems that the triangulated equivalence between K (Proj( R )) and D ( VF ( R )) is much less involved than the one between K (Proj( R )) and D (Flat( R )).6. Quillen equivalent models for D (Flat( X )) Setup:
Throughout this section X will denote a quasi-compact and semi-separatedscheme. If U = { U , . . . , U m } is an affine open cover of X and α = { i , . . . , i k } isa finite sequence of indices in the set { , . . . , m } (with i < · · · < i k ), we write U α = U i ∩ · · · ∩ U i k for the corresponding affine intersection.In [16] Murfet shows that the derived category of flat quasi-coherent sheaveson X , D (Flat( X )), constitutes a good replacement of the homotopy category ofprojectives for non-affine schemes, because in case X = Spec( R ) is affine, thecategories D (Flat( X )) and K (Proj( R )) are triangulated equivalent. There is amodel for D (Flat( X )) in Ch( Qcoh ( X )) given by the triple M flat = (Ch(Flat( X ) , W , dg (Cot( X ))) . (see [10, Corollary 4.1]). We devote this section to provide a general method toproduce model categories M in Ch( Qcoh ( X )) which are Quillen equivalent to M flat .In particular this implies that the homotopy category Ho( M ) and D (Flat( X )) aretriangulated equivalent. Theorem 6.1.
Let X be a scheme and let P be a property of modules and A its associated class of modules. Assume that A ⊆
Flat , and that the followingconditions hold: (1)
The class A is Zariski-local. (2) For each R = O X ( U ) , U ∈ U , the pair ( A R , B R ) is a hereditary cotorsionpair generated by a set. (3) For each R = O X ( U ) , U ∈ U , every flat A R -periodic module is trivial. (4) j ∗ ( A qc( U α ) ) ⊆ A qc( X ) , for each α ⊆ { , . . . , m } .Then there is an abelian model category structure M A qc in Ch(
Qcoh ( X )) givenby the triple (Ch( A qc ) , W , dg ( B )) . If we denote by D ( A qc ) the homotopy categoryof M A qc , then the categories D (Flat( X )) and D ( A qc ) are triangulated equivalent,induced by a Quillen equivalence between the corresponding model categories. Incase X = Spec( R ) is affine, D ( A R ) is triangulated equivalent to K (Proj( R )) . Before proving the theorem, let us focus on one particular instance of it: if wetake A = VF (the class of very flat modules) the theorem gives us that D (Flat( X ))and D ( VF ( X )) are triangulated equivalent. This generalizes to arbitrary schemes[21, Corollary 5.4.3], where such a triangulated equivalence is obtained for a semi-separated Noetherian scheme of finite Krull dimension. Corollary 6.1.
For any scheme X , the categories D (Flat( X )) and D ( VF ( X )) aretriangulated equivalent. Let us prove Theorem 6.1. We firstly require the following useful lemma.
Lemma 6.2.
Suppose A is as in Theorem 6.1 (possibly without satisfying condition(3)). Then for any M • ∈ Ch(Flat( X )) there exists a short exact sequence → K • → F • → M • → , where F • ∈ Ch( A qc( X ) ) and K • ∈ g Flat( X ) . UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 11
Proof.
We essentially follow the proof of [22, Lemma 4.1.1]; the main difference isthat instead of sheaves, we are dealing with complexes of sheaves. Starting withthe empty set, we gradually construct such a short exact sequence with the desiredproperties manifesting on larger and larger unions of sets from U , reaching X in afinite number of steps.Assume that for an open subscheme T of X we have constructed a short exactsequence 0 → L • → G • → M • → h ∗ ( G • ) belongs toCh( A qc( T ) ) ( h : T ֒ → X being the inclusion map) and L • ∈ g Flat( X ). Let U ∈ U (with inclusion map j : U ֒ → X ); our goal is to construct a short exact sequence0 → L ′• → G ′• → M • → U ∪ T .Let us note that the adjoint pairs of functors on sheaves ( j ∗ , j ∗ ), ( h ∗ , h ∗ ) yieldcorresponding adjoint pairs of functors on complexes of sheaves.Pick a short exact sequence 0 → K ′• → Z • → j ∗ ( G • ) → U , where Z • ∈ Ch( A qc( U ) ) = Ch( A O U ( U ) ) and K ′• ∈ Ch( A O U ( U ) ) ⊥ , i.e. special precover in the category of complexes of O U ( U )-modules. In this (affine) setting we know from [20] that K ′• ∈ g Flat( U ), since K ′• ∈ Ch(Flat( U )) ∩ Ch( A O U ( U ) ) ⊥ ⊆ Ch(Flat( U )) ∩ Ch(Proj( U )) ⊥ . Using thedirect image functor, we get 0 → j ∗ ( K ′• ) → j ∗ ( Z ′• ) → j ∗ j ∗ ( G • ) → X . Since U ∈ U is affine, j ∗ is an exact functor taking flats to flats and also preserving A by condition (4), so j ∗ ( K ′• ) ∈ g Flat( X ), whence j ∗ ( Z • ) stays in Ch( A qc ). Nowconsidering the pull-back with respect to the adjunction morphism G • → j ∗ j ∗ ( G • ),one gets a new short exact sequence ending in G • ; let G ′• be its middle term:0 / / j ∗ ( K ′• ) / / G ′• / / (cid:15) (cid:15) G • / / (cid:15) (cid:15) / / j ∗ ( K ′• ) / / j ∗ ( Z ′• ) / / j ∗ j ∗ ( G • ) / / G ′• are in Ch( A qc( U ∪ T ) ); this is sufficientto check on U and T separately. Firstly, j ∗ ( G ′• ) ∼ = Z • , which is in Ch( A qc( U ) ) byconstruction. On the other hand, the complex j ∗ ( G • ), when further restricted to U ∩ T , is in Ch( A qc( U ∪ T ) ) ( A being Zariski-local class), and the same holds forthe complex K ′• by the resolving property of A . The embedding U ∩ T ֒ → T isan affine morphism (by semi-separatedness) and preserving A by (4), so j ∗ ( K ′• ) ∈ Ch( A qc( T ) ). Therefore G ′• , as an extension of j ∗ ( K ′• ) by G • , belongs to Ch( A qc( T ) ),too.Finally, the kernel K • of the composition of morphisms G ′• → G • → M • isan extension of L • and j ∗ ( K ′• ), hence a complex from g Flat( X ). This proves theexistence of the short exact sequence from the statement. (cid:3) Proof of Theorem 6.1.
First of all we notice that the class A qc contains a family ofgenerators for Qcoh ( X ); this is just a variation of the idea used in the proof of [21,Lemma 4.1.1], where we replace the class of very flats by A (and do not care aboutthe kernel of the morphisms), which is possible thanks to property (4).Then, by [6, Corollary 3.15] we get in Qcoh ( X ) the complete hereditary cotorsionpair ( A qc , B ) generated by a set. Thus by [10, Theorem 4.10] we get the abelianmodel structure M qc A qc = (Ch( A qc ) , W , dg ( B )) in Ch( Qcoh ( X )) given by the twocomplete hereditary cotorsion pairs:(Ch( A qc ) , Ch( A qc ) ⊥ ) and ( e A qc , dg ( B )) . Since A qc ⊆ Flat( X ), we get the corresponding induced cotorsion pairs in Ch(Flat( X ))(with the induced exact structure from Flat( X )):(Ch( A qc ) , Ch( A qc ) ⊥ ∩ Ch(Flat( X ))) and ( e A qc , dg ( B ) ∩ Ch(Flat( X ))) . To see that e.g. the former one is indeed a cotorsion pair, we have to check thatCh( A qc ) = ⊥ (Ch( A qc ) ⊥ ∩ Ch(Flat( X ))) ∩ Ch(Flat( X )). The inclusion “ ⊆ ” is clear.To see the other one, pick X • ∈ ⊥ (Ch( A qc ) ⊥ ∩ Ch(Flat( X ))) ∩ Ch(Flat( X )) andconsider a short exact sequence 0 → B • → A • → X • → A • ∈ Ch( A qc ) and B • ∈ Ch( A qc ) ⊥ . As A qc ⊆ Flat( X ) and Ch(Flat( X )) is a resolving class, we inferthat B • ∈ Ch(Flat( X )). Thus the sequence splits and X • is a direct summand of A • , hence an element of Ch( A qc ). The proof for the latter cotorsion pair goes in asimilar way.Now we will apply [10, Lemma 4.3] to these two complete cotorsion pairs in thecategory Ch(Flat( X )) and to the thick class W = g Flat( X ) in Ch(Flat( X )). So weneed to check that the following conditions hold:(i) e A qc = Ch( A qc ) ∩ g Flat( X ).(ii) Ch( A qc ) ⊥ ∩ Ch(Flat( X )) ⊆ g Flat( X ).Since every flat A R -periodic module is trivial and the classes A and Flat are Zariski-local, we immediately infer that every flat A qc -periodic quasi-coherent sheaf istrivial. Thus, from Proposition 4.2, we get condition (i). So let us see condition (ii).Let L • ∈ Ch( A qc ) ⊥ ∩ Ch(Flat( X )). Since the pair (Ch(Flat( X )) , Ch(Flat( X )) ⊥ )in Ch( Qcoh ( X )) has enough injectives, there exists an exact sequence,0 → L • → P • → M • → , with P • ∈ Ch(Flat( X )) ⊥ and M • ∈ Ch(Flat( X )). Now, since L • ∈ Ch(Flat( X )),we get that P • ∈ Ch(Flat( X )) ∩ Ch(Flat( X )) ⊥ = ^ FlatCot( X ). By Lemma 6.2,there exists an exact sequence0 → K • → F • → M • → , where F • ∈ Ch( A qc ) and K • ∈ g Flat( X ). Now, we take the pull-back of P • → M • and F • → M • , so we get a commutative diagram:0 (cid:15) (cid:15) (cid:15) (cid:15) K • (cid:15) (cid:15) K • (cid:15) (cid:15) / / L • / / Q • / / (cid:15) (cid:15) / / F • / / (cid:15) (cid:15) / / L • / / P • / / (cid:15) (cid:15) M • / / (cid:15) (cid:15)
00 0In the middle column, the complexes K • and P • belong to g Flat( X ). Therefore, thecomplex Q • also belongs to g Flat( X ). Since F • ∈ Ch( A qc ) and L • ∈ Ch( A qc ) ⊥ ,the exact sequence in the middle row splits. So, L • ∈ g Flat( X ) as desired.Therefore by [10, Lemma 4.3] we have the exact model structure in Ch(Flat( X ))given by the triple UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 13 M Ch flat A qc = (Ch( A qc ) , g Flat( X ) , dg ( B ) ∩ Ch(Flat( X ))) . Since it has the same class of trivial objects, this model is Quillen equivalent to theflat model in Ch(Flat( X )), M Ch flat flat = (Ch(Flat( X )) , g Flat( X ) , dg (Cot( X )) ∩ Ch(Flat( X ))) . This is, in turn, the restricted model of the model M flat = (Ch(Flat( X ) , W , dg (Cot( X )))in Ch( Qcoh ( X )) with respect to the exact category Ch(Flat( X )) of cofibrant ob-jects. Thus, M flat and M Ch flat flat are canonically Quillen equivalent. To finish theproof, let us show that the model M Ch flat A qc is Quillen equivalent to M A qc = (Ch( A qc( X ) ) , W , dg ( B )) . But this model is canonically Quillen equivalent to its restriction to the cofibrantobjects, i.e. M Ch A qc A qc = (Ch( A qc ) , e A qc , dg ( B ) ∩ Ch( A qc )) . Finally the Quillen equivalent cofibrant restricted model of M Ch flat A qc = (Ch( A qc ) , g Flat( X ) , dg ( B ) ∩ Ch(Flat( X )))is given by the triple(Ch( A qc ) , g Flat( X ) ∩ Ch( A qc ) , dg ( B ) ∩ Ch( A qc ))) , which by condition (i) above is precisely the previous model M Ch A qc A qc . In summary,we have the following chain of Quillen equivalences among the several models, M flat ≃ M Ch flat flat ≃ M Ch flat A qc ≃ M Ch A qc A qc ≃ M A qc . The first and the last models give our desired Quillen equivalence. (cid:3)
Recall from [4] that M ∈ Qcoh ( X ) is an infinite-dimensional vector bundle if,for each U ∈ U , the O X ( U )-module M ( U ) is projective. We will denote by Vect( X )the class of all infinite-dimensional vector bundles on X . In case Vect( X ) contains agenerating set of Qcoh ( X ), we know from [6, Corollary 3.15 and 3.16] that the pair(Vect( X ) , B ) (where B := Vect( X ) ⊥ ) is a complete cotorsion pair generated by aset. It is hereditary, because the class Vect( X ) is resolving. Thus by [10, Theorem4.10] we get the abelian model structure M vect = (Ch(Vect( X )) , W , dg ( B )) inCh( Qcoh ( X )) given by the two complete hereditary cotorsion pairs:(Ch(Vect( X )) , Ch(Vect( X )) ⊥ ) and ( g Vect( X ) , dg ( B )) . We will denote by D (Vect( X )) its homotopy category.We are now in position to prove Corollary 2 in the Introduction. Corollary 6.2.
Let X be a scheme with enough infinite-dimensional vector bun-dles. Then the categories D (Flat( X )) and D (Vect( X )) are triangle equivalent, theequivalence being induced by a Quillen equivalence between the corresponding modelcategories.Proof. The proof will follow by showing that D (Vect( X )) and D ( VF ( X )) are Quillenequivalent, and then by applying Corollary 6.1. To this end, we will prove that themodel structures M vect and M VF have the same trivial objects. To achieve this,by [8, Theorem 1.2], it suffices to show that the trivial fibrant and cofibrant objectsof one structure are trivial also in the other structure. This assertion is clearlysatisfied by the trivial cofibrants of M vect and trivial fibrants of M VF , as g Vect( X ) ⊆ f VF ( X ) and Ch( VF ( X )) ⊥ ⊆ Ch(Vect( X )) ⊥ . Now let V • ∈ f VF ( X ); since there are enough infinite-dimensional vector bundles,the cotorsion pair ( g Vect( X ) , dg ( B )) has enough projectives, hence there is a shortexact sequence 0 → Q • → P • → V • → P • ∈ g Vect( X ). Restricting this to an open affine subset of X , we obtaina short exact sequence with a complex of projective modules in the middle andending in a complex of very flat modules, and the objects of cycles also belongingto the respective classes. Since the projective dimension of very flat modules doesnot exceed 1, it follows that Q • has also projective cycles after this restriction,hence Q • ∈ g Vect( X ). We conclude that V • , being a factor of two trivial objects, isitself trivial in M vect .Finally, pick M • ∈ Ch(Vect( X )) ⊥ . Using the completeness of the cotorsion pair(Ch( VF ( X )) , Ch( VF ( X )) ⊥ ), we obtain a short exact sequence0 → K • → V • → M • → V • ∈ Ch( VF ( X )) and K • ∈ Ch( VF ( X )) ⊥ . As K • is trivial in M vect , it sufficesto show that V • is trivial, too. Furthermore, Ch( VF ( X )) ⊥ ⊆ Ch(Vect( X )) ⊥ impliesthat in fact, V • ∈ Ch(Vect( X )) ⊥ . So as above, construct a short exact sequence0 → Q • → P • → V • → , this time with P • ∈ Ch(Vect( X )) and Q • ∈ Ch(Vect( X )) ⊥ . The same local argu-ment as above shows that Q • ∈ Ch(Vect( X )), and we also have P • ∈ Ch(Vect( X )) ⊥ (being an extension of two objects from the class). Hence V • is a factor of two com-plexes from the class Ch(Vect( X )) ∩ Ch(Vect( X )) ⊥ , which is a subclass of g Vect( X )and consequently f VF ( X ), therefore consisting of trivial objects of M VF . (cid:3) Finally, the last consequence is also an application of Theorem 6.1 for the classof very flat quasi-coherent sheaves. It follows from Gillespie [10, Theorem 4.10]
Corollary 6.3.
There is a recollement D ac ( VF ( X )) j / / D ( VF ( X )) s s k k w / / D ( X ) t t j j Remark 6.3.
Murfet and Salarian deal in [18] with a suitable generalization oftotal acyclicity for schemes. Namely, they define the category D F-tac (Flat( X )) of F-totally acyclic complexes in D (Flat( X )) and prove that, in case X = Spec( R )is affine and R is Noetherian of finite Krull dimension, D F-tac (Flat( X )) is trian-gle equivalent to K tac (Proj( R )) (the homotopy category of totally acyclic com-plexes of projective modules) showing that D F-tac (Flat( X )) also constitutes a goodreplacement of K tac (Proj( R )) in a non-affine context. An analogous version ofTheorem 6.1 allows to restrict the equivalence between D (Flat( X )) and D ( A qc )to their corresponding categories of F-totally acyclic complexes D F-tac ( A qc ) and D F-tac (Flat( X )). In particular, the full subcategory D F-tac ( VF ( X )) of F-totallyacyclic complexes of very flat quasi-coherent sheaves in D ( VF ( X )) is triangle equiv-alent with Murfet’s and Salarian’s derived category of F-totally acyclic complexesof flats. acknowledgements We would like to thank Jan ˇSˇt’ov´ıˇcek for many useful comment and discussionsduring the preparation of this manuscript. We would also like to thank LeonidPositselski for his inspiring work [21] and for sharing his knowledge on the subjectof very flat modules. The terminology used in [18] is D tac (Flat( X )). UILLEN EQUIVALENT MODELS FOR THE DERIVED CATEGORY OF FLATS 15
References [1] L. Alonso Tarr´ıo, A. Jerem´ıas L´opez, and J. Lipman.
Local homology and cohomology ofschemes,
Ann. sci. ´Ecole Norm. Sup. (4) (1997), 1–39.[2] D. Benson and K. Goodearl, Periodic flat modules, and flat modules for finite groups , Pac. J.Math. (1) (2000), 45–67.[3] D. Bravo, J. Gillespie, and M. Hovey,
The stable module category of a general ring, preprint(2014), arXiv:1405.5768v1; 38 pp.[4] V. Drinfeld,
Infinite–dimensional vector bundles in algebraic geometry: an introduction , in‘The Unity of Mathematics’, Birkh¨auser, Boston 2006, pp. 263–304.[5] S. Estrada, X. Fu, and A. Iacob,
Totally acyclic complexes,
J. Algebra. (2017), 300–319.[6] S. Estrada, P.A. Guil Asensio, M. Prest, and J. Trlifaj,
Model category structures arising fromDrinfeld vector bundles,
Adv. Math. (2012), 1417–1438.[7] S. Estrada, P.A. Guil Asensio, and J. Trlifaj,
Descent of restricted flat Mittag-Leffler modulesand locality for generalized vector bundles,
Proc. Amer. Math. Soc. (2014), 2973–2981.[8] J. Gillespie,
How to construct a Hovey triple from two cotorsion pairs , Fund. Mathematicae (3) (2015), 281–289.[9] J. Gillespie,
Model structures on exact categories , J. Pure App. Alg. (2011), 2892–2902.[10] J. Gillespie,
Models for mock homotopy categories of projectives , Homol. Homotopy App. (1) (2016), 247–263.[11] R. G¨obel, and J. Trlifaj, Approximations and Endomorphism Algebras of Modules , 2nd rev.ext. ed., W. de Guyter, Berlin 2012.[12] P. Gross,
Tensor generators on schemes and stacks , preprint, available at arXiv:1306.5418v2.[13] M. Hovey,
Cotorsion pairs, model category structures, and representation theory , Math. Zeit., (2002), 553-592.[14] D. Herbera, and J. Trlifaj,
Almost free modules and Mittag–Leffler conditions,
Adv. Math. (2012), 3436–3467.[15] J. de Jong et al.,
The Stacks Project,
Version 7016ab5. Available at http://stacks.math.columbia.edu/download/book.pdf [16] D. Murfet,
The mock homotopy category of projectives and Grothendieck duality Derived ,PhD Thesis. Available at [17] D. Murfet,
Ample sheaves and ample families
Totally acyclic complexes over noetherian schemes,
Adv. Math. (2011), 1096–1133.[19] A. Neeman,
Some adjoints in homotopy categories,
Ann. Math., (2010), 2142–2155.[20] A. Neeman,
The homotopy category of flat modules, and Grothendieck duality,
Invent. Math. (2008), 255–308.[21] L. Positselski,
Contraherent cosheaves , preprint, available at arXiv:1209.2995v4.[22] L. Positselski,
Two kinds of derived categories, Koszul duality, and comodule-contramodulecorrespondence , preprint, available at arXiv:0905.2621v12.[23] L. Positselski, and A. Slavik,
Flat morphisms of finite presentation are very flat , preprint,available at arXiv:1708.00846v1.[24] M. Raynaud, and L. Gruson,
Crit`eres de platitude et de projectivit´e,
Invent. Math. (1971),1–89.[25] J. ˇSaroch, Approximations and Mittag-Leffler conditions – the tools , preprint, available atarXiv:1612.01138.[26] A. Sl´avik, and J. Trlifaj,
Very flat, locally very flat, and contraadjusted modules,
J. PureAppl. Algebra (2016), 3910–3926.[27] J. ˇSˇt’ov´ıˇcek,
Deconstructibility and the Hill lemma in Grothendieck categories , Forum Math. (2013), 193–219.[28] B. Totaro, The resolution property for schemes and stacks,
J. Reine Angew. Math. 577 (2004),1–22.[29] R. Vakil,
Math 216: Foundations of Algebraic Geometry , 2013. Available at http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf (S.E.) Departamento de Matem´aticas, Universidad de Murcia, 30100 Murcia, Spain
E-mail address : [email protected] (A.S.) Charles University, Faculty of Mathematics and Physics, Department of Al-gebra, Sokolovsk´a 83, 186 75 Prague 8, Czech Republic E-mail address ::