THE FLAT MODEL STRUCTURE ON Ch (O) JAMES GILLESPIE Abstract.Let Ch(O) be the category of chain complexes of O-modules on a topological space T (where O is a sheaf of rings on T). We put a Quill* *en model structure on this category in which the cofibrant objects are buil* *t out of flat modules. More precisely, these are the dg-flat complexes. Dually* *, the fibrant objects will be called dg-cotorsion complexes. We show that this* * model structure is monoidal, solving the previous problem of not having any mo* *noidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category Sh(O)of O-modules. I.e., we ha* *ve a natural way to define the functors Extand Torin Sh(O). 1.Introduction Let (O, T ) be a ringed space. That is, T is a topological space and O is a sheaf of (commutative) rings (with 1) on T . In algebraic geometry we study the category Sh(O) , of sheaves of O-modules. This category is a popular example of a Grothendieck category. Consequently, it has a lot of properties similar to t* *he category Rmod , of R-modules, where R is a commutative ring with identity. One main difference though is that Sh(O) doesn't have enough projective objects. Th* *is causes some difficulties when doing homotopy theory on the chain complex cate- gory Ch (O). (I.e., doing homological algebra in Sh(O) .) Surely we can compute ExtnO(-, -) in Sh(O) since Grothendieck categories have enough injectives. How- ever, since we also have a tensor product on Sh(O), we would also like a TornO(* *-, -) functor analogous to the situation in Rmod . (In fact, Sh(O) is the same as Rm* *od , when we take T to be the one point space and let O be the constant sheaf with O(U) = R for all U T .) Since we do not have enough projectives we can not define it in the same way we do for Rmod . The idea then is to use flat mod- ules since they are tensor exact. But a flat resolution of a sheaf S need not * *be unique up to homotopy equivalence. Consequently, we can't uniquely define Tor using such flat resolutions. Edgar Enochs and L. Oyonarte recently showed we can define Torby showing the existence of flat resolutions for which all the kernel* *s are cotorsion [EO01 ]. Considering such resolutions allows us to show that any shea* *f S has a unique resolution up to homotopy equivalence. We will deal with the problem in a more general way, and from a slightly diff* *erent perspective. We take the point of view that wherever there exists a notion of h* *omo- topy in some category D, then there should be a Quillen model structure [Qui67] on D describing it. Furthermore, if D is a closed symmetric monoidal category (* *i.e., if D has a tensor product), then ideally we would like a monoidal model structu* *re ____________ Date: January 3, 2004. The author thanks Mark Hovey of Wesleyan University and Edgar Enochs of the * *University of Kentucky. 1 2 JAMES GILLESPIE on D. Loosely speaking, this just means the model structure is compatible with the tensor product. It was shown in [Gil03] that there is a new monoidal model structure on Ch (R) which we call the "flat model structure". The point of this paper is to general* *ize this result to Ch (O). Again, this is a much more interesting and useful result* * since Sh(O) doesn't have enough projectives. In model category terms, this means there is no projective model structure on Ch (O) analogous to the well known projecti* *ve model structure on Ch (R). There is an "injective model structureö n complexes of sheaves, but it is not monoidal [Joy84]. One useful corollary to having this flat model structure is that it will impl* *icitly deal with the "flat resolution problem" mentioned above. The model structure wi* *ll provide the "right typeö f resolution, simply by taking a cofibrant replacemen* *t of the module S. In short, the flat model structure on Ch (O) gives us the proper framework to define the derived functors Ext and Tor using flat resolutions. The method boils down to taking resolutions by flat O-modules so that the kernels a* *re cotorsion. So the method of Enochs and Oyonarte mentioned above is encompassed in Quillen's idea of a monoidal model category. 2.Preliminaries This section is meant to serve as a reference for definitions, notation, and * *the- orems that we will refer to in later sections of the paper. In particular, we * *will discuss cotorsion pairs, which Hovey showed in [Hov02 ] are closely related to * *model structures on abelian categories. For the rest of the paper we assume that the * *reader has a basic understanding of model categories (see [DS95 ],[Hov99 ], and [Qui67* *]), Grothendieck categories (see [Sten75]), and sheaves (see [Har77]). For much more information on cotorsion pairs see [EJ01]. Let D be an abelian category. A cotorsion pair (also called a cotorsion theor* *y) is a pair of classes of objects (A, B) of D such that A? = B and A = ?B. Here A? is the class of objects X 2 D such that Ext1(A, X) = 0 for all A 2 A, and simil* *arly ?B is the class of objects X 2 D such that Ext1(X, B) = 0 for all B 2 B. Two simple examples of cotorsion pairs in Rmod are (P, A) and (A, I) where P is the class of projectives, I is the class of injectives and A is the class of all R-* *modules. Another example of a cotorsion theory in Rmod is (F, C) where F is the class * *of flat modules and C are the so-called cotorsion modules. For a reference on coto* *rsion modules see [Xu96 ] and [EJ01]. The cotorsion pair is said to have enough projectives if for any X 2 D there * *is a short exact sequence 0 -!B -!A -!X -!0 where B 2 B and A 2 A. We say it has enough injectives if it satisfies the dual statement. It is worth making a note* * on this terminology. The phrase "enough projectivesä nd "enough injectives" is standard in reference to cotorsion pairs. Unfortunately we also use the phrase "enough projectves/injectives" in reference to a category. This should never be confus* *ing since from the context we are always referring to either a category or a cotors* *ion pair. Note however that saying that the projective cotorsion pair, (P, A), has enough projectives is equivalent to saying that the category has enough project* *ives. So in fact the terminology applied to a cotorsion theory is just a generalizati* *on of the usual terminology. In addition however, for any class of objects F in an ab* *elian category D, the author will use the terminology enough F-objects to mean for any object X 2 D there exists an F 2 F and an epimorphism F -! X. Thus if (F, C) is THE FLAT MODEL STRUCTURE ON Ch(O) 3 the "flat" cotorsion pair described above, saying we have enough F-objects means we can find a surjection F -! M where F is flat. But we say (F, C) has enough projectives to mean there exists a short exact sequence 0 -!C -!F -! M -!0 where C 2 C and F 2 F. The central cotorsion pair we study in this paper is (F, C) where F is the se* *t of all flat O-modules and C = F? is the class of cotorsion O-modules. An O-module F is called flat if the functor F O - is an exact functor. Recall that for two* * O- modules S1 and S2, we define S1 O S2 to be the sheafification of the presheaf w* *hich assigns each open set U T to S1(U) O(U)S2(U). We always assume our rings are commutative with identity 1, so that S1(U) O(U) S2(U) is an O(U)-module and therefore S1 O S2 is a sheaf of O-modules. Like most properties of sheaves, flatness is a "stalkwise property". That is, F is flat iff Fp is a flat Op-modu* *le for each p 2 T . This follows from the well-known isomorphism (S1 O S2)p ~=(S1)p Op (S2)p , which can be found in [Lit82]. In Sh(O), there are enough flat objects even tho* *ugh there are not enough projective objects. This fact is not hard and the proof ca* *n be found as Proposition 1.2 in [Har66]. We say a cotorsion theory is complete if it has enough projectives and enough injectives. The three cotorsion pairs in Rmod described above are all examples* * of complete pairs. Proving that the "flat" pair (F, C) is complete is nontrivial * *and two different proofs were recently given by the three authors of [BBE01 ]. Simi* *larly we have the "flat" cotorsion pair on Sh(O), the category of sheaves of O-modules where O is a ringed space on T . This cotorsion pair is also complete as follo* *ws from [EO01 ]. The next theorem is a special case of Theorem 2.2 from [Hov02 ] which relates cotorsion pairs to model structures on abelian categories. It is this method wh* *ich we will use to find the flat model structure on Ch (O). Theorem 2.1. (Hovey) Let D be an abelian category with all small limits and colimits and Ch(D) be the category of chain complexes on D. Let E be the class of all exact chain complexes and let Q and R be classes of chain complexes such that (Q, R \ E) and (Q \ E, R) are complete cotorsion theories. Then there exis* *ts a model structure on Ch(D) , where the weak equivalences are H*-isomorphisms, the cofibrations are the monomorphisms whose cokernels are in Q and the fibrations are the epimorphisms whose kernels are in R. Furthermore, in case D is a closed symmetric monoidal category, Hovey gives conditions on the classes Q and R which guarantee that the model structure is monoidal. So to find the flat model structure we will need to come up with the classes Q and R. This leads us to the following definition which first appeared in [Gil03]. Definition 2.2. Let (A, B) be a cotorsion pair on an abelian category D. Let X be a chain complex. (1) X is called an A complex if it is exact and ZnX 2 A for all n. (2) X is called a B complex if it is exact and ZnX 2 B for all n. (3) X is called a dg-A complex if Xn 2 A for each n, and every chain map f :X -!B with B a B-complex, is homotopic to zero. 4 JAMES GILLESPIE (4) X is called a dg-B complex if Xn 2 B for each n, and every chain map f :A -!X with A an A-complex, is homotopic to zero. We denote the class of A complexes by Aeand the class of dg-A complexes by dgAe. Similarly, the B complexes are denoted by eBand the class of dg-B complex* *es are denoted by dgBe. These definitions are inspired from the dg-projective comp* *lexes and projective complexes in Ch (R), induced by the projective cotorsion pair. Now let G be a Grothendieck category and (A, B) be a cotorsion pair. Corol- lary 3.8 of [Gil03] tells us that if G has enough A-objects then we have the in* *duced cotorsion pairs (Ae, dgBe) and (dgAe, eB) of chain complexes. A cotorsion pair (A, B) is called hereditary if one of the following hold: (1) A is resolving. That is, A is closed under taking kernels of epis. (2) B is coresolving. That is, B is closed under taking cokernels of monics. (3) Exti(A, B) = 0 for any A 2 A and B 2 B and i 1. See [GR99 ] for a proof that these are equivalent. If a cotorsion pair (A, B) is hereditary and our category D has enough projec* *tives and injectives, then dgAe\ E = Aeand dgBe\ E = eB, where E be the class of all exact chain complexes [Gil03]. In this case we say the induced cotorsion pairs are compatible, and we have cotorsion pairs (dgAe, dgBe\ E) and (dgAe\ E, dgBe). This makes the situation reminiscent of the hypothesis of Theorem 2.1 by making Q = dgAe and R = dgBe. But for a general Grothendieck category G, we may not have enough projectives. Therefore, to get the analogous result for heredit* *ary cotorsion pairs in G we will need a slight modification. This will be provided* * in Corollary 3.7. The typical cotorsion pairs one comes across in practice are hereditary. For example, all of the previous examples of cotorsion pairs are hereditary. For t* *he "flat" cotorsion pair (F, C) in Sh(O), a typical tensor product argument (like * *the one found in Proposition 3.4 of [Lan97]) will show that this is an hereditary c* *otorsion pair. 3. A simplification for Grothendieck categories For the rest of the paper we let (F, C) denote the "flat" cotorsion pair in t* *he category Sh(O) . As we have already noted, it follows from [Gil03] that we have two induced cotorsion pairs in Ch (O), which we denote (dgFe, eC) and (Fe, dgCe* *). We will call the complexes in the class eF, flat, and the complexes in the class d* *gFe, dg- flat. Similarly, we will use the terminology cotorsion and dg-cotorsion. To obt* *ain the "flat" model structure we will use Hovey's Theorem 2.2 from [Hov02 ], which we have reprinted in section 2 in a more convenient form as Theorem 2.1. Thus we wish to show first that (dgFe, eC) and (Fe, dgCe) are complete and second th* *at (dgFe, eC) and (Fe, dgCe) are compatible. By compatible we mean that dgFe\ E = * *eF and dgCe\ E = eCwhere E is the class of exact complexes. In this section we solve the problem of showing the induced cotorsion pair (dgFe, eC) is complete. We also show that (dgFe, eC) and (Fe, dgCe) are compat* *ible. Since the solutions are very general and could possibly be used for other situa* *tions we prove it for an arbitrary Grothendieck category with reasonable hypotheses on the cotorsion pair. THE FLAT MODEL STRUCTURE ON Ch(O) 5 In particular, we let (A, B) be an hereditary cotorsion pair and G be a Groth* *endieck category with enough A-objects. Again, this means that for any object X 2 G, we have an epimorphism A -! X where A 2 A. We will also assume that (A, B) is cogenerated by a set S. A cotorsion pair (A, B) is cogenerated by a set if ther* *e is a set S A (so not just a class) such that S? = C. This idea is fundamental to the study of cotorsion pairs. It will come up again in section 4. These hypothe* *ses are all satisfied by the "flat" cotorsion pair in Sh(O). For example, to see th* *at the cotorsion pair is cogenerated by a set, see the proof of Theorem 3.1 in [EO01 ]. One thing that makes a Grothendieck category feel more öc ncrete" is the fact that every object is ~-presented where ~ is some infinite cardinal. Since we wi* *ll use this idea heavily we start with some definitions. Definition 3.1. Let ~ be a cardinal number. A nonempty category K is called ~-filtered if every small subcategory S K with |mor (S)| ~ is the base of a cocone. (Here mor(S) is the set of morphisms of S.) We call a functor F :K -!D a ~-filtered functor if K is a ~-filtered category. The colimit of a ~-filtered* * functor F :K -!D is called a ~-filtered colimit. One specialization of the idea of a ~-filtered category is the notion of a ~-* *filtered ordinal. (For example, see [Hov99 ] pp. 28.) Of course this is an ordinal ff in* * which for any S ff with |S| ~, we have supS < ff (or to say the same thing, [S 2 * *ff). It is a fact that given any infinite cardinal ~, the smallest ~-filtered limit * *ordinal is ~+ , the cardinal successor of ~. Also, any successor cardinal ~0> ~ is ~-filte* *red. Given a functor F :K -!D and an object X 2 D, we will denote the composition of Hom D(X, -) with F by Hom D(X, F ). Recall that there is a natural map colimHom D (X, F ) -!Hom D (X, colimF ) (assuming all of these colimits exist) induced by the universal property of a c* *olimit. Definition 3.2. Let ~ be a cardinal and let X be an object of a category D. We say X is ~-presented (or ~-small) if for any ~-filtered functor F :K -! D (whose colimit exists in D), the natural map colimHom D(X, F ) -! Hom D(X, colimF ) is an isomorphism. In particular, suppose ff is a ~-filtered ordinal and X0 X1 . . .Xfl . . . is an increasing sequence of objects indexed by ff. If X is ~-presented, then a* *ny map X -! [Xflfactors through some Xfl. Also, notice that if ~0 ~, then a ~-presented object X 2 D is also ~0-presented. Now lets recall the following lemma of Grothendieck. See [Gro57] for a proof. Lemma 3.3. (Grothendieck) If G is a Grothendieck category, then an object I 2 G is injective iff for each subobject V U, where U is a generator, and each mor* *phism f :V -! I, f extends to a morphism U -!I. In a Grothendieck category, G, every object is ~-presented for some ~ [Hov01 * *]. Also the class of subobjects of an object is actually a set.(See [Sten75] pp 94* *). Now if we let U be a generator for G, then each subobject V U is ~V -presented for some cardinal ~V . Let ~ = sup{~V : V U}. Then clearly all U and all of its subobjects are ~-presented. Now if we let K be a ~-filtered category, and consi* *der a functor F :K -!G where F (d) is injective for all i 2 K, then for any subobje* *ct V U, and arrow i -!j in K, we get a commutative diagram 6 JAMES GILLESPIE Hom (U, F (i))----!Hom (V, F (i))----!0 ?? ? y ?y Hom (U, F (k))----!Hom (V, F (k))----!0 with exact rows. Thus colimHom (U, F ) -!colimHom (V, F ) -!0 is exact. And since K is ~-filtered, the direct limits commute to give us Hom (U, colimF ) -!Hom (V, colimF ) -!0 is exact. So Grothendieck's Lemma 3.3 implies the following lemma. Lemma 3.4. Let G be a Grothendieck category. Then there exists a cardinal ~ such that for every ~-filtered functor F :K -!G with F (j) injective for all j * *2 K, we have colimF injective. Lemma 3.5. Let G be a Grothendieck category and let (A, B) be a cotorsion pair cogenerated by a set S. Then there exists a cardinal ~0such that for every ~0-f* *iltered functor F :K -!G with F (j) 2 B for all j 2 K, we have colimF 2 B. Proof.Since (A, B) is cogenerated by a set S, there actually exists a single ob* *ject A which cogenerates (A, B). I.e. Ext1(A, B) = 0 iff B 2 B. Let ~ be as in the l* *ast lemma and let ~0 ~ be a cardinal for which A is ~0-presented. Now let K be a ~0-filtered category and let F :K -! G be a functor such that F (j) 2 B for all j 2 K. It follows from Corollary 6.6 of [Hov02 ], that in a Grothendieck category we may take injective coresolutions functorially. There- fore, we may take a functorial injective coresolution of the diagram F (K) as s* *hown below: Ij ----! Ij0 x? x ? ?? F (j)----! F (j0) Applying the functor Hom (A, -) to the diagram of injective complexes {Ij}j2K gives us (in the obvious way), a diagram of complexes {Hom (A, Ij)}j2K By hy- pothesis Ext1(A, F (j)) = H1 Hom (A, Ij) = 0 for all j 2 K. So it follows that H1[colimj2KHom (A, Ij)] = 0. But since K is ~-filtered, the last lemma tells us that colimF -! colimj2KIj is an injective coresolution. Also, since K is ~0-fil* *tered, Hom (A, colimj2KIj) ~=colimj2KHom (A, Ij). Therefore, Ext1(A, colimF ) = H1[Hom (A, colimj2KIj] = 0. For a moment, let D be any cocomplete abelian category (not necessarily Groth* *en- dieck). Though it is an abuse of notation, we will also let D denote the class of objects of D. Then for a complete cotorsion theory (A, B), clearly (A, B) = (A \ D, B) = (A, B \ D). So Hovey's Theorem 2.2 from [Hov02 ] gives us a (trivi* *al) model structure on the category D with the cofibrations being the injective maps i with coki 2 A and the fibrations being the surjective maps p with kerp 2 B. In THE FLAT MODEL STRUCTURE ON Ch(O) 7 particular, any map f :X -! Y factors as f = pi where i is a cofibration and p * *is a fibration. We will use this to prove the next proposition. The author learned* * the idea of the proof from Edgar Enochs. Proposition 3.6. Let (A, B) be a cotorsion pair in a Grothendieck category G wi* *th enough A-objects. Furthermore assume that (A, B) is cogenerated by a set. Then the cotorsion pair of complexes (dgAe, eB) has enough injectives. Proof.The assumption that G has enough A-objects is only used to guarantee that (dgAe, eB) is indeed a cotorsion pair. (See Corollary 3.8 of [Gil03].) Let X be any complex. We want to embed X as 0 -! X -! B -! A -! 0 where B 2 Beand A 2 dgAe. First let ~ be any ordinal and we will describe a general embedding method. Later we will specify an ordinal which will prove the proposition. Let (nk)k2N be any sequence which maps N bijectively onto Z. Set X0 = X. Now dn1 factor X0n1--! Zn1-1(X0) as pn1in1 where pn1: Yn01-!Zn1-1(X0) is a surjection with kerpn12 B and in1: X0n1-!Yn01is an injection with cokin12 A. Then set in1d 0 pn1 0 X1 = . .-.!X0n1+1---!Yn1 --! Xn1-1 -!. ... Then X1 satisfies (1) Zn1(X1) = kerpn1 (2) Hn1-1(X1) = 0 (3) X0 X1 and X1=X0 = . .-.!0 -!Yn01=X0n1-!0 -!. .i.s a dg-A complex. Using this method, we continue by transfinite induction to build an increasing sequence of chain complexes. Suppose ff is an ordinal and suppose that we have already constructed X = X0 X1 X2 . .X.fi . . . for all fi < ff. If ff is a limit ordinal then we set ff = [fi 0. So ExtnO(F, Hom (S, Q)) = 0 for n > 0. Proposition 5.6. A complex of O-modules F is dg-flat if and only if Fn is flat * *for each n, and F O E is exact for each exact complex E. Proof.()) Let F be dg-flat and E any exact complex. We wish to show F E is exact. By Lemma 5.5, E+ is cotorsion. So the global sections complex Hom (F, E+ ) = Hom (F, E+ )(T ) is exact. But using adjoint associativity Hom (F, E+ ) = Hom (F, Hom (E, S(Q)) ~=Hom (F E, S(Q)) = (F E)+ . So the global sections complex (F E)+ (T ) = . .-.!Sh((F E)n, Q) -!Sh ((F E)n+1, Q) -!. . . must be an exact sequence (of O(T )-modules). Now one argues as in the proof of Lemma 5.2 to conclude that F E is exact. (() Suppose that we have a chain complex F in which every Fn is a flat O- module and F E is exact for each exact complex E. By Corollary 3.7 we can find a short exact sequence 0 -!F -! C -!F 0-!0 where C is cotorsion and F 0is dg-flat. We claim that C is in fact flat. Using Lemma 5.4, it is enough to show that C+ is exact and dg-injective. Certainly C+ is exact since C is exact. Also Cn is flat since it is an extension of Fn and Fn0. This tells us that (C+ )n is injec* *tive. To finish showing C+ is dg-injective we need to show that every map E -!C+ , where E is exact, is homotopic to zero. This follows from showing that the global sec* *tions complex Hom (E, C+ ) = Hom (E, C+ )(T ) is exact. Now for any complex E we have a short exact sequence 0 -!F E -!C E -!F 0 E -!0 since 0 -! F -! C -! F 0-! 0 is "degreewise pure". Furthermore, if E is exact, then by hypothesis F E is exact and since we have proved the ()) direction of the Proposition, F 0 E is exact too. Therefore, by the fundamental lemma of homological algebra, C also has the property that C E is exact whenever E is exact. THE FLAT MODEL STRUCTURE ON Ch(O) 19 Using this property, we go back to the problem of showing the global sections complex Hom (E, C+ ) is exact whenever E is exact: For such an E, we have Hom (E, C+ ) = Hom (E, Hom (C, S(Q)) ~=Hom (E C, S(Q)) = (E C)+ and the complex on the right is clearly exact. In fact, (E C)+ must be preshe* *af exact by the comments made after the proof of Lemma 5.3. Therefore, the global sections complex Hom (E, C+ ) = Hom (E, C+ )(T ) is exact. This concludes the proof that C is flat. Now for any cotorsion complex C0, the sequence 0 -!Hom (F 0, C0) -!Hom (C, C0) -!Hom (F, C0) -!0 is exact. Indeed, in degree n we have the exact sequence Y Y Y Sh(O)(Ck, C0k+n) -! Sh (O)(Fk, C0k+n) -! Ext1O(Fk0, C0k+n) = 0. k2Z k2Z k2Z Since F 0and C are both dg-flat we see that Hom (F 0, C0) and Hom (C, C0) are e* *xact. Thus the fundamental lemma of homological algebra implies Hom (F, C0) is exact, so that F is dg-flat. Finally, we may now prove that the flat model structure is monoidal. Theorem 5.7. Let (F, C) be the flat cotorsion pair in the category Sh(O). Then the induced model structure on Ch (O) is monoidal with respect to the usual ten* *sor product of chain complexes. Proof.We use Hovey's criteria listed in Proposition 5.1. 1) Cofibrations are monomorphisms with dg-flat cokernels. In particular, the cokernel is flat in each degree. Therefore each cofibration is a pure injection* * in each degree. (See for example, the proof of Lemma XVI.3.1 in [Lan97].) 2) Let X, Y be dg-flat. We wish to show X Y is dg-flat. It is easy to s* *ee that (X Y )n is flat for each n. Now given an exact complex E, we know by Proposition 5.6 that Y E is exact, and likewise X (Y E) is exact. But then (X Y ) E is exact and using Proposition 5.6 again we get that X Y is dg-f* *lat. 3) Let X be dg-flat and Y flat. We wish to show X Y is flat. Recall that a complex is flat if and only if it is dg-flat and exact. 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