K-THEORY AND DERIVED EQUIVALENCES

DANIEL DUGGER AND BROOKE SHIPLEY

Abstract. We show that if two rings have equivalent derived categories then they have the same algebraic K-theory. Similar results are given for G-theory, 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 14 7. Many-generators version of proofs 17 Appendix A. Proof of Proposition 3.7 20 References 22

1. Introduction

Algebraic K-theory began as a collection of elaborate invariants for a ring R. Quillen was able to construct these in [Q2] by feeding the category of finitelygenerated projective R-modules into the so-called Q-construction; this showed in particular that K-theory was a Morita invariant of the ring. The Q-construction could take as input any category with a sensible notion of exact sequence, and Waldhausen later generalized things even further. He realized that the same kind of invariants can be defined for a very broad class of homotopical situations (cf. [Wa], which used ‘categories with cofibrations 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 really depends on the whole input category or just on the associated homotopy category, where the weak equivalences have been inverted. In the algebraic case this asks whether the K-theory of a ring depends only on the associated derived category. In this paper we answer the latter question in the affirmative; if one is given the derived category of a ring, 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: October, 2003; 1991 AMS Math. Subj. Class.: 19D99, 18E30, 55U35.

Second author supported in part by an NSF grant.

We now give a more detailed description of the results. If R is a ring, let D_{R} denote the derived category of unbounded chain complexes of R-modules. Recall that D_{R} is a triangulated category in a standard way [Wb, 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 rings for which D_{R} and D_{S} 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 K_{n}(R) is the nth homotopy group of a certain space K(R) produced by one’s favorite K-theory machine. Let D_{c}(R) denote the full subcategory of D_{R} consisting of the perfect complexes—that is, those complexes which are isomorphic in D_{R} to a bounded complex of finitely-generated projectives. (The ‘c’ is for ‘compact’, a term which is defined in Example 3.5).

Theorem B. If R and S are rings such that D_{c}(R) and D_{c}(S) are equivalent as triangulated categories, then K_{n}(R) ≅ K_{n}(S) for all n ≥ 0. Even more, one has a weak equivalence of K-theory spaces K(R) ≃ K(S).

The K_{0} part of this result is very simple (see [R1, 9.3]), and so our contribution is the extension to higher K-theory. We should mention that one can even weaken the hypotheses somewhat, to require only an equivalence between D_{c}(R) and D_{c}(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 Rmodules (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 associated homotopy category by D_{b}(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, G_{n}(R) ≅ G_{n}(S) for all n ≥ 0.

(b) If D_{b}(mod- R) is triangulated-equivalent to D_{b}(mod- S), then G(R) ≃ G(S).

Results like these for K_{0} first appeared in SGA6. For higher K-theory the most notable reference is the work of Neeman [N1], which first appeared in the mid 80s. 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 this in the case of abelian categories (cf. [N1, Thm. 7.1, p336]), and so for instance can construct G(R) from D_{b}(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 above.

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 accentuate is that our proofs of the above theorems are all quite simple. The only ‘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 streamlines 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 3.7), and

(2) If two rings R and S are derived equivalent then tilting theory shows that their 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 3.12. 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 it is not hard to prove (in fact, considering the extra structure in the model category seems to simplify the classical tilting theory proofs). The reason it is surprising 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. Equivalent 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 higher 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 has an unbounded derived category D_{A}, and we’ll say that A and B are derived equivalent if D_{A} is triangulated-equivalent to D_{B}. Let K_{c}(A) denote the Waldhausen K-theory of the compact objects in Ch(A). It turns out that the space K_{c}(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 ⊕_{α} hom_{A}(P, X_{α}) ⟶ hom_{A}(P, ⊕_{α}X_{α}) is a bijection for every set of objects {X_{α}}_{α}. Gabriel [G, V.1] has classified the abelian categories which are equivalent to categories of modules over a ring: these are the co-complete 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 basic 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 Ch(A) and Ch(B) are equivalent as pointed model categories.

(b) If A and B are derived equivalent, then K_{c}(A) ≃ K_{c}(B).

Neeman [N1, 7.1] has proven that if A and B are small abelian categories for which D_{b}(A) is triangulated-equivalent to D_{b}(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 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 abelian 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 possible. The sections previous to that build up the necessary machinery, but with most of the technical proofs postponed until later. Section 3 recasts Waldhausen Ktheory 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 Quillen 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: C_{n} → C_{n−1}, and the shift operator is denoted as ΣC: it is the chain complex with (ΣC)_{n} = C_{n−1}.

Throughout this paper we deal with right modules (and our rings are not necessarily commutative). Everything could be translated to left modules as well, but because of the usual conventions for composing maps, right modules are what naturally 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). Likewise, Proj- R is the category of all projective R-modules and proj- R is the subcategory 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.

After writing this paper we learned of [B], which doesn’t overlap with any of our results but nevertheless deals with similar issues. If the reader enjoys the present paper, we highly recommend [B] as a nice companion.

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 examples: the category of topological spaces, the category of simplicial sets, and the category of chain complexes over a given ring. In this section we recall the basic axioms of model categories, and state the main facts we need in the body of the paper. [DwS], [H] and [Ho] are good references for this material.

Definition 2.1. A model category is a category M equipped with three distinguished classes of maps: the weak equivalences, the cofibrations, and the fibrations. Cofibrations are depicted as ↣, fibrations as ↠, and weak equivalences as ⟶^{∼}.

Maps which are both cofibrations and weak equivalences are called trivial cofibrations, and denoted by ↣^{∼}; trivial fibrations are defined similarly. The following axioms are required: Axiom 1: M is complete and co-complete. 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 third.

Axiom 3: (Retract axiom) A retract of a weak equivalence (respectively cofibration,

fibration) is again a weak equivalence (respectively cofibration, fibration).

Axiom 4: (Lifting axiom) Suppose

[figure: A; X; 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 is 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 ↣^{∼} B ↠ X and as A ↣ Y −↠^{∼} 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 respect to X → Y. In Quillen’s original definition [Q1] the factorizations in Axiom 5 were not required to be functorial; this extra requirement has since become standard, however, as in [H] and [Ho]. It is often possible to get by without this assumption, but this is also usually more trouble than it’s worth.

Example 2.2. In this paper we only deal explicitly with model categories on categories of chain complexes.

(a) The category Ch_{+}(R) of 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 monomorphisms with degreewise projective cokernels; see [Q1, II p. 4.11, Remark 5], [DwS, Sec. 7]. This model structure on Ch_{+}(R) is referred to as the projective model structure since there are other model structures on Ch_{+}(R).

(b) The category Ch(R) of unbounded chain complexes over a ring R also has a (projective) model structure with weak equivalences the homology isomorphisms, and fibrations the epimorphisms; see [Ho, 2.3.11], [SS1, 5]. Every cofibration is still a degreewise split injection and the cokernel is levelwise projective, but not all such degreewise split injections are cofibrations.

(c) Ch(R) has another model structure with the same weak equivalences, but where the cofibrations are the monomorphisms. The fibrations are harder to describe, but any fibration is a degreewise surjection with levelwise-injective kernel. This is the injective model structure on Ch(R). In this paper we only need to use the projective model structure on Ch(R), however.

When M is a model category, one may formally invert the weak equivalences W to obtain the category-theoretic localization W^{−1}M. This is the homotopy category of M, written Ho M; see [Q1, I.1], [DwS, 6.2]. Since the weak equivalences in Ch(R) are the quasi-isomorphisms, the homotopy category Ho Ch(R) is equivalent to the (unbounded) derived category D_{R} (cf. [Wb, 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 Σ. For topological spaces this is ordinary suspension, whereas for Ch_{+}(R) and Ch(R) it is the functor sending a chain complex C to the shift ΣC with (ΣC)_{n} = C_{n−1}. As the example of Ch_{+}(R) shows, this functor need not 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 Ch(R) this of course specializes to the usual triangulation on D_{R}.

Definition 2.3. A Quillen map of model categories M → N consists of a pair of adjoint functors L: M ⇄ N : R such that L preserves cofibrations and trivial cofibrations (it is equivalent to require that R preserves fibrations and trivial 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 ⇄ 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_{+}(S) and Ch(R) → Ch(S).

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 functors are adjoint equivalences of the homotopy categories. This is equivalent to Quillen’s original definition by [Ho, 1.3.13]. More generally we say that M and N are Quillen equivalent if they are connected by a zig-zag of Quillen equivalences, and we write M ≃_{Q} N. As one simple example, the identity functors give a Quillen equivalence Ch^{proj}(R) → Ch^{inj}(R) between the projective and injective model structures on Ch(R). Two pointed model categories are ∗Quillen equivalent if they are connected by a zig-zag of Quillen equivalences between pointed model categories.

Remark 2.5. In general, having a Quillen equivalence of model categories is much stronger than just having an equivalence between the associated homotopy categories. This is because of the added structure required for a Quillen map; functors 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 example, 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 structures [DK, 5.4], [Ho, 5.6.2]. There are simple topological examples—see [SS2, 3.2.1], for instance—of stable model categories which have the same triangulated homotopy category, but nevertheless are not Quillen equivalent. In Remark 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 Ch(R), Quillen equivalence is not a stronger notion than triangulated equivalence of homotopy categories. In some sense this happens because rings are determined by ‘first order’ information—compared, for example, to differential graded rings which 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 associated model categories of chain complexes Ch(R) and Ch(S) are ∗Quillen equivalent.

This theorem cannot be extended to cover the case where R or S is a differential graded algebra; we give an example in [DS] which is discussed briefly in Remark 6.8. In Corollary 7.7 we do give a certain extension of this theorem to abelian categories, however. The situation is a little confusing, because these two sentences may seem contradictory; Remark 7.8 explains why they are not.

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 advantage of streamlining the theory somewhat.

Let M be a pointed model category with initial object ∗. An object A is called cofibrant if ∗ ↣ 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 ↣ B and A → X are maps in U, then the pushout B ⨿_{A} X (computed

in M) belongs to U. The proof of the following is just a matter of chasing through the definitions:

Lemma 3.1. Any Waldhausen subcategory of M, equipped with the notions of cofibrations 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 axiom [Wa, p. 327].

The lemma says that we may apply Waldhausen’s S_{•}-construction [Wa, 1.3] to obtain a simplicial category wS_{•}(U). Taking the nerve in every dimension gives a simplicial space [n] ↦ N(wS_{n}(U)), and K(U) is defined to be loops on the realization of this simplicial space: K(U) = Ω|NwS_{•}(U)|. One defines the algebraic K-groups of U by K_{n}(U) = π_{n}(K(U)).

Remark 3.2. There are potential set-theoretic difficulties if the Waldhausen subcategory is not small, and this is particularly a problem with the ‘complete’ Waldhausen subcategories that we work with below. Fortunately it is never really a problem—we refer the reader to the nice Remark in [Wa, p. 379]. Waldhausen’s solution (a), which works in all cases of interest in this paper, can be implemented by assuming the model category is combinatorial, and by putting a bound on the size of the objects in the Waldhausen subcategory. Information about combinatorial model categories can be found in [D, Sections 2,7].

We give a partial description of wS_{•}(U) here, because we need it later in Appendix A. Let wF_{n}(U) denote the category whose objects are sequences {A} of cofibrations A_{0} ↣ A_{1} ↣ ⋯ ↣ A_{n} in U, and whose morphisms are commutative diagrams {A} → {A′} in which every map A_{n} → A′_{n} is a weak equivalence. One can almost make [n] ↦ wF_{n−1}(U) into a simplicial category (where wF_{−1}(U) is interpreted as the trivial category with one object and an identity map) by defining

d_{i}([A_{0} ↣ A_{1} ↣ ⋯ ↣ A_{n}]) = \begin{cases}[A_{0} ↣ ⋯ ↣ Â_{i} ↣ ⋯ ↣ A_{n}] & if i ≠ 0, \\ [A_{1}/A_{0} ↣ A_{2}/A_{0} ↣ ⋯ ↣ A_{n}/A_{0}] & if i = 0. \end{cases}

The difficulty is that with this definition the simplicial identities do not hold on the nose, in the end because there are different possible choices for the quotients A_{i}/A_{0} (they are canonically isomorphic, but still different). The category wS_{n}(U) is equivalent to wF_{n−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 basic ideas in the present paper can all be understood by pretending that wS_{n}(U) is just wF_{n−1}(U). The only time the details of wS_{n}(U) are needed is in the Appendix.

Example 3.3. Let R be a ring. The following are Waldhausen subcategories of Ch(R) (as is easily verified).

(1) U_{K} = {all bounded complexes of finitely-generated projectives}.

(2) U_{G} = {all bounded below complexes C of finitely-generated projectives such that H_{k}(C) ≠ 0 for only finitely-many values of k}.

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 3.4. K(U_{K}) ≃ K(R), and if R is Noetherian then K(U_{G}) ≃ G(R).

Proof. A reference for K(U_{K}) ≃ 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 Ch(R) 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 Ch(R) consisting of all chain complexes quasiisomorphic to an element of V. One can check that if R is Noetherian then U_{G} consists precisely of the cofibrant objects in W. Then [TT, 1.9.8] shows that U_{G} ↪ W and V ↪ W induce equivalences of K-theory spaces. □

Example 3.5. If T is a triangulated category with infinite coproducts (usually called infinite sums in this context), an object X ∈ T is called compact if the natural map ⊕_{α}T (X, Z_{α}) → T (X, ⊕_{α}Z_{α}) is a bijection for every set {Z_{α} ∈ T }. If M is a stable model category, it is easy to check 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 M_{c} ⊆ M consisting of all compact, cofibrant objects is a complete Waldhausen category.

We are especially interested in this for the case M = Ch(R), where a theorem of Bökstedt-Neeman [BN, 6.4] identifies the compact objects as the perfect complexes, i.e. the complexes which are quasi-isomorphic to a bounded complex of finitelygenerated projectives.

Example 3.6. 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 structure inherited from the category of simplicial sets [Q1, II.3] by forgetting the retraction over X (cf. [H, 7.6.5]). Take U to be the subcategory consisting of those retractive 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 \overline{U} for the full subcategory of M consisting of all cofibrant objects which are weakly equivalent to an object in U. From Example 3.5 above it follows that (Ch(R))_{c} = \overline{U_{K}} (where U_{K} is from Example 3.3). Call the Waldhausen category U complete if U = \overline{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-defined map K(U) → K(V).

Proposition 3.7. Suppose that (L, R) is a Quillen equivalence, and that U is a complete Waldhausen subcategory of M. Let V = \overline{LU}; that is, V consists of all cofibrant objects which are weakly equivalent to an object in L(U). Then V is a complete 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. It is worth noting that, since the original appearance of this paper, the above proposition was given an interesting generalization in [TV, Prop. 1.1].

Remark 3.8. The proposition also works in the following way. Let Q be a cofibrant replacement functor for M; for example one can take the map ∗ → X and apply the functorial factorization ∗ ↣ QX −↠^{∼} X in M to define Q. Similarly, let F be a fibrant-replacement functor for N with Y ↣^{∼} F Y ↠ ∗ 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 ∈ V, and let U = \overline{RV}. Then U is a complete Waldhausen subcategory of M, and \overline{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 3.9. Let M and N be pointed 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 subcategory of N, and there is an induced zig-zag of weak equivalences between K(U) and K(V).

Corollary 3.10. A Quillen equivalence M → N between stable model categories induces a weak equivalence of K-theory spaces K(M_{c}) ⟶^{∼} K(N_{c}), where M_{c} and N_{c} denote the subcategories of cofibrant, compact objects.

Proof. Write the functors of the Quillen equivalence as (L, R). The derived functors 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 N_{c} ⊇ \overline{LM_{c}}; it basically gives the opposite inclusion as well, but we now explain this in more detail.

If X is in N_{c}, let F X be a fibrant-replacement X ↣^{∼} F X ↠ ∗ in N and let Q(RF X) be a cofibrant-replacement ∗ ↣ Q(RF X) −↠^{∼} 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 ∈ M_{c}. Yet LQRF X is weakly equivalent to X, and so X ∈ \overline{LM_{c}}. At this point we have shown N_{c} = \overline{LM_{c}}, and so we can just apply Proposition 3.7. □

Corollary 3.11. If Ch(R) and Ch(S) 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(U_{K}), and U_{K} = (Ch(R))_{c}. All the intermediate model categories in the zig-zag must be stable because ‘stability’ is preserved under ∗Quillen equivalence. Therefore Corollary 3.10 applies. □

Remark 3.12. The above corollary has two improvements over similar results in the literature. The first is that we are allowing a zig-zag of ∗Quillen equivalences, rather than just an equivalence Ch(R) → Ch(S); in particular, note that our zigzag could conceivably pass through very non-algebraic model categories. For just a single Quillen equivalence Ch(R) → Ch(S), the closest result in the literature seems to be [TT, 1.9.8]. In that result, however, the functor L: Ch(R) → Ch(S) is required to be complicial, meaning in part that it is induced via prolongation from a functor 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 the [TT] result is not applicable.

4. Tilting Theory

In this section we determine the algebraic content of having a Quillen equivalence between Ch(R) and Ch(S) for rings R and S. A nice, complete answer can be given in terms of tilting theory. Originally tilting theory only dealt with derived equivalences, but it turns out that for rings derived equivalence and Quillen equivalence coincide.

We begin with a classical analogue of tilting theory, namely Morita theory. Morita theory describes necessary and sufficient conditions for when two categories of modules are equivalent. Call a (right) R-module P a strong generator if hom_{R}(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 [Wb, 9.5]. For (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 Rhom_{R}(P, P) bimodule. □

Now we turn to the analogue of Morita theory for categories of chain complexes, 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 tilting theory to apply to more general derived equivalences of abelian categories. We extend both sets of results to give Quillen equivalences underlying the derived equivalences. 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 under the suspension functor, and (iii) has the property that if two objects of a distinguished 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 coproducts of sets of objects [N2, 1.5.1, 3.2.6]. An object P in T is a (weak) generator if the only localizing subcategory of T which contains P is T . Although this definition looks much different than the definition of a strong generator, it is not. If P is compact (see Example 3.5 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 these are equivalent). Here T (−, −)_{∗} denotes the graded maps with T (X, Y)_{n} = T (Σ^{n}X, Y).

An object P ∈ Ch(R) is called a tilting complex if it is a bounded complex of finitely-generated projectives, a generator of D_{R}, and D_{R}(P, P)_{∗} is concentrated in degree zero [R1, Def. 6.5]. Here is our generalization of Rickard’s result [R1, Thm 6.4]:

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:

Ch(R) ≃_{Q} Ch(S).

(2) The unbounded derived categories are triangulated equivalent:

D_{R} ≃_{Δ} D_{S}.

(3) The naive homotopy categories of bounded chain complexes of finitely generated projective R and S-modules are triangulated equivalent:

K_{b}(proj- R) ≃_{Δ} K_{b}(proj- S).

(4) The model category Ch(R) has a tilting complex P whose endomorphism ring in D_{R} is isomorphic to S: D_{R}(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 D_{b}(Mod- R) ≃ D_{b}(Mod- S). He defined two rings to be ‘derived equivalent’ if any of these conditions holds. We defined ‘derived equivalent’ to mean (2), and so the result shows that our use agrees with Rickard’s. Note that [R1, 6.4] gives several other equivalent 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 categories induces an equivalence of triangulated homotopy categories [Q1, I.4 Theorem 3], so (1) implies (2). Any triangulated equivalence restricts to an equivalence between the respective subcategories of compact objects. Since K_{b}(proj-R) is equivalent to the full subcategory of compact objects in D_{R} by [BN, Prop. 6.4], (2) implies (3).

Now we assume condition (3) and choose a triangulated equivalence between K_{b}(proj-R) and K_{b}(proj-S). Let S[0] be the free S-module on one generator, viewed as a complex in Ch(S) concentrated in dimension zero; let T be its image in K_{b}(proj- R). We have D_{R}(T, T) ≅ D_{S}(S[0], S[0]) ≅ S. Since S[0] generates K_{b}(proj-S), T generates K_{b}(proj-R). Since R[0] is a generator of D_{R} and R[0] ∈ K_{b}(proj- R), the only localizing subcategory of D_{R} containing K_{b}(proj- R) is D_{R}; so T generates D_{R}. Hence T is a tilting complex and condition (4) holds. □

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 required 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 trivial 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).

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 removed 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 symmetric 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 first three theorems cited in the introduction.

Proof of Theorem B. This follows from Corollary 3.11 together with the equivalence of Parts 1 and 3 in Theorem 4.2. Note that D_{c}(R) and K_{b}(proj- R) are two names for the same thing, by [BN, 6.4]. □

Proof of Theorem A. If D_{R} and D_{S} are equivalent as triangulated categories, then so are their full subcategories of compact objects. So Theorem B applies. This also follows from Corollary 3.11 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 right. Consider a function C which assigns to each ring R a subcategory of D_{R}. We say that the assignment preserves equivalences if every triangulated equivalence F: D_{R} → D_{S} restricts to an equivalence between C(R) and C(S).

Here is some new notation: D_{h+}(Mod- R) denotes the full subcategory of D_{R} consisting of chain complexes with bounded below homology, and D_{hb}(Mod- R) denotes the full subcategory of complexes with bounded homology. One can similarly define K_{hb}(proj- R), etc. The notation K_{+,hb}(proj- R) means the intersection of K_{+}(proj- R) and K_{hb}(proj- R). It is an easy exercise to check that D_{h+}(Mod- R) = K_{+}(Proj- R) and D_{hb}(Mod- R) = D_{b}(Mod- R).

Proposition 5.1. The assignments R ↦ C(R) preserve equivalences, where C(R) is any of the following:

K_{b}(proj- R), K_{+}(Proj- R) = D_{h+}(Mod- R), D_{h−}(Mod- R),
D_{hb}(Mod- R) = D_{b}(Mod- R), K_{+}(proj- R), K_{+,hb}(proj- R).

Proof. The result [BN, 6.4] identifies K_{b}(proj- R) with the subcategory of compact objects in D_{R}. Any equivalence D_{R} → D_{S} must preserve direct sums, and so it takes compact objects to compact objects.

A complex X lies in D_{h+}(Mod- R) if and only if it satisfies the following property: for any compact object A, there exists an N such that D_{R}(Σ^{−k}A, X) = 0 for k > N. Since triangulated equivalences preserve compact objects and the suspension, they preserve these objects as well.

Similarly, a complex X lies in D_{h−}(Mod- R) if and only if for any compact object A, there exists an N such that D_{R}(Σ^{k}A, X) = 0 for all k > N. The same argument as above applies. For D_{hb}(Mod- R), note that this is just the intersection of D_{h+}(Mod- R) and D_{h−}(Mod- R).

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 intersection of K_{+}(proj- R) and D_{hb}(Mod- R). □

K_{+,hb}(proj- R) is the full subcategory of D_{R} 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 D_{c}(R) ⊆ K_{+,hb}(proj- R) ⊆ D_{R}. Note that K_{+,hb}(proj- R) is the image in D_{R} of the Waldhausen subcategory U_{G}(R) ⊆ Ch(R). It is an easy exercise to check that when R is right-Noetherian one has K_{+,hb}(proj- R) = D_{b}(mod- R), where the latter denotes the full subcategory of D_{R} consisting of the bounded complexes of finitely-generated modules.

Theorem C follows immediately from the following more comprehensive statement:

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 D_{b}(mod- R) ≃_{Δ} D_{b}(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 uses 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 Ch(R) and Ch(S). On the homotopy categories, this gives us a chain of triangulated equivalences between D_{R} and D_{S}. Proposition 5.1 says that this triangulated equivalence between D_{R} and D_{S} restricts to an equivalence between K_{+,hb}(proj- R) and K_{+,hb}(proj- S). So the complete Waldhausen subcategory U_{G}(R) is carried to U_{G}(S) via the various adjoint functors in the chain of Quillen equivalences. One can now use Corollary 3.9 to deduce that K(U_{G}(R)) ≃ K(U_{G}(S)). That is, G(R) ≃ G(S).

For (c), recall that when R is Noetherian D_{b}(mod- R) is just another name for K_{+,hb}(proj- R), and the same for S. So if D_{b}(mod- R) ≃ D_{b}(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 from [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- hom_{A}(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 Freyd; see Section 7.

We begin by defining a chain complex of morphisms between any two chain complexes. For M, N in Ch(R) define Hom_{R}(M, N) in Ch(ℤ) by

Hom_{R}(M, N)_{n} = ∏_{k} hom_{R}(M_{k}, N_{n+k}).

The differential for Hom_{R}(M, N) is given by df_{n} = d_{N}f_{n} + (−1)^{n+1}f_{n}d_{M}. This structure gives an enrichment of Ch(R) over Ch(ℤ). So instead of an endomorphism ring, an object in Ch(R) has a differential graded ring of endomorphisms.

Definition 6.2. The tensor product of X and Y in Ch(ℤ) is defined by

(X ⊗ Y)_{n} = ⨁_{k} X_{k} ⊗ Y_{n−k}

where d(x_{p} ⊗ y_{q}) = dx_{p} ⊗ y_{q} + (−1)^{p}x_{p} ⊗ dy_{q}. A differential graded algebra is a chain complex A in Ch(ℤ) with a multiplication μ: A ⊗ A ⟶ A which is associative and unital [Wb, 4.5.2]. A (right) differential graded module M over a differential graded algebra A is a chain complex M with an associative and unital action α: M ⊗ A ⟶ A. Denote the category of such modules by Mod-A.

For any P in Ch(R) let End_{R}(P) = Hom_{R}(P, P). Notice that End_{R}(P) is a differential graded ring with the product structure coming from composition. Also, for any X ∈ Ch(R) the complex Hom_{R}(P, X) is a right differential graded End_{R}(P)- module with the action given by precomposition. So Hom_{R}(P, −) induces a functor from Ch(R) to Mod- End_{R}(P). Its left adjoint is denoted − ⊗_{End_{R}(P)} P. This left adjoint can be defined as the coequalizer that the notation suggests using the evaluation map Hom_{R}(P, P) ⊗ P ⟶ P.

Our differential graded analogue of Gabriel’s theorem produces a Quillen equivalence of model categories instead of an equivalence of categories. So before stating it we need to establish the model category structure on a category of differential 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 homology and the fibrations are the surjections. The cofibrations are then determined to be the maps with the left-lifting-property with respect to the trivial fibrations.

We can now state the following differential graded version of Gabriel’s theorem.

Theorem 6.4. Let P in Ch(R) be a bounded complex of finitely generated projectives. If P is a (weak) generator for Ch(R), then there is a Quillen equivalence

Mod- End_{R}(P) ⟶ Ch(R)

in which the right-adjoint is the functor Hom_{R}(P, −).

Before proving this theorem we need the following lemma.

Lemma 6.5. Let M, N ∈ Ch(R). Then H_{∗} Hom_{R}(M, N) ≅ D_{R}(M, N)_{∗} when M is cofibrant.

Proof. It is easy to see in general that H_{n} Hom_{R}(M, N) ≅ H_{0} Hom_{R}(Σ^{n}M, N) ≅ [Σ^{n}M, N] where [−, −] denotes chain-homotopy-classes of maps. When A is cofibrant one has that D_{R}(A, B) ≅ [A, B] (since all objects are fibrant in Ch(R)), and so we can write

H_{n} Hom_{R}(M, N) ≅ [Σ^{n}M, N] ≅ D_{R}(Σ^{n}M, N) = D_{R}(M, N)_{n}.
□

Proof of Theorem 6.4. For any complex of projectives P, Hom_{R}(P, −) preserves surjections (fibrations) and hence is exact. We next show that Hom_{R}(P, −) preserves trivial fibrations; since Hom_{R}(P, −) is exact, we only need to show that H_{∗} Hom_{R}(P, K) = 0 when H_{∗}K = 0 and apply this to the kernel K of the trivial fibration. P is cofibrant by [Ho, 2.3.6] because P is a bounded complex of projectives. Thus, by Lemma 6.5, if K is acyclic then H_{∗} Hom_{R}(P, K) ≅ D_{R}(P, K)_{∗} ≅ 0. Hence, the functor Hom_{R}(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 induces total derived functors on the level of homotopy categories [Q1, I.4]. Denote these derived functors by RHom_{R}(P, −) and − ⊗_{End_{R}(P)}^{L} P respectively.

Since Ch(R) and Mod- End_{R}(P) are stable model categories, both total derived functors preserve shifts and triangles in the homotopy categories, i.e., they are exact functors of triangulated categories by [Q1, I.4 Prop. 2]. Since −⊗_{End_{R}(P)}^{L} P is a left adjoint it commutes with coproducts. To see that RHom_{R}(P, −) commutes with coproducts it is enough to show that D_{End_{R}(P)}(End_{R}(P), RHom_{R}(P, −)) commutes with coproducts since End_{R}(P) is a compact generator of Mod- End_{R}(P). By adjointness, this functor is isomorphic to D_{R}(End_{R}(P)⊗_{End_{R}(P)}^{L}P, −) which in turn is isomorphic to D_{R}(P, −) since End_{R}(P) is cofibrant. Since P is compact [BN, 6.4] this functor commutes with coproducts.

Now consider the full subcategories of those M in Ho(Mod- End_{R}(P)) and X in D_{R} respectively for which the unit of the adjunction

η : M ⟶ RHom_{R}(P, M ⊗_{End_{R}(P)}^{L} P)

or the counit of the adjunction

ν : RHom_{R}(P, X) ⊗_{End_{R}(P)}^{L} X ⟶ X

are isomorphisms. Since both derived functors are exact and preserve coproducts, these are localizing subcategories. The map η is an isomorphism on the free module End_{R}(P) and the map ν is an isomorphism on P. Since the free module End_{R}(P) generates the homotopy category of End_{R}(P)-modules and P generates Ch(R), 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 H_{k}(A) ≅ H_{k}(R) for all k. (That is, if and only if H_{k}(A) = 0 for k ≠ 0 and H_{0}(A) ≅ R.)

Proof. Given H_{k}(A) ≅ H_{k}(R) for all k, then there are quasi-isomorphisms of DGAs A ⟵ A⟨0⟩ ⟶ H_{0}(A) ≅ R. Here A⟨0⟩ is the (-1)-connected cover of A with A⟨0⟩_{k} = 0 for k < 0, A⟨0⟩_{k} = A_{k} for k > 0 and A⟨0⟩_{0} = Z_{0}A the zero cycles. □

Proposition 6.7. Any quasi-isomorphism A ⟶ B of differential graded algebras induces a Quillen equivalence Mod-A → Mod- B.

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 scalars and the left adjoint is −⊗_{A} B. [SS1, 4.3] and [Hi, 3.3.1] both show that this adjoint pair is a Quillen equivalence. □

Completion of the proof of Theorem 4.2. We must show that (4) ⇒ (1), so suppose that Ch(R) has a tilting complex T. Then T satisfies the hypotheses of Theorem 6.4, hence Ch(R) is ∗Quillen equivalent to the category of modules over the differential graded algebra End_{R}(T). Since T is a bounded complex of projectives, it is cofibrant by [Ho, 2.3.6]; hence from Lemma 6.5 we have H_{∗} End_{R}(T) ≅ D_{R}(T, T)_{∗} ≅ S concentrated in dimension zero. By Lemma 6.6 this implies that End_{R}(T) is quasiisomorphic to S. Thus the categories of End_{R}(T)-modules and right differential graded S-modules (Ch(S)) are ∗Quillen equivalent by Proposition 6.7:

Ch(R) ≃_{Q} Mod- End(T) ≃_{Q} Ch(S).
□

Remark 6.8. We have now shown that when R and S are rings, their model categories of dg-modules are ∗Quillen equivalent if and only if the associated homotopy 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 derived equivalent DGAs whose model categories of dg-modules are not Quillen equivalent. Fixing a prime p, the first DGA is ℤ⟨e, x⟩/(ex + xe = x^{2}, e^{2} = 0) where e, x both have degree 1, and de = p, dx = 0. The second DGA is ℤ/p[x, x^{−1}], where again x has degree 1 and dx = 0. For a proof that these are derived equivalent, see [DS].

The example is based on [Sc] which considers model categories underlying the stable category of modules over the Frobenius rings R = ℤ/p^{2} and R′ = (ℤ/p)[ϵ]/ϵ^{2}. The homotopy categories are triangulated equivalent, but when p is odd the corresponding K-theory groups are non-isomorphic at K_{4} (when p = 2 it turns out they are actually non-isomorphic at K_{3}; we thank W. van der Kallen for referring us to the relevant computations). So by Corollary 3.10 these model categories cannot be Quillen equivalent. In [DS] we give a simpler proof of this by studying certain endomorphism DGAs (which turn out to be the two specified above), where the difference can be detected in the second Postnikov sections instead of in K_{4}.

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 categories is in [F, 5.3H]. For derived equivalences, Keller [Kr, 8.2] gave the corresponding extension of Rickard’s work [R1, 6.4]. As always, our purpose is just to upgrade the derived equivalences to Quillen equivalences.

As in Section 6, before moving to a differential graded setting we first recall the classical setting. Define a ring with many objects to be a small Ab-category (a category enriched over abelian groups); this terminology makes sense because an Ab-category with one object corresponds to a ring, with composition corresponding to the ring multiplication. Given a ring with many objects \underline{R}, a (right) \underline{R}-module M is a contravariant additive functor from \underline{R} to Ab. This means that for any two objects P, P′ in \underline{R} there are maps M(P′) ⊗ \underline{R}(P, P′) ⟶ M(P). The category of right \underline{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 \underline{End}_{A}(P) to be the full subcategory of A (enriched over Ab) with object set P. The following theorem from [F, 5.3H] classifies abelian categories with a set of strong 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- \underline{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 = Ch(ℤ).) Since a Ch-category with one object is a differential graded algebra, one may think of 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′ of R there is a structure map of chain complexes M(P′) ⊗ R(P, P′) ⟶ M(P). See [Ky, 1.2] or [Bo, 6.2] for more details.

Notice that Ch(R) and Ch(\underline{R}) are both Ch-categories, where R is a ring and \underline{R} is a ring with many objects. The enrichment of Ch(R) over Ch was discussed in the previous section. Since any two \underline{R}-modules have an abelian group of morphisms hom\underline{R}(M, N), the enrichment for Ch(\underline{R}) 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′) = Hom_{C}(P, P′). We let

Hom_{C}(P, −) : C ⟶ Mod- E(P)

denote the functor given by Hom_{C}(P, Y)(P) = Hom_{C}(P, Y).

Note that if P = {P } has a single element, then E(P) is determined by the single differential graded ring End_{C}(P) = Hom_{C}(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 difference 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 subcategory 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 detect when an object is trivial; see [SS2, 2.2.1]. Note that a (possibly infinite) coproduct of a set of generators is still a generator, but is not necessarily compact.

Theorem 7.3. Let \underline{R} be a ring with many objects and P a set in Ch(\underline{R}) of bounded complexes of finitely generated projectives. If P is a set of (weak) generators for Ch(\underline{R}) then there is a Quillen equivalence

Mod- E(P) ⟶ Ch(\underline{R})

in which the right adjoint is the functor Hom\underline{R}(P, −).

Note that for every object r ∈ \underline{R} there is a corresponding ‘free module’ F \underline{R} ; r given by F \underline{R} ; r(s) = \underline{R}(s, r). A projective \underline{R}-module is finitely-generated if it is a direct summand of a module ⊕_{i}F \underline{R} ; r_{i}, where the sum is finite. And as usual, we denote the homotopy category of Ch(\underline{R}) by D\underline{R}. We need the following lemma:

Lemma 7.4. The compact objects in D\underline{R} are those complexes which are quasiisomorphic to a bounded complex of finitely-generated projective \underline{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\underline{R}(P, −) takes fibrations and trivial fibrations in Ch(\underline{R}) to fibrations and trivial fibrations in Ch for any bounded complex of projectives P. So the functor Hom\underline{R}(P, −) preserves fibrations and trivial fibrations. Thus, together with its 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 coproducts. 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 η or the counit of the adjunction ν are isomorphisms are localizing subcategories.

Note that for each object P in P there is a free E(P)-module F_{P}^{E(P)} defined by F_{P}^{E(P)}(P′) = E(P)(P′, P), and these generate the homotopy category of E(P)- modules. For every P ∈ P the E(P)-module Hom\underline{R}(P, P) is isomorphic to the free module F_{P}^{E(P)} by inspection, and F_{P}^{E(P)} ⊗_{E(P)} P is isomorphic to P since they represent the same functor on Ch(\underline{R}). Thus, η is an isomorphism on every free module and ν is an isomorphism on every object of P. Since the free modules F_{P}^{E(P)} generate the homotopy category of E(P)-modules and the objects of P generate Ch(\underline{R}), the localizing subcategories where η and ν 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 set 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 \underline{R} and \underline{S}. The tilting complex is replaced by a set of tiltors 𝕋 with H_{∗}E(𝕋) ≅ \underline{S}.

The proof is given below, but first we state some easy consequences:

Corollary 7.6. Two rings-with-many-objects \underline{R} and \underline{S} are derived equivalent if and only if their associated model categories of chain complexes Ch(\underline{R}) and Ch(\underline{S}) are Quillen equivalent.

Using Theorem 7.1 we get the following corollary as well. Given an abelian category A satisfying the hypotheses of Theorem 7.1, choose a set of small, projective, strong generators P. Let \underline{A} = \underline{End}_{A}(P) be the associated ring-with-many-objects. Freyd’s theorem says that A is equivalent to Mod- \underline{A}, and so Ch(A) is equivalent to Ch(\underline{A}). In particular, one gets a projective model structure on Ch(A) by lifting the one on Ch(\underline{A}) across the equivalence; see [SS3, 6.1]. The next result is now an immediate 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 if their associated model categories of chain complexes Ch(A) and Ch(B) are ∗Quillen equivalent.

Remark 7.8. Warning: Let M and N be two stable model categories whose underlying categories are abelian, with sets of small, strong, projective generators. 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 false; see [Sc], [DS]. Note in particular that it does not apply to the model category Mod- R where R is a DGA; the problem is that Ho(Mod- R) is not the same as Ho(Ch(Mod- R)).

Proof of Theorem D. Part (a) is the above corollary. Part (b) is immediate from (a) and Corollary 3.10. □

Proof of Theorem 7.5. The proof that condition (1) implies condition (2) and condition (2) implies condition (3) follows just as in Theorem 4.2.

Now assume condition (3) and fix a triangulated equivalence between K_{b}(proj-\underline{R}) and K_{b}(proj-\underline{S}). For s any object in \underline{S}, consider the module F \underline{S} ; s as a complex concentrated in dimension zero; let T_{s} be its image in K_{b}(proj-\underline{R}). Since the objects in \{F_{s}\}_{s∈\underline{S}} generate K_{b}(proj-\underline{S}), the objects in 𝕋 = \{T_{s}\}_{s∈\underline{S}} generate K_{b}(proj-\underline{R}). But K_{b}(proj-\underline{R}) generates D_{R}, so 𝕋 generates D_{R} as well. By Lemma 7.4 the objects of 𝕋 are compact in D_{R}. Finally, we also have H_{∗}E(𝕋) ≅ H_{∗}E(\{F \underline{S} ; s\}) ≅ \underline{S}. So 𝕋 is a set of tiltors.

If we are given a set of tiltors 𝕋 for Ch(\underline{R}), then by Theorem 7.3 Ch(\underline{R}) is Quillen equivalent to the category of modules over the endomorphism category E(𝕋). Since H_{∗}E(𝕋) ≅ \underline{S} is concentrated in dimension zero, E(𝕋) is quasi-isomorphic to \underline{S} by an extension of Lemma 6.6. Thus the categories of differential graded E(𝕋)- modules and differential graded \underline{S}-modules are Quillen equivalent by [SS3, 6.1] which generalizes Proposition 6.7:

Ch(\underline{R}) ≃_{Q} Mod- E(𝕋) ≃_{Q} Ch(\underline{S})

□

Appendix A. Proof of Proposition 3.7

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 = \overline{(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 ∈ V then X ≃ LA for some A ∈ U, and then QRF X ≃ QRF LA ≃ A. Since U is complete and A ∈ U, it follows that QRF X ∈ 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 ↣ B and A → X are maps in N where A, B, X ∈ V, we must show that the pushout B ⨿_{A} X is also in V.

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 ↣ Z −↠^{∼} QRF B. Then Z ∈ U and so the pushout P = Z ⨿_{QRF A} QRF X is also in U, because U is a Waldhausen subcategory of M. This pushout is weakly equivalent to the homotopy pushout (see [DwS, 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 cofibrant; see [Ho, 5.2.6]. Since Z −↠^{∼} 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 ⨿_{A} X (since A → B is a cofibration and all the objects A, B, X are cofibrant). So B ⨿_{A} X is weakly equivalent to LP, and is therefore in \overline{LU}. □

Let wU denote the subcategory consisting of all weak equivalences in U, and write 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 complete), and comes equipped with a natural weak equivalence QX → X. This shows that the induced map NQ: N(wU) → N(wU) is homotopic to the identity [Se, 2.1]. Similarly, NF: N(wV) → N(wV) is homotopic to the identity.

The functor QRF: N → M maps V to U, as was remarked prior to the previous lemma. There are natural transformations LQRF → LRF → F and Q → QRL → QRF L, and the composites are weak equivalences. It follows that the compositions NL ∘ N(QRF) and N(QRF) ∘ NL are homotopic to the respective identity maps, and so are part of a homotopy equivalence. □

Let Δ_{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 the maps increase dimension. The category of diagrams M^{Δ_{n}} has a corresponding Reedy model structure [Ho, 5.2.5] in which a map X_{•} → Y_{•} is a weak equivalence (respectively fibration) if and only if each X_{n} → Y_{n} is a weak equivalence (respectively fibration). A map is a cofibration if and only if all the maps X_{n} ⨿_{X_{n−1}} Y_{n−1} → Y_{n} are cofibrations. In particular, an object X_{•} is cofibrant if and only if the maps X_{n−1} → X_{n} are all cofibrations; by a simple recursion, this implies that all the X_{i}’s are cofibrant as well.

Let U_{n} denote the full subcategory of M^{Δ_{n}} consisting of cofibrant diagrams whose objects all belong to U. It is easy to see that U_{n} is a complete Waldhausen subcategory of M^{Δ_{n}}. The functors (L, R) prolong to functors (L, R): M^{Δ_{n}} → N^{Δ_{n}}, and this is still a Quillen equivalence. We need the following:

Lemma A.3. Any diagram in V_{n} is weakly equivalent to one in L(U_{n}). That is, V_{n} = \overline{LU_{n}}.

Proof. Let X_{•} = [X_{0} → ⋯ → X_{n}] be an object in V_{n}. Then each X_{i} is in V, and so QRF X_{i} lies in U (as was shown above Lemma A.1). Consider the object QRF X_{•} = [QRF X_{0} → ⋯ → QRF X_{n}]. This need not be cofibrant in M^{Δ_{n}}, but we can still take its cofibrant replacement—call this new object C_{•}. Each C_{i} is a cofibrant object weakly equivalent to QRF X_{i}, and is therefore in U; so C_{•} is in U_{n}. We have a sequence of maps LC_{•} → LQRF X_{•} → F X_{•} ← X_{•}, all of which are objectwise weak equivalences, and so X_{•} is weakly equivalent to an object in LU_{n}. □

The category w(U_{n}) is exactly the category wF_{n}(U) defined in Section 3. So there is a ‘forgetful’ functor wS_{n}(U) → w(U_{n−1}): in the notation of [Wa, 1.3] it sends an object {A_{ij}} to the sequence A_{01} ↣ A_{02} ↣ ⋯ ↣ A_{0n}. This functor is easily seen to be an equivalence of categories (see [Wa, bottom of p. 328]).

Proof of Proposition 3.7. Recall that K(U) is defined as the geometric realization of a simplicial space [n] ↦ N(wS_{n}(U)). It is therefore enough to show that L induces weak equivalences N(wS_{n}(U)) → N(wS_{n}(V)) at each level. There is a commutative diagram

\xymatrix{wS_{n}(U) \ar[d] \ar[r] & wS_{n}(V) \ar[d] \\ wU_{n−1} \ar[r] & wV_{n−1}}

and the vertical maps are equivalences of categories. So it suffices to show that the maps N(wU_{n}) → N(wV_{n}) are weak equivalences. But this follows from Lemma A.2 applied to the complete Waldhausen subcategories U_{n} and V_{n} = \overline{LU_{n}} of M^{Δ^{n}}. □

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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