K-THEORY AND DERIVED EQUIVALENCES DANIEL DUGGER AND BROOKE SHIPLEY Abstract.We show that if two rings have equivalent derived categories th* *en they have the same algebraic K-theory. Similar results are given for G-t* *heory, and for a large class of abelian categories. Contents 1. Introduction 1 2. Model category preliminaries 4 3. K-theory and model categories 7 4. Tilting Theory 10 5. Proofs of the main results 12 6. Derived equivalence implies Quillen equivalence 13 7. Many generators version of proofs 16 Appendix A. Proof of Proposition 3.6 19 References 21 1.Introduction Algebraic K-theory began as a collection of elaborate invariants for a ring R. Quillen [Q2 ] constructed these by feeding the category of finitely-generated p* *rojec- tive R-modules into the so-called Q-construction. In fact, the Q-construction c* *an take as input any category with a sensible notion of exact sequence. Waldhausen later realized in [Wa ] that the same kind of invariants can be defined for a v* *ery broad class of homotopical situations (Waldhausen used `categories with cofibra- tions and weak equivalences'). To define the algebraic K-theory of a ring using* * the Waldhausen approach, one takes as input the category of bounded chain complexes of finitely-generated projective modules. As soon as one understands this perspective it becomes natural to ask whether the Waldhausen K-theory construction really depends on the whole input category or just on the associated homotopy category (where the weak equivalences have been inverted). Or, in the algebraic case, one asks whether the K-theory of a r* *ing depends only on the the associated derived category. In this paper we answer the latter question in the affirmative; if one is given the derived category of a r* *ing, together with its triangulation_but without knowing which ring it is_then it is theoretically possible to recover the algebraic K-theory of the ring. ____________ Date: September 6, 2002; 1991 AMS Math. Subj. Class.: 19D99, 18E30, 55U35. Second author supported in part by an NSF grant. 1 2 DANIEL DUGGER AND BROOKE SHIPLEY We now give a more detailed description of the results. If R is a ring, let DR denote the derived category of unbounded chain complexes of R-modules. Recall that DR is a triangulated category in a standard way [We , 10.4]. Also, let K*(* *R) denote the algebraic K-groups of R. Our first theorem is the following: Theorem A. If R and S are two rings for which DR and DS are equivalent as triangulated categories, then their algebraic K-groups are isomorphic: K*(R) ~= K*(S). When the hypothesis of the theorem holds we say that R and S are derived equivalent, and so the result says that derived equivalent rings have isomorphic K-theories. (This definition of `derived equivalent' is not manifestly the same* * as that of [R1 , Def 6.5], but they do in fact agree_see Theorem 4.2). We can actually state somewhat stronger results. Recall that Kn(R) is the nth homotopy group of a certain space K(R) produced by ones favorite K-theory machine. Let Dc(R) denote the full subcategory of DR consisting of the perfect complexes_that is, those complexes which are isomorphic in DR to a bounded complex of finitely-generated projectives. (The `c' is for `compact', a term wh* *ich is defined in Example 3.4). Theorem B. If R and S are rings such that Dc(R) and Dc(S) are equivalent as triangulated categories, then Kn(R) ~=Kn(S) for all n 0. Even more, one has a weak equivalence of K-theory spaces K(R) ' K(S). The K0 part of this result is very simple (see [R1 , 9.3]), and so our contri* *bution is the extension to higher K-theory. We should mention that one can even weaken the hypotheses somewhat, to require only an equivalence between Dc(R) and Dc(S) which commutes with the shift or suspension functor; see Remark 4.4. There are similar results for the G-theory of a ring. Recall that when R is Noetherian G(R) is the Quillen K-theory of the category of finitely-generated R- modules (as opposed to finitely-generated projectives); see [Sr, Chapter 5]. In terms of the Waldhausen machinery, it is the algebraic K-theory of the category* * of bounded chain complexes of finitely-generated R-modules_we denote the associ- ated homotopy category by Db(mod- R). Theorem C. Suppose that R and S are Noetherian rings. (a)If R and S are derived equivalent, then G(R) ' G(S); in particular, Gn(R) ~= Gn(S) for all n 0. (b)If Db(mod- R) is triangulated-equivalent to Db(mod- S), then G(R) ' G(S). Results along these lines first appeared in the work of Neeman [N1 ]. Neeman has had the much more ambitious goal of actually constructing the algebraic K- theory space directly from the derived category. It seems he has accomplished t* *his in the case of abelian categories (cf. [N1 , Thm. 7.1, p336]), and so for insta* *nce can construct G(R) from Db(mod- R) when R is Noetherian. Using this result Neeman is able to prove Theorem C(b), and from this he is able to deduce Theorem B in the case of regular rings (because for regular rings one has G*(R) ~=K*(R)). Theorem B in the above generality is new, however, as are the other results abo* *ve. Neeman's work is quite long and intricate, and it has sometimes been met with a certain amount of suspicion_mostly because experts just did not believe that K-theory could depend only on the derived category. The point we would like to DERIVED EQUIVALENCES 3 accentuate is that our proofs of the above theorems are all quite simple. The o* *nly `new' tool which enters the mix is the use of model categories. Although model categories are not often used in these contexts, their use effectively streamli* *nes our work. There are two main points underlying the above theorems: (1)Any equivalence of model categories yields a weak equivalence of K-theory spaces (see Proposition 3.6), and (2)If two rings R and S are derived equivalent then tilting theory shows that t* *heir model categories of chain complexes Ch R and Ch S are in fact equivalent as model categories (see Theorem 4.2). The first observation can be seen as an improvement of [TT , 1.9.8], see Remark* * 3.11. The second is a more structured version of [R1 , 6.4] and [R2 , 3.3, 5.1]; note* * that unlike [R2 ], we do not require any flatness hypotheses. See also [SS2, 5.1.1, * *B.1]. The observation in (2) is definitely surprising, although it turns out that i* *t is not hard to prove (in fact, considering the extra structure in the model catego* *ry seems to simplify the classical tilting theory proofs). The reason it is surpr* *ising is that the derived category of R is the `homotopy category' of Ch R, and this usually represents only first-order information in the model category. Equivale* *nt model categories have equivalent homotopy categories, but it almost never works the other way around. So something special happens when dealing with chain complexes over a ring; the first order information here determines all of the h* *igher order information. Note that this does not happen in arbitrary `abelian' model categories. See also Remarks 2.5 and 6.8. We state one last theorem along these lines, where we replace the category of* * R- modules by any rich enough abelian category. Of course any abelian category A h* *as an unbounded derived category DA , and we'll say that A and B are derived equiv- alent if DA is triangulated-equivalent to DB. Let Kc(A) denote the Waldhausen K-theory of the compact objects in ChA . It turns out that the space Kc(Mod- R) is just K(R). Recall that if A is an abelian category, we say that an object P is a strong generator if X = 0 whenever hom A(P, X) = 0; when A has arbitrary coproducts, the object P is called small if ffhomA (P, Xff) -! homA (P, ffXff) is a bijec* *tion for every set of objects {Xff}ff. Gabriel [G , V.1] has classified the abelian * *categories which are equivalent to categories of modules over a ring: these are the co-com* *plete abelian categories with a single strong generator. Freyd [F , 5.3H] generalized* * this to include the case of many generators; see Theorems 6.1 and 7.1. Using these b* *asic tools, we can extend our above statements to prove the following: Theorem D. Let A and B be co-complete abelian categories which have sets of small, projective, strong generators. Then (a)A and B are derived equivalent if and only if ChA and ChB are equivalent as model categories. (b)If A and B are derived equivalent, then Kc(A) ' Kc(B). Neeman [N1 , 7.1] has proven that if A and B are small abelian categories for which Db(A) is triangulated-equivalent to Db(B), then K(A) ' K(B) where K(A) denotes the Quillen K-theory of the exact category A. There is little overlap between this result and the above one: the abelian categories in Theorem D have 4 DANIEL DUGGER AND BROOKE SHIPLEY infinite direct sums, so it follows from the Eilenberg-Swindle that K(A) and K(* *B) are both trivial. We do not know how to apply our methods to the kinds of abeli* *an categories Neeman deals with. One final note: The reader may have noticed that we have always talked about K-theory spaces, rather than K-theory spectra. In fact, all of the results in * *this paper hold when restated in terms of spectra, and there is no difference in the* * proofs. We have chosen to avoid the added complications in an attempt to streamline the presentation. 1.1. Organization. The proofs of Theorems A-C are given in Section 5, and the paper has been structured so that the reader can get to them as soon as possibl* *e. The sections previous to that build up the necessary machinery, but with most of the technical proofs postponed until later. Section 3 recasts Waldhausen K- theory as an invariant for model categories, and proves that it is preserved by Quillen equivalences. Section 4 explains what tilting theory has to say about Q* *uillen equivalences between model categories of chain complexes. Finally, in Section 7* * we develop the many-generators version of tilting theory, and prove Theorem D. 1.2. Notation and terminology. Being topologists, our convention is to always work with chain complexes C* rather than cochain complexes. So the differentials have the form d: Cn ! Cn-1, and the shift operator is denoted as C: it is the chain complex with ( C)n = Cn-1. Throughout this paper we deal with right modules (and our rings are not nec- essarily commutative). Everything could be translated to left modules as well, * *but because of the usual conventions for composing maps, right modules are what nat- urally arise in some of our results; see Theorems 6.1 and 6.4 for example. Mod-* * R denotes the category of all R-modules, whereas mod- R denotes the category of finitely-generated R-modules (we only use this when R is right-Noetherian). Lik* *e- wise, Proj-R is the category of all projective R-modules and proj-R is the subc* *at- egory of finitely-generated projectives. Finally, if C is a category then we write C(X, Y ) for Hom C(X, Y ). 1.3. Acknowledgments. We are grateful to Mike Mandell and Stefan Schwede for several helpful conversations related to this paper. 2. Model category preliminaries A model category is a category equipped with certain extra structures which allow one to `do homotopy theory'. The theory is based on three standard exampl* *es: the category of topological spaces, the category of simplicial sets, and the ca* *tegory of chain complexes over a given ring. In this section we recall the basic axio* *ms of model categories, and state the main facts we need in the body of the paper. [DwSp ], [Hi] and [Ho ] are good references for this material. Definition 2.1.A model category is a category M equipped with three distin- guished classes of maps: the weak equivalences, the cofibrations, and the fibra* *tions. Cofibrations are depicted as æ, fibrations as i, and weak equivalences as -~!. Maps which are both cofibrations~and weak equivalences are called trivial cofib* *ra- tions, and denoted by æ ; trivial fibrations are defined similarly. The follow* *ing axioms are required: Axiom 1: M is complete and co-complete. DERIVED EQUIVALENCES 5 Axiom 2: (Two-out-of-three axiom) If f :A ! B and g :B ! C are maps in M and any two of f, g, and gf are weak equivalences, then so is the thir* *d. Axiom 3: (Retract axiom) A retract of a weak equivalence (respectively cofibrat* *ion, fibration) is again a weak equivalence (respectively cofibration, fibr* *ation). Axiom 4: (Lifting axiom) Suppose Af_____//flfflX | | | fflffl|fflfflfflffl| B _____//Y is a square (in which A ! B is a cofibration and X ! Y is a fibration). Then if either of the two vertical maps is a weak equivalence, there i* *s a lifting B ! X making the diagram commute. Axiom 5: (Factorization axiom) Any map A ! X may be functorially factored in ~ ~ two ways, as A æ B i X and as A æ Y -i X. Suppose maps A ! B and X ! Y are given. When any square as in Axiom 4 has a lifting B ! X, we say that A ! B has the left-lifting-property with respe* *ct to X ! Y . Example 2.2. In this paper we only deal explicitly with model categories on cat- egories of chain complexes. (a)The category Ch+Rof non-negatively graded chain complexes over a ring R has a model structure where the weak equivalences are the maps inducing homology isomorphisms (the quasi-isomorphisms), the fibrations are the maps which are surjective in positive degrees, and the cofibrations are the monomorphis* *ms with degreewise projective cokernels; see [Q1 , II p. 4.11, Remark 5], [DwSp* * , Sec. 7]. This model structure on Ch+Ris referred to as the projective model structure since there are other model structures on Ch+R. (b)The category ChR of unbounded chain complexes over a ring R also has a (projective) model structure with weak equivalences the homology isomor- phisms, and fibrations the epimorphisms; see [Ho , 2.3.11], [SS1, 5]. Every cofibration is still a degreewise split injection and the cokernel is levelw* *ise pro- jective, but not all such degreewise split injections are cofibrations. (c)ChR has another model structure with the same weak equivalences, but where the cofibrations are the monomorphisms. The fibrations are harder to describ* *e, but any fibration is a degreewise surjection with levelwise-injective kernel* *. This is the injective model structure on ChR . In this paper we only need to use * *the projective model structure on ChR , however. When M is a model category, one may formally invert the weak equivalences W to obtain the category-theoretic localization W-1M. This is the homotopy cate- gory of M, written Ho M; see [Q1 , I.1], [DwSp , 6.2]. Since the weak equivalen* *ces in ChR are the quasi-isomorphisms, the homotopy category Ho ChR is equivalent to the (unbounded) derived category DR (cf. [We , Example 10.3.2]). A model category is called pointed if the initial object and terminal object * *are the same. The homotopy category of any pointed model category turns out to have a suspension functor . For topological spaces this is ordinary suspensio* *n, whereas for Ch+Rand ChR it is the functor sending a chain complex C to the shift C with ( C)n = Cn-1. As the example of Ch+Rshows, this functor need not 6 DANIEL DUGGER AND BROOKE SHIPLEY be an equivalence. When it is an equivalence we say that M is a stable model category, and in this case Ho M becomes a triangulated category in a natural way [Ho , 7.1]. (When M is not stable, Ho M only has a `partial' triangulation; see [Q1 , I.2, I.3], [Ho , 6.5] for details). For ChR this of course specializ* *es to the usual triangulation on DR . Definition 2.3.A Quillen map of model categories M ! N consists of a pair of adjoint functors L: M Æ N :R such that L preserves cofibrations and triv- ial cofibrations (it is equivalent to require that R preserves fibrations and t* *rivial fibrations). In this case the pair (L, R) is also called a Quillen pair. Example 2.4. Let R ! S be a map of rings. The adjoint pair of functors L: Mod- R Æ Mod- S :R defined by L(M) = M R S and R(N) = Hom R(S, N) prolongs to an adjoint pair between categories of chain complexes. One readily checks that these prolongations are Quillen maps Ch+R! Ch+Sand ChR ! ChS. A Quillen map induces adjoint total derived functors between the homotopy categories [Q1 , I.4]. The map is a Quillen equivalence if the total derived fu* *nctors are adjoint equivalences of the homotopy categories. This is equivalent to Quil* *len's original definition by [Ho , 1.3.13]. More generally we say that M and N are Qu* *illen equivalent if they are connected by a zig-zag of Quillen equivalences, and we w* *rite M 'Q N . As one simple example, the identity functors give a Quillen equivalence ChprojR! ChinjRbetween the projective and injective model structures on ChR . Remark 2.5. In general, having a Quillen equivalence of model categories is mu* *ch stronger than just having an equivalence between the associated homotopy cate- gories. This is because of the added structure required for a Quillen map; func* *tors on the homotopy categories may not lift to the model category level, and even if they do they may not be compatible with the model category structures. For ex- ample, it follows from [Q1 , I.4 Thm. 3] that Quillen maps between stable model categories induce triangulated functors between the homotopy categories. Quillen maps preserve even more structure, for example the simplicial mapping space str* *uc- tures [DK80 , 5.4], [Ho , 5.6.2]. There are simple topological examples_see [S* *S2, 3.2.1], for instance_of stable model categories which have the same triangulated homotopy category, but which are nevertheless not Quillen equivalent. In Re- mark 6.8 we discuss another example (based on [Sc]) which is entirely algebraic. The following theorem shows that for the special case of the model categories ChR , Quillen equivalence is not a stronger notion than triangulated equivalenc* *e of homotopy categories. In some sense this happens because rings are determined by `first order' information_compared, for example, to differential graded rings w* *hich are not. This is proved in Section 6 as Theorem 4.2 (Parts 1 and 2). Theorem 2.6. Two rings R and S are derived equivalent if and only if their as- sociated model categories of chain complexes ChR and ChS are Quillen equivalent. This theorem cannot be extended to cover the case where R or S is a different* *ial graded algebra; we give an example in [DS ] which is discussed a little in Rema* *rk 6.8. In Corollary 7.7 we do give a certain extension of this theorem to abelian cate* *gories, however. The situation is a little confusing, because these two sentences may s* *eem contradictory. They are not though; see Remark 7.8. DERIVED EQUIVALENCES 7 3. K-theory and model categories In [Wa , Section 1] Waldhausen defined a notion of category with cofibrations* * and weak equivalences and showed how to construct a K-theory space from such data. The purpose of this section is to adapt Waldhausen's machinery to the context of model categories. This is almost entirely straightforward, but it has the advan* *tage of streamlining the theory somewhat. Let M be a pointed model category with initial object *. An object A is called cofibrant if * æ A is a cofibration. By a Waldhausen subcategory of M we mean a full subcategory U with the properties that (i)U contains the initial object *; (ii)Every object of U is cofibrant; (iii)If A æ B and A ! X are maps in U, then the pushout B qA X (computed in M) belongs to U. The proof of the following is just a matter of chasing through the definition* *s: Lemma 3.1. Any Waldhausen subcategory of M, equipped with the notions of cofi- brations and weak equivalences from M, is a `category with cofibrations and weak equivalences' in the sense of [Wa , 1.2]; also, it satisfies the saturation axi* *om [Wa , p. 327]. The lemma says that we may apply Waldhausen's So-construction [Wa , 1.3] to obtain a simplicial category wSo(U). Taking the nerve in every dimension gives a simplicial space [n] 7! N(wSn(U)), and K(U) is defined to be loops on the realization of this simplicial space: K(U) = |NwSo(U)|. One defines the algebr* *aic K-groups of U by Kn(U) = ßn(K(U)). We give a partial description of wSo(U) here, because we need it later in Ap- pendix A. Let wFn(U) denote the category whose objects are sequences {A} of cofibrations A0 æ A1 æ . .æ.An in U, and whose morphisms are commutative diagrams {A} ! {A0} in which every map An ! A0nis a weak equivalence. One can almost make [n] 7! wFn-1(U) into a simplicial category (where wF-1(U) is interpreted as the trivial category with one object and an identity map) by def* *ining ( [A0 æ . .æ.^Aiæ . .æ.An] ifi 6= 0, di [A0 æ A1 æ . .æ.An] = [A1=A0 æ A2=A0 æ . .æ.An=A0] ifi = 0. The difficulty is that with this definition the simplicial identities do not ho* *ld on the nose, in the end because there are different possible choices for the quoti* *ents Ai=A0 (they are canonically isomorphic, but still different). The category wSn(* *U) is equivalent to wFn-1(U), but is slightly `fatter' in a way that allows one to* * make the face and degeneracy maps commute on the nose. The reader is referred to [Wa , p. 328] for the precise definition_it should be noted, though, that the b* *asic ideas in the present paper can all be understood by pretending that wSn(U) is j* *ust wFn-1(U). The only time the details of wSn(U) are needed is in the Appendix. Example 3.2. Let R be a ring. The following are Waldhausen subcategories of ChR (as is easily verified). (1)UK = {all bounded complexes of finitely-generated projectives}. (2)UG = {all bounded below complexes C of finitely-generated projectives such that Hk(C) 6= 0 for only finitely-many values of k}. 8 DANIEL DUGGER AND BROOKE SHIPLEY Let K(R) and G(R) denote the Quillen K-theory spaces for the exact categories of finitely-generated projectives and finitely-generated modules, respectively.* * Then we have: Lemma 3.3. K(UK ) ' K(R), and if R is Noetherian then K(UG ) ' G(R). Proof.A reference for K(UK ) ' K(R) is [TT , 1.11.7]. For the G-theory, the reference is [TT , 3.11.10, 3.12, 3.13]_however, since the terminology of that * *paper is fairly cumbersome, we repeat the proof for the reader's convenience. Let V denote the subcategory of ChR consisting of all bounded complexes of finitely-generated modules; [TT , 1.11.7] shows that K(V) is the same as G(R). * *Let W denote the subcategory of ChR consisting of all chain complexes quasi-isomorp* *hic to an element of V. One can check that if R is Noetherian then UG consists prec* *isely of the cofibrant objects in W. Then [TT , 1.9.8] shows that UG ,! W and V ,!_W induce equivalences of K-theory spaces. |__| Example 3.4. If T is a triangulated category with infinite sums, an object X 2 T is called compact if the natural map ffT (X, Zff) ! T (X, ffZff) is a bijecti* *on for every collection {Zff2 T }. If M is a stable model category, it is easy to chec* *k that the homotopy category Ho M has all infinite sums. We'll say that an object in M is compact if its image in Ho M is compact. The subcategory Mc M consisting of all compact, cofibrant objects is a complete Waldhausen category. We are especially interested in this for the case M = ChR , where a theorem of Bökstedt-Neeman [BN , 6.4] identifies the compact objects as the perfect comple* *xes, i.e. the complexes which are quasi-isomorphic to a bounded complex of finitely- generated projectives. Example 3.5. Waldhausen never explicitly used model categories, but he could have been working in this context all along. Waldhausen developed his machinery to apply to the following case. Let X be a simplicial set, and let (X # sSet # * *X) denote the category of retractive spaces over X. This has a natural model struc* *ture inherited from the category of simplicial sets [Q1 , II.3] by forgetting the re* *traction over X (cf. [Hi, 7.6.5]). Take U to be the subcategory consisting of those retr* *active spaces X ,! Z ! X for which the map X ,! Z is obtained by attaching finitely many simplices. This is a Waldhausen subcategory, and the associated K-theory space is denoted A(X); see [Wa , 2.1]. __ If U is a subcategory of M, write U for the full subcategory of M consisting * *of all cofibrant objects which are weakly_equivalent to an object in U. From Example 3* *.4 above it follows that (ChR )c = UK (where_UK is from Example 3.2). Call the Waldhausen category U complete if U = U. Suppose that (L, R): M ! N is a Quillen map of pointed model categories. Let U and V be Waldhausen subcategories of M and N such that L maps U into V. Since L preserves cofibrations, one checks easily that it induces a well-define* *d map K(U) ! K(V). Proposition 3.6.Suppose that (L, R) is a Quillen_equivalence, and that U is a complete Waldhausen subcategory of M. Let V = LU _i.e., V consists of all cofi- brant objects which are weakly equivalent to an object in L(U). Then V is a com* *plete Waldhausen subcategory of N , and L: K(U) ! K(V) is a weak equivalence. The proof is simple but long winded, so we defer it to an appendix. DERIVED EQUIVALENCES 9 Remark 3.7. The proposition also works in the following way. Let Q be a cofibr* *ant replacement functor for M; for example~one can take the map * ! X and apply the functorial factorization * æ QX -i X in M to~define Q. Similarly, let F be a fibrant-replacement functor for N with Y æ F Y i * for Y in N . Suppose that V is a complete Waldhausen subcategory of N . Define_RV to be the set of all objects of the form QRF X where X 2 V,_and_let U = RV . Then U is a complete Waldhausen subcategory of M, and LU = V. The functor L induces a map K(U) ! K(V), and the proposition says this is an equivalence. So we have actually proven: Corollary 3.8.Let M and N be model categories connected by a zig-zag of Quillen equivalences. Let U be a complete Waldhausen subcategory of M, and let V consist of all cofibrant objects in N which are carried into U by the composite of the * *derived functors of the Quillen equivalences. Then V is a complete Waldhausen subcatego* *ry of N , and there is an induced zig-zag of weak equivalences between K(U) and K(* *V). Corollary 3.9.A Quillen equivalence M ! N between stable model categories induces a weak equivalence of K-theory spaces K(Mc) -~!K(Nc), where Mc and Nc denote the subcategories of cofibrant, compact objects. Proof.Write the functors of the Quillen equivalence as (L, R). The derived func* *tors of L and R induce an equivalence between the homotopy categories, and so in particular_they take compact objects to compact objects. This clearly implies Nc LMc ; it basically gives the opposite inclusion as well, but we now explain this in more detail. ~ If X is in Nc, let F X be a fibrant-replacement X æ~ F X i * in N and let Q(RF X) be a cofibrant-replacement * æ Q(RF X) -i RF X of RF X in M. Because the derived functors of L and R take compact objects to compact objects, QRF X must still be_compact_i.e.,_QRF X 2 Mc. Yet LQRF X is_weakly_ equivalent to X, and so X 2 LMc . At this point we have shown Nc = LMc , and_ so we can just apply Proposition 3.6. |__| Corollary 3.10.If ChR and ChS are Quillen equivalent (perhaps through a zig-zag of Quillen equivalences), then K(R) ' K(S). Proof.We have already remarked that K(R) ' K(UK ), and UK = (ChR )c. All the intermediate model categories in the zig-zag must be stable because `stability'* * is_ preserved under Quillen equivalence. Therefore Corollary 3.9 applies. * *|__| Remark 3.11. The above corollary has two improvements over similar results in the literature. The first is that we are allowing a zig-zag of Quillen equivale* *nces, rather than just an equivalence ChR ! ChS; in particular, note that our zig-zag could conceivably pass through very non-algebraic model categories. For just a single Quillen equivalence ChR ! ChS, the closest result in the literature seem* *s to be [TT , 1.9.8]. In that result, however, the functor L: ChR ! ChS is required * *to be complicial, meaning in part that it is induced via prolongation from a funct* *or Mod- R ! Mod- S. In some of our applications L is the functor which tensors with a chain complex of projectives (rather than just a single projective), and so t* *he [TT ] result is not applicable. 10 DANIEL DUGGER AND BROOKE SHIPLEY 4.Tilting Theory In this section we determine the algebraic content of having a Quillen equiva* *lence between ChR and ChS for rings R and S. A nice, complete answer can be given in terms of tilting theory. Originally tilting theory only dealt with derived e* *quiva- lences, but it turns out that for rings derived equivalence and Quillen equival* *ence coincide. We begin with a classical analogue of tilting theory, namely Morita theory. Morita theory describes necessary and sufficient conditions for when two catego* *ries of modules are equivalent. Call a (right) R-module P a strong generator if homR (P, X) = 0 implies X = 0 for any (right) R-module X. Theorem 4.1. (Morita Theory) Given rings R and S, the following conditions are equivalent: 1. The categories of (right) modules over R and S are equivalent. 2. There is an R-S bimodule M and an S-R bimodule N such that M S N ~=R as R-bimodules and N R M ~=S as S-bimodules. 3. There is a (right) R-module P which is finitely-generated, projective and a strong generator such that hom R(P, P ) ~=S. Proof.We only give a brief sketch because this is classical, see [We , 9.5]. Fo* *r (2) implies (1), the functors - R M :Mod- R -! Mod- S and - S N :Mod- S -! Mod- R give the inverse equivalences. For (1) implies (3), given an equivalence F : Mod-S -! Mod- R one may take P = F (S). For (3) implies (2), take N = P since P is a hom R(P, P )-R bimodule and take M = hom R(P, R) which is an_R- homR (P, P ) bimodule. |__| Now we turn to the analogue of Morita theory for categories of chain complexe* *s, called `tilting theory'. This analogue was developed by Rickard in [R1 , 6.4] * *to classify derived equivalences of rings. Later, Keller [Kr , 8.2] broadened tilt* *ing the- ory to apply to more general derived equivalences of abelian categories. We ext* *end both sets of results to give Quillen equivalences underlying the derived equiva* *lences. Theorem 4.2 below extends Rickard's work, whereas the generalization to abelian categories is considered in Section 7. These results can also be used to remove certain flatness assumptions in [R2 , 3.3, 5.1]. Let T be a triangulated category. Recall that a full subtriangulated category S is a full subcategory which is (i) closed under isomorphisms, (ii) closed und* *er the suspension functor, and (iii) has the property that if two objects of a disting* *uished triangle in T lie in S then so does the third object. When T has infinite sums,* * a full subtriangulated category is called localizing if it is closed under coprod* *ucts of sets of objects [N2 , 1.5.1, 3.2.6]. A complex P in T is a (weak) generator if * *the only localizing subcategory of T which contains P is T . Although this definiti* *on looks much different than the definition of a strong generator, it is not. If * *P is compact (see Example 3.4 for a definition), then P is a (weak) generator if and only if T (P, X)* = 0 implies X is trivial (see [SS2, 2.2.1] for a proof that t* *hese are equivalent). Here T (-, -)* denotes the graded maps with T (X, Y )n = T ( nX, Y* * ). An object P 2 ChR is called a tilting complex if it is a bounded complex of finitely-generated projectives, a generator of DR , and DR (P, P )* is concentr* *ated in degree zero [R1 , Def. 6.5]. Here is our generalization of Rickard's result [R1* * , Thm 6.4]: DERIVED EQUIVALENCES 11 Theorem 4.2. (Tilting theorem) The following conditions are equivalent for rings R and S: 1. There is a zig-zag of Quillen equivalences between the model categories of chain complexes of R- and S-modules: ChR 'Q ChS. 2. The unbounded derived categories are triangulated equivalent: DR ' DS. 3. The naive homotopy categories of bounded chain complexes of finitely gener- ated projective R and S-modules are triangulated equivalent: Kb(proj-R) ' Kb(proj-S). 4. The model category ChR has a tilting complex P whose endomorphism ring in DR is isomorphic to S: DR (P, P ) ~=S. Remark 4.3. Rickard [R1 , 6.4] showed that (3) and (4) are equivalent and that both these are equivalent to having a triangulated equivalence Db(Mod- R) ' Db(Mod- S). He defined two rings to be `derived equivalent' if any of these con* *di- tions holds. We defined `derived equivalent' to mean (2), and so the result sho* *ws that our use agrees with Rickard's. Note that [R1 , 6.4] gives several other eq* *uiva- lent conditions involving variations of the derived category; see Proposition 5* *.1 as well. Proof of (1) ) (2) ) (3) ) (4).Every Quillen equivalence of stable model cate- gories induces an equivalence of triangulated homotopy categories [Q1 , I.4 The- orem 3], so (1) implies (2). Any triangulated equivalence restricts to an equi* *va- lence between the respective subcategories of compact objects. Since Kb(proj-R) is equivalent to the full subcategory of compact objects in DR by [BN , Prop. 6* *.4], (2) implies (3). Now we assume condition (3) and choose a triangulated equivalence between Kb(proj-R) and Kb(proj-S). Let S[0] be the free S-module on one generator, viewed as a complex in ChS concentrated in dimension zero; let T be its im- age in Kb(proj-R). We have DR (T, T ) ~=DS(S[0], S[0]) ~=S. Since S[0] gener- ates Kb(proj-S), T generates Kb(proj-R). Since R[0] is a generator of DR and R[0] 2 Kb(proj-R), the only localizing subcategory of DR containing Kb(proj-R) * * __ is DR ; so T generates DR . Hence T is a tilting complex and condition (4) hold* *s. |__| The real content of the theorem, of course, is the proof that (4) ) (1). This* * is given in Section 6, after we have developed a little more machinery. Remark 4.4. We could have put one more intermediary condition in Theorem 4.2. Instead of a triangulated equivalence (in either (2) or (3)) we could have requ* *ired only an equivalence of categories which commutes with the shift or suspension functor. Such an equivalence would preserve compact objects and preserve the graded maps D(-, -)*. It would also preserve the property of being a compact generator, since an object is a compact generator if and only if it detects tri* *vial objects by [SS2, 2.2.1]. Thus, such equivalences preserve tilting complexes. * *We do not have very interesting examples of such equivalences, though (other than triangulated equivalences). 12 DANIEL DUGGER AND BROOKE SHIPLEY Remark 4.5. The two tilting theory results in this paper, Theorem 4.2 and its analogue Theorem 7.5, also appear in disguised form in [SS2]. Chain complexes do not satisfy the stated hypotheses of the tilting theorem in [SS2, 5.1.1], but in [SS2, Appendix B.1] chain complexes are shown to be Quillen equivalent to a model category which does satisfy the stated hypotheses. So Theorems 4.2 and 7.5 can be considered as special cases of [SS2, 5.1.1]. Here, though, we have remov* *ed all hypotheses and the proofs are much simplified_they only use categories of chain complexes, whereas the proofs in [SS2] require the use of the new symmetr* *ic monoidal category of symmetric spectra [HSS ]. 5.Proofs of the main results If you accept the basic results stated so far, it becomes easy to prove the f* *irst three theorems cited in the introduction. Proof of Theorem B.This follows from Corollary 3.10 together with the equivalen* *ce of Parts 1 and 3 in Theorem 4.2. Note that Dc(R) and Kb(proj-R) are two names_ for the same thing, by [BN , 6.4]. |__| Proof of Theorem A.If DR and DS are equivalent as triangulated categories, then so are their full subcategories of compact objects. So Theorem B applies. This * *also__ follows from Corollary 3.10 and the equivalence of Parts 1 and 2 in Theorem 4.2* *. |__| We now turn our attention to the proof of Theorem C, which is the G-theory result. We begin with a proposition which is fairly interesting in its own rig* *ht. Consider a function C which assigns to each ring R a subcategory of DR . We say that the assignment preserves equivalences if every triangulated equivalence F :DR ! DS restricts to an equivalence between C(R) and C(S). Here is some new notation: Dh+(Mod- R) denotes the full subcategory of DR consisting of chain complexes with bounded below homology, and Dhb(Mod- R) de- notes the full subcategory of complexes with bounded homology. One can similarly define Khb(proj-R), etc. The notation K+,hb(proj-R) means the intersection of K+ (proj-R) and Khb(proj-R). It is an easy exercise to check that Dh+(Mod- R) = K+ (Proj-R) and Dhb(Mod- R) = Db(Mod- R). Proposition 5.1.The assignments R 7! C(R) preserve equivalences, where C(R) is any of the following: Kb(proj-R), K+ (Proj-R) = Dh+(Mod- R), Dh-(Mod- R), Dhb(Mod- R) = Db(Mod- R), K+ (proj-R), K+,hb(proj-R). Proof.The result [BN , 6.4] identifies Kb(proj-R) with the subcategory of compa* *ct objects in DR . Any equivalence DR ! DS must preserve direct sums, and so it takes compact objects to compact objects. A complex X lies in Dh+(Mod- R) if and only if it satisfies the following pro* *perty: for any compact object A, there exists an N such that DR ( -kA, X) = 0 for k > * *N. Since triangulated equivalences preserve compact objects and the suspension, th* *ey preserve these objects as well. Similarly, a complex X lies in Dh-(Mod- R) if and only if for any compact object A, there exists an N such that DR ( kA, X) = 0 for all k > N. The same argument as above applies. For Dhb(Mod- R), note that this is just the intersec* *tion of Dh+(Mod- R) and Dh-(Mod- R). DERIVED EQUIVALENCES 13 The case of K+ (proj-R) is harder, but was proven by Rickard_see the first paragraph in the proof of [R1 , 8.1]. Finally, K+,hb(proj-R) is just the_inters* *ection_ of K+ (proj-R) and Dhb(Mod- R). |__| K+,hb(proj-R) is the full subcategory of DR consisting of complexes which are quasi-isomorphic to a bounded-below complex of finitely-generated projectives, and which also have bounded homology. So one has the inclusions Dc(R) K+,hb(proj-R) DR . Note that K+,hb(proj-R) is the image in DR of the Wald- hausen subcategory UG (R) ChR . It is an easy exercise to check that when R is right-Noetherian one has K+,hb(proj-R) = Db(mod- R), where the latter denotes the full subcategory of DR consisting of the bounded complexes of finitely-gene* *rated modules. Theorem C follows immediately from the following more comprehensive state- ment: Theorem 5.2. Let R and S be right-Noetherian. (a)If R and S are derived equivalent, then G(R) ' G(S). (b)R and S are derived equivalent if and only if K+,hb(proj-R) and K+,hb(proj-S) are equivalent as triangulated categories. (c)If Db(mod- R) ' Db(mod- S), then G(R) ' G(S) and K(R) ' K(S). Proof.Part (b) is entirely due to Rickard [R1 , 8.1,8.2]. (Note that Rickard u* *ses cochain complexes whereas we use chain complexes, and writes K-,b(proj-R) for what we call K+,hb(proj-R), etc.) For (a), suppose that R and S are derived equivalent. Then Theorem 4.2 says that there is a chain of Quillen equivalences between ChR and ChS. On the ho- motopy categories, this gives us a chain of triangulated equivalences between DR and DS. Proposition 5.1 says that this triangulated equivalence between DR and DS restricts to an equivalence between K+,hb(proj-R) and K+,hb(proj-S). So the complete Waldhausen subcategory UG (R) is carried to UG (S) via the various ad- joint functors in the chain of Quillen equivalences. One can now use Corollary * *3.8 to deduce that K(UG (R)) ' K(UG (S)). That is, G(R) ' G(S). For (c), recall that when R is Noetherian Db(mod- R) is just another name for K+,hb(proj-R), and the same for S. So if Db(mod- R) ' Db(mod- S) then by (b) __ R and S are derived equivalent; so we can apply (a) and Theorem B. |__| 6. Derived equivalence implies Quillen equivalence In this section we prove the Tilting Theorem 4.2. The only difficult part of * *this theorem follows from a differential graded analogue of the following result fro* *m [G , V.1]. This can also be viewed as another perspective on Morita theory. Theorem 6.1. (Gabriel) Let A be a co-complete abelian category with a small, projective, strong generator P . Then the functor hom A(P, -): A -! Mod- homA (P, P ) is the right adjoint of an equivalence of categories. There is also a version of this theorem for a set of small generators, due to F* *reyd; see Section 7. 14 DANIEL DUGGER AND BROOKE SHIPLEY We begin by defining a chain complex of morphisms between any two chain complexes. For M, N in ChR define HomR (M, N) in ChZ by Y HomR (M, N)n = homR(Mk, Nn+k). k The differential for HomR (M, N) is given by dfn = dN fn + (-1)n+1fndM . This structure gives an enrichment of ChR over ChZ. So instead of an endomorphism ring, an object in ChR has a differential graded ring of endomorphisms. Definition 6.2.The tensor product of X and Y in ChZ is defined by M (X Y )n = Xk Yn-k k where d(xp yq) = dxp yq+ (-1)pxp dyq. A differential graded algebra is a chain complex A in ChZ with an associative and unital multiplication ~: A A -! A [We , 4.5.2]. A (right) differential graded module M over a differential grad* *ed algebra A is a chain complex M with an associative and unital action ff: M A -! A. Denote the category of such modules by Mod-A. For any P in ChR let EndR (P ) = HomR (P, P ). Notice that EndR (P ) is a dif- ferential graded ring with the product structure coming from composition. Also, for any X 2 ChR the complex HomR (P, X) is a right differential graded EndR (P * *)- module with the action given by precomposition. So HomR (P, -) induces a functor from ChR to Mod- EndR (P ). Its left adjoint is denoted - EndR(P)P . This le* *ft adjoint can be defined as the coequalizer that the notation suggests using the * *eval- uation map HomR (P, P ) P -! P . Our differential graded analogue of Gabriel's theorem produces a Quillen equi* *v- alence of model categories instead of an equivalence of categories. So before s* *tating it we need to establish the model category structure on a category of different* *ial graded modules. The following proposition is proved in [Hi, 2.2.1, 3.1] and in * *[SS1, 4.1.1]. Proposition 6.3.Let A be a DGA. The category Mod- A has a model category structure where the weak equivalences are the maps inducing an isomorphism in h* *o- mology and the fibrations are the surjections. The cofibrations are then determ* *ined to be the maps with the left-lifting-property with respect to the trivial fibra* *tions. We can now state the following differential graded version of Gabriel's theor* *em. Theorem 6.4. Let P in ChR be a bounded complex of finitely generated projectiv* *es. If P is a (weak) generator for ChR , then there is a Quillen equivalence Mod- EndR (P ) ----! ChR in which the right-adjoint is the functor HomR (P, -). Before proving this theorem we need the following lemma. Lemma 6.5. Let M, N 2 ChR . Then H*HomR (M, N) ~=DR (M, N)* when M is cofibrant. Proof.It is easy to see in general that Hn Hom R(M, N) ~=Hn Hom R( nM, N) ~= [ nM, N] where [-, -] denotes chain-homotopy-classes of maps. When A is cofi- brant one has that DR (A, B) ~=[A, B] (since all objects are fibrant in ChR ), * *and so DERIVED EQUIVALENCES 15 we can write Hn Hom R(M, N) ~=[ nM, N] ~=DR ( nM, N) = DR (M, N)n. |___| Proof of Theorem 6.4.For any complex of projectives P , HomR (P, -) preserves surjections (fibrations) and hence is exact. We next show that HomR (P, -) pre- serves trivial fibrations; since HomR (P, -) is exact, we only need to show that H*HomR (P, K) = 0 when H*K = 0 and apply this to the kernel K of the trivial f* *i- bration. P is cofibrant by [Ho , 2.3.6] because P is a bounded complex of proje* *ctives. Thus, by Lemma 6.5, if K is acyclic then H*HomR (P, K) ~=DR (P, K)* ~=0. Hence, the functor HomR (P, -) preserves fibrations and trivial fibrations; see also [* *Ho , 4.2.13]. So its left adjoint is a Quillen map, and therefore the adjoint pair i* *nduces total derived functors on the level of homotopy categories [Q1 , I.4]. Denote t* *hese derived functors by RHomR (P, -) and - LEndR(P)P respectively. Since Ch R and Mod- EndR (P ) are stable model categories, both total derived functors preserve shifts and triangles in the homotopy categories, i.e., they a* *re exact functors of triangulated categories by [Q1 , I.4 Prop. 2]. Since - LEndR(P)P is* * a left adjoint it commutes with coproducts. To see that RHomR (P, -) commutes with coproducts it is enough to show that DEndR(P)(End R(P ), RHomR (P, -)) commutes with coproducts since End R(P ) is a compact generator of Mod- EndR (P ). By adjointness, this functor is isomorphic to DR (End R(P ) LEndR(P)P, -) which in* * turn is isomorphic to DR (P, -) since End R(P ) is cofibrant. Since P is compact [B* *N , 6.4] this functor commutes with coproducts. Now consider the full subcategories of those M in Ho(Mod- End R(P )) and X in DR respectively for which the unit of the adjunction j : M ----! RHomR (P, M LEndR(P)P ) or the counit of the adjunction : RHomR (P, X) LEndR(P)X ----! X are isomorphisms. Since both derived functors are exact and preserve coproducts, these are localizing subcategories. The map j is an isomorphism on the free mod* *ule EndR (P ) and the map is an isomorphism on P . Since the free module EndR (P ) generates the homotopy category of EndR (P )-modules and P generates ChR , the * * __ derived functors are inverse equivalences of the homotopy categories. * *|__| Before completing the proof of the Tilting Theorem, here are two important statements. Lemma 6.6. Suppose that A is a DGA and R is a ring (considered as a DGA concentrated in degree zero). Then A and R are quasi-isomorphic if and only if Hk(A) ~= Hk(R) for all k. (That is, if and only if Hk(A) = 0 for k 6= 0 and H0(A) ~=R.) Proof.Given Hk(A) ~=Hk(R) for all k, then there are quasi-isomorphisms of DGAs A- A<0> -! H0(A) ~= R. Here A<0> is the (-1)-connected cover of A with * *__ A<0>k = 0 for k < 0, A<0>k = Ak for k > 0 and A<0>0 = Z0A the zero cycles. |* *__| Proposition 6.7.Any quasi-isomorphism A -! B of differential graded algebras induces a Quillen equivalence Mod-A ! Mod- B. 16 DANIEL DUGGER AND BROOKE SHIPLEY Proof.Any map f :A -! B induces a Quillen adjoint pair between Mod-A and Mod- B, just as in Example 2.4. The right adjoint is given by restriction of sc* *alars and the left adjoint is - A B. [SS1, 4.3] shows that this adjoint pair_is_a Qu* *illen equivalence. |__| Completion of the proof of Theorem 4.2.We must show that (4) ) (1), so suppose that ChR has a tilting complex T . Then T satisfies the hypotheses of Theorem 6* *.4, hence ChR is Quillen equivalent to the category of modules over the differential graded algebra EndR(T ). Since T is a bounded complex of projectives, it is cof* *ibrant by [Ho , 2.3.6]; hence from Lemma 6.5 we have H*EndR (T ) ~= DR (T, T )* ~= S concentrated in dimension zero. By Lemma 6.6 this implies that EndR(T ) is quas* *i- isomorphic to S. Thus the categories of EndR (T )-modules and right differenti* *al graded S-modules (ChS) are Quillen equivalent by Proposition 6.7: ChR 'Q Mod- End(T ) 'Q ChS. |___| Remark 6.8. We have now shown that when R and S are rings, their model cate- gories of dg-modules are Quillen equivalent if and only if the associated homot* *opy categories are triangulated equivalent. This is false if R and S are allowed to* * be DGAs rather than rings, essentially because the analog of Lemma 6.6 fails: the quasi-isomorphism type of an arbitrary DGA is not determined by its homology (not even if you include all its Massey products, see [S, A.3]). In [DS ] we give an explicit example of two DGAs which are derived equivalent, but where the model categories of dg-modules are not Quillen equivalent. The example is based on [Sc] which considers model categories underlying the stable category of modules over the Frobenius rings R = Z=p2 and R0= Z=p[ffl]=ffl2. The homotopy categories are triangulated equivalent but the corresponding K-theory groups are non-isomorphic at K4. So by Corollary 3.9 these model categories can* *not be Quillen equivalent. In [DS ] we give a simpler proof of this by studying cer* *tain endomorphism DGAs, where we can detect the difference in the second Postnikov sections instead of in K4. 7. Many generators version of proofs In this section we generalize the work in Section 6 to the case where we have a set of generators instead of just one. Here the analogue for abelian categor* *ies is in [F , 5.3H]. For derived equivalences, Keller [Kr , 8.2] gave the correspo* *nding extension of Rickard's work [R1 , 6.4]. As always, our purpose is just to upgra* *de the derived equivalences to Quillen equivalences. As in Section 6, before moving to a differential graded setting we first reca* *ll the classical setting. Define a ring with many objects to be a small Ab-categor* *y (a category enriched over abelian groups); this terminology makes sense because an Ab-category with one object corresponds to a ring, with composition correspondi* *ng to the ring multiplication. Given a ring with many objects R_, a (right) R_-mod* *ule M is a contravariant additive functor from R_to Ab. This means that for any two objects P, P 0in R_there are maps M(P 0) R_(P, P 0) -! M(P ). The category of right R_-modules is an abelian category. If A is an abelian category and P is a set of objects, say that P is a set of strong generators if X = 0 whenever hom A(P, X) = 0 for every P in P. Define End_A(P) to be the full subcategory of A (enriched over Ab) with object set P. * *The DERIVED EQUIVALENCES 17 following theorem from [F , 5.3H] classifies abelian categories with a set of s* *trong generators: Theorem 7.1. (Freyd) Let A be a co-complete abelian category with a set of small, projective, strong generators P. Then the functor hom A(P, -): A -! Mod- End_A(P) is the right adjoint of an equivalence of categories. In order to generalize this result to a more homotopical setting, we need to replace Ab-categories with Ch-categories (categories enriched over Ch = ChZ.) S* *ince a Ch-category with one object is a differential graded algebra, one may think o* *f a small Ch-category as a DGA with many objects. Given a small Ch-category R, a (right) R-module M is a contravariant Ch-functor from R to Ch. This means that for any two objects P, P 0of R there is a structure map of chain complexes M(P 0) R(P, P 0) -! M(P ). See [Ky , 1.2] or [B , 6.2] for more details. Notice that ChR and ChR_are both Ch-categories, where R is a ring and R_is a ring with many objects. The enrichment of ChR over Ch was discussed in the previous section. Since any two R_-modules have an abelian group of morphisms homR_(M, N), the enrichment for ChR_follows similarly. Definition 7.2.Let P be a set of objects in a Ch-category C. We denote by E(P) the full subcategory of C (enriched over Ch) with objects P, i.e., E(P)(P, P 0)* * = HomC (P, P 0). We let HomC (P, -) : C ----! Mod- E(P) denote the functor given by HomC (P, Y )(P ) = HomC (P, Y ). Note that if P = {P } has a single element, then E(P) is determined by the si* *ngle differential graded ring EndC(P ) = HomC (P, P ). In [SS3, 6.1] it is established that there is a (projective) model structure * *on the category Mod- E(P) of E(P)-modules: the weak equivalences are the maps which induce quasi-isomorphisms at each object and the fibrations are the epimorphisms (at each object). Now we can state the differential graded analogue of Freyd's theorem; the dif- ference is that here we have weak generators and a Quillen equivalence instead * *of strong generators and a categorical equivalence. A set of objects P in a stable model category C is a set of (weak) generators if the only localizing subcatego* *ry of Ho(C) which contains P is Ho(C). As mentioned above Theorem 4.2, when the elements of P are compact then they generate Ho(C) if and only if they can dete* *ct when an object is trivial; see [SS2, 2.2.1]. Note that a (possibly infinite) co* *product of a set of generators is still a generator, but is not necessarily compact. Theorem 7.3. Let R_be a ring with many objects and P a set in ChR_of bounded complexes of finitely generated projectives. If P is a set of (weak) generators* * for ChR_then there is a Quillen equivalence Mod- E(P) ----! ChR_ in which the right adjoint is the functor Hom R_(P, -). Note that for every object r 2 R_there is a corresponding `free module' FrR_g* *iven by FrR_(s) = R_(s, r). A projective R_-module is finitely-generated if it is a* * direct 18 DANIEL DUGGER AND BROOKE SHIPLEY summand of a module iFrR_i, where the sum is finite. And as usual, we denote t* *he homotopy category of ChR_by DR_. We need the following lemma: Lemma 7.4. The compact objects in DR_are those complexes which are quasi- isomorphic to a bounded complex of finitely-generated projective R_-modules. Proof.This follows from [Kr , 5.3]. |___| Proof of Theorem 7.3.Just as in the proof of Theorem 6.4, one can check that Hom R_(P, -) takes fibrations and trivial fibrations in ChR_to fibrations and t* *riv- ial fibrations in Ch for any bounded complex of projectives P . So the functor Hom R_(P, -) preserves fibrations and trivial fibrations. Thus, together with i* *ts left adjoint - E(P)P, it forms a Quillen pair. We proceed as in the proof of Theorem 6.4. The induced total derived functors are again exact functors of triangulated categories which commute with coproduc* *ts. Here we use the fact that each P is compact to show the right adjoint commutes with coproducts. The full subcategories for which the unit of the adjunction j * *or the counit of the adjunction are isomorphisms are localizing subcategories. Note that for each object P in P there is a free E(P)-module FPE(P)defined by FPE(P)(P 0) = E(P)(P 0, P ), and these generate the homotopy category of E(P* *)- modules. For every P 2 P the E(P)-module Hom R_(P, P ) is isomorphic to the free module FPE(P)by inspection, and FPE(P) E(P)P is isomorphic to P since they represent the same functor on ChR_. Thus, j is an isomorphism on every free mod* *ule and is an isomorphism on every object of P. Since the free modules FPE(P)gene* *rate the homotopy category of E(P)-modules and the objects of P generate ChR_, the localizing subcategories where j and are isomorphisms are the whole homotopy * * __ categories. This implies that the adjoint pair is a Quillen equivalence. * * |__| Finally, we can write down a many-objects version of Theorem 4.2. If P is a s* *et of (weak) generators with each element P a bounded complex of finitely generated projectives and H*E(P) is concentrated in degree zero, then we call P a set of tiltors. The following theorem is a generalization of Keller's work [Kr , 8.2]: Theorem 7.5. (Many-objects tilting theorem) Theorem 4.2 holds when the rings R and S are replaced by rings-with-many-objects R_and S_. The tilting com* *plex is replaced by a set of tiltors T with H*E(T) ~=S_. The proof is given below, but first we state some easy consequences: Corollary 7.6.Two rings-with-many-objects R_ and S_are derived equivalent if and only if their associated model categories of chain complexes ChR_and ChS_are Quillen equivalent. Using Theorem 7.1 we get the following corollary as well. Given an abelian ca* *t- egory A satisfying the hypotheses of Theorem 7.1, choose a set of small, projec* *tive, strong generators P. Let A_= End_A(P) be the associated ring-with-many-objects. Freyd's theorem says that A is equivalent to Mod- A_, and so ChA is equivalent * *to ChA_. In particular, one gets a projective model structure on ChA by lifting th* *e one on ChA_across the equivalence; see [SS3, 6.1]. The next result is now an immedi* *ate consequence of Corollary 7.6. Corollary 7.7.Let A and B be co-complete abelian categories with sets of small, projective, strong generators. Then A and B are derived equivalent if and only DERIVED EQUIVALENCES 19 if their associated model categories of chain complexes ChA and ChB are Quillen equivalent. Remark 7.8. Warning: Let M and N be two stable model categories whose un- derlying categories are abelian, with sets of small, strong, projective generat* *ors. The above corollary does not say that M and N are Quillen equivalent if and only if Ho (M) and Ho (N ) are triangulated equivalent. This statement is fals* *e; see [Sc], [DS ]. Note in particular that it does not apply to the model catego* *ry Mod- R where R is a DGA; the problem is that Ho(Mod- R) is not the same as Ho(ChMod-R ). Proof of Theorem D.Part (a) is the above corollary. Part (b) is immediate_from_ (a) and Corollary 3.9. |__| Proof of Theorem 7.5.The proof that condition (1) implies condition (2) and con- dition (2) implies condition (3) follows just as in Theorem 4.2. Now assume condition (3) and fix a triangulated equivalence between Kb(proj-R* *_) and Kb(proj-S_). For s any object in S_, consider the module FsS_as a complex concentrated in dimension zero; let Ts be its image in Kb(proj-R_). Since the o* *bjects in {Fs}s2S_generate Kb(proj-S_), the objects in T = {Ts}s2S_generate Kb(proj-R_* *). But Kb(proj-R_) generates DR , so T generates DR as well. By Lemma 7.4 the objects of T are compact in DR . Finally, we also have H*E(T) ~=H*E({FsS_}) ~=S* *_. So T is a set of tiltors. If we are given a set of tiltors T for ChR_, then by Theorem 7.3 ChR_is Quill* *en equivalent to the category of modules over the endomorphism category E(T). Since H*E(T) ~= S_is concentrated in dimension zero, E(T) is quasi-isomorphic to S_ by an extension of Lemma 6.6. Thus the categories of differential graded E(T)- modules and differential graded S_-modules are Quillen equivalent by [SS3, 6.1] which generalizes Proposition 6.7: ChR_ 'Q Mod- E(T) 'Q ChS_ |___| Appendix A. Proof of Proposition 3.6 Recall that M and N are pointed model categories, (L, R): M ! N is_a_Quillen equivalence, U is a complete Waldhausen subcategory of M, and V = (LU). (Note that L, being a left adjoint, must preserve the initial object). We must show that V is a complete Waldhausen subcategory of M and that the induced map L: K(U) ! K(V) is a weak equivalence. For the remainder of this section, let F be a fibrant-replacement functor in N and let Q be a cofibrant-replacement functor in M. Note that the functor QRF :N ! M takes V into U: for if X 2 V then X ' LA for some A 2 U, and then QRF X ' QRF LA ' A. Since U is complete and A 2 U, it follows that QRF X 2 U as well. Lemma A.1. V is a complete Waldhausen subcategory of N . Proof.The only point which takes work is axiom (iii) for Waldhausen categories. So if A æ B and A ! X are maps in N where A, B, X 2 V, we must show that the pushout B qA X is also in V. 20 DANIEL DUGGER AND BROOKE SHIPLEY Consider the maps QRF A ! QRF B and QRF A ! QRF X. All the domains and~codomains of these maps are in U. Factor QRF A ! QRF B as QRF A æ Z -i QRF B. Then Z 2 U and so the pushout P = Z qQRFA QRF X is also in U, because U is a Waldhausen subcategory of M. This pushout is weakly equivalent to the homotopy pushout (see [DwSp , 10]) of Z QRF A ! QRF X, because QRF A ! Z is a cofibration and all the~objects Z, QRF A, and QRF X are cofi- brant; see [Ho , 5.2.6]. Since Z -i QRF B, P is also weakly equivalent to the homotopy pushout of the diagram QRF B QRF A ! QRF X. Finally, any left Quillen functor L preserves homotopy pushouts, in the sense that LP is weakly equivalent to the homotopy pushout of LQRF B LQRF A LQRF X. The latter homotopy pushout is weakly equivalent to the homotopy pushout of B A ! X, which in turn is just weakly equivalent to the pushout B qA X (since A ! B is a cofibration and all the objects A,_B,_X are cofibrant)* *._ So B qA X is weakly equivalent to LP , and is therefore in LU . |* *__| Let wU denote the subcategory consisting of all weak equivalences in U, and w* *rite N(wU) for the nerve of this category. The functor L induces a map wU ! wV. Lemma A.2. NL: N(wU) ! N(wV) is a weak equivalence of spaces. Proof.First of all, the functor Q: M ! M maps U into itself (because U is com- plete), and comes equipped with a natural transformation QX ! X. This shows that the induced map NQ: NU ! NU is homotopic to the identity [Se, 2.1]. Sim- ilarly, NF :NV ! NV is homotopic to the identity. The functor QRF :N ! M maps V to U, as was remarked prior to the pre- vious lemma. There are natural transformations LQRF ! LRF ! F , and Q ! QRL ! QRF L. It follows that the compositions NL O N(QRF ) and N(QRF ) O NL are homotopic to the respective identity maps, and so are part_ of a homotopy equivalence. |__| Let n denote the category consisting of n composable arrows 0 ! 1 ! . .!.n. This may be given the structure of a Reedy category [Ho , 5.2.1] in which all t* *he maps increase dimension. The category of diagrams M n has a corresponding Reedy model structure [Ho , 5.2.5] in which a map Xo ! Yo is a weak equivalence (resp* *ec- tively fibration) if and only if each Xn ! Yn is a weak equivalence (respective* *ly fibration). A map is a cofibration if and only if all the maps Xn qXn-1 Yn-1 ! * *Yn are cofibrations. In particular, an object Xo is cofibrant if and only if the * *maps Xn-1 ! Xn are all cofibrations; by a simple recursion, this implies that all the Xi's are cofibrant as well. Let Un denote the full subcategory of M n consisting of cofibrant diagrams whose objects all belong to U. It is easy to see that Un is a complete Waldhaus* *en subcategory of M n . The functors (L, R) prolong to functors (L, R): M n ! N n, and this is still a Quillen equivalence. We need the following: Lemma_A.3. Any diagram in Vn is weakly equivalent to one in L(Un)_i.e., Vn = LUn. Proof.Let Xo = [X0 ! . .!.Xn] be an object in Vn. Then each Xi is in V, and so QRF Xi lies in U (as was shown above Lemma A.1). Consider the object QRF Xo = [QRF X0 ! . .!.QRF Xn]. This need not be cofibrant in M n , but we can still take its cofibrant replacement_call this new object Co. Each Ci is* * a cofibrant object weakly equivalent to QRF Xi, and is therefore in U; so Co is in DERIVED EQUIVALENCES 21 Un. We have a sequence of maps LCo ! LQRF Xo ! F Xo Xo, all of which are objectwise weak equivalences, and so Xo is weakly equivalent to an_object in LUn. |__| The category w(Un) is exactly the category wFn(U) defined in Section 3. So there is a `forgetful' functor wSn(U) ! w(Un-1): in the notation of [Wa , 1.3] * *it sends an object {Aij} to the sequence A01 æ A02 æ . .æ.A0n. This functor is easily seen to be an equivalence of categories (see [Wa , bottom of p. 328]). Proof of Proposition 3.6.Recall that K(U) is defined as the geometric realizati* *on of a simplicial space [n] 7! N(wSn(U)). It is therefore enough to show that L induces weak equivalences N(wSn(U)) ! N(wSn(V)) at each level. There is a commutative diagram wSn(U) _____//wSn(V) | | | | |fflffl fflffl| wUn-1 _____//_wVn-1 and the vertical maps are equivalences of categories. So it suffices to show th* *at the maps N(wUn) ! N(wVn) are weak equivalences. But this follows from_LemmanA.2 __ applied to the complete Waldhausen subcategories Un and Vn = LUn of M . |__| References [BN] M. Bökstedt, A. Neeman, Homotopy limits in triangulated categories Composi* *tio Math. 86 (1993), 209-234. [B] F. Borceux. Handbook of categorical algebra II, Categories and structures,* * Cambridge University Press, 1994. [DS] D. Dugger and B. Shipley, Stable module categories and differential graded* * rings, in prepa- ration. [DK80]W. G. Dwyer and D. M. Kan, Function complexes in homotopical algebra, Top* *ology 19 (1980), 427-440. [DwSp]W. G. Dwyer and J. Spalinski, Homotopy theories and model categories, Han* *dbook of algebraic topology (Amsterdam), North-Holland, Amsterdam, 1995, pp. 73-126. [F] P. Freyd, Abelian categories, Harper and Row, New York, 1964. [G] P. Gabriel, Des cate'gories abe'liennes, Bull. Soc. Math. France 90 (1962)* *, 323-448. [Hi] V. Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (199* *7), no. 10, 3291-3323. [Hi] P. Hirschhorn, Model categories and their localizations, 2002 preprint. [Ho] M. Hovey, Model categories, Mathematical Surveys and Monographs, 63, Ameri* *can Math- ematical Society, Providence, RI, 1999, xii+209 pp. [HSS]M. Hovey, B. Shipley, and J. Smith, Symmetric spectra, J. Amer. Math. Soc.* * 13 (2000), 149-208. [Kr] B. Keller, Deriving DG categories, Ann. Sci. 'Ecole Norm. Sup. (4) 27 (199* *4), 63-102. [Ky] G. M. Kelly, Basic concepts of enriched category theory, Cambridge Univ. P* *ress, Cam- bridge, 1982, 245 pp. [N1] A. Neeman, K-theory for triangulated categories I(A): Homological functors* *, Asian J. Math. Vol 1, no. 2 (1997), 330-417. [N2] A. Neeman, Triangulated categories, Annals of Mathematics Studies, 148 Pri* *nceton Uni- versity Press, Princeton, NJ, 2001, viii+449. [Q1] D. G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, 43, Spri* *nger-Verlag, 1967. [Q2] D. G. Quillen, Higher algebraic K-theory. I. Algebraic K-theory, I: Higher* * K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pp. 85-147. * *Lecture Notes in Math., 341, Springer, Berlin 1973. 22 DANIEL DUGGER AND BROOKE SHIPLEY [R1] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2)* * 39 (1989), 436-456. [R2] J. Rickard, Derived equivalences as derived functors, J. London Math. Soc.* * (2) 43 (1991), 37-48. [Se] G. Segal, Classifying spaces and spectral sequences, Inst. Hautes tudes Sc* *i. Publ. Math. 34 (1968), 105-112. [SS1]S. Schwede and B. Shipley, Algebras and modules in monoidal model categori* *es, Proc. London Math. Soc. 80 (2000), 491-511. [SS2]S. Schwede and B. Shipley, Stable model categories are categories of modul* *es, to appear in Topology. http://www.math.purdue.edu/~bshipley/ [SS3]S. Schwede and B. Shipley, Equivalences of monoidal model categories, Prep* *rint, 2002. http://www.math.purdue.edu/~bshipley/ [Sc] M. Schlichting, A note on K-theory and triangulated categories, to appear * *in Inv. Math. http://www.math.uiuc.edu/~mschlich/ [S] B. Shipley, An algebraic model for rational S1-equivariant homotopy theory* *, Quart. J. Math 53 (2002), 87-110. [Sr] V. Srinivas, Algebraic K-theory, Progress in Mathematics 90, Birkhauser, B* *oston, 1996. [TT] R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and o* *f derived categories, The Grothendieck Festschrift, Vol. III, 247-435, Progr. Math.,* * 88, Birkhuser Boston, Boston, MA, 1990. [Wa] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topol* *ogy (New Brunswick, N.J., 1983), 318-419, Lecture Notes in Math., 1126, Springer, B* *erlin, 1985. [We] C. Weibel, An introduction to homological algebra Cambridge Studies in Adv* *anced Math- ematics, 38, Cambridge University Press, Cambridge, 1994, xiv+450 pp. Department of Mathematics, University of Oregon, Eugene, OR 97403, USA E-mail address: ddugger@math.uoregon.edu Department of Mathematics, Purdue University, W. Lafayette, IN 47907, USA E-mail address: bshipley@math.purdue.edu