KTHEORY 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 Ktheory. Similar results are given for Gt*
*heory,
and for a large class of abelian categories.
Contents
1. Introduction 1
2. Model category preliminaries 4
3. Ktheory 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 Ktheory began as a collection of elaborate invariants for a ring R.
Quillen [Q2 ] constructed these by feeding the category of finitelygenerated p*
*rojec
tive Rmodules into the socalled Qconstruction. In fact, the Qconstruction 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 Ktheory of a ring using*
* the
Waldhausen approach, one takes as input the category of bounded chain complexes
of finitelygenerated projective modules.
As soon as one understands this perspective it becomes natural to ask whether
the Waldhausen Ktheory 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 Ktheory 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 Ktheory 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 Rmodules. Recall
that DR is a triangulated category in a standard way [We , 10.4]. Also, let K*(*
*R)
denote the algebraic Kgroups 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 Kgroups 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
Ktheories. (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 Ktheory
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 finitelygenerated 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 Ktheory 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 Ktheory. 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 Gtheory of a ring. Recall that when R is
Noetherian G(R) is the Quillen Ktheory of the category of finitelygenerated R
modules (as opposed to finitelygenerated projectives); see [Sr, Chapter 5]. In
terms of the Waldhausen machinery, it is the algebraic Ktheory of the category*
* of
bounded chain complexes of finitelygenerated Rmodules_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 triangulatedequivalent 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
Ktheory 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 Ktheory
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 firstorder 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 triangulatedequivalent to DB. Let Kc(A) denote the Waldhausen
Ktheory 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 cocom*
*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 cocomplete 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 triangulatedequivalent to Db(B), then K(A) ' K(B) where K(A)
denotes the Quillen Ktheory 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 EilenbergSwindle 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
Ktheory spaces, rather than Ktheory 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 AC 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 manygenerators 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 ! Cn1, and the shift operator is denoted as C: it is the
chain complex with ( C)n = Cn1.
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 Rmodules, whereas mod R denotes the category of
finitelygenerated Rmodules (we only use this when R is rightNoetherian). Lik*
*e
wise, ProjR is the category of all projective Rmodules and projR is the subc*
*at
egory of finitelygenerated 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 cocomplete.
DERIVED EQUIVALENCES 5
Axiom 2: (Twooutofthree 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

 
fflfflfflfflfflffl
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 leftliftingproperty 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 nonnegatively graded chain complexes over a ring R has a
model structure where the weak equivalences are the maps inducing homology
isomorphisms (the quasiisomorphisms), 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 levelwiseinjective 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 categorytheoretic localization W1M. 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 quasiisomorphisms, 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 = Cn1. 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 zigzag 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. Ktheory and model categories
In [Wa , Section 1] Waldhausen defined a notion of category with cofibrations*
* and
weak equivalences and showed how to construct a Ktheory 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 Soconstruction [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
Kgroups 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! wFn1(U) into a simplicial category (where wF1(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 wFn1(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
wFn1(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 finitelygenerated projectives}.
(2)UG = {all bounded below complexes C of finitelygenerated projectives such
that Hk(C) 6= 0 for only finitelymany values of k}.
8 DANIEL DUGGER AND BROOKE SHIPLEY
Let K(R) and G(R) denote the Quillen Ktheory spaces for the exact categories
of finitelygenerated projectives and finitelygenerated 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 Gtheory, 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
finitelygenerated 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 quasiisomorp*
*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 Ktheory 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
BökstedtNeeman [BN , 6.4] identifies the compact objects as the perfect comple*
*xes,
i.e. the complexes which are quasiisomorphic 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 Ktheory
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 welldefine*
*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 fibrantreplacement 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 zigzag 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 zigzag 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 Ktheory 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 fibrantreplacement X æ~ F X i * in N and
let Q(RF X) be a cofibrantreplacement * 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 zigzag
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 zigzag 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 zigzag of Quillen equivale*
*nces,
rather than just an equivalence ChR ! ChS; in particular, note that our zigzag
could conceivably pass through very nonalgebraic 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) Rmodule P a strong generator if
homR (P, X) = 0 implies X = 0 for any (right) Rmodule 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 RS bimodule M and an SR bimodule N such that M S N ~=R
as Rbimodules and N R M ~=S as Sbimodules.
3. There is a (right) Rmodule P which is finitelygenerated, 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 : ModS ! 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
finitelygenerated 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 zigzag of Quillen equivalences between the model categories of
chain complexes of R and Smodules:
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 Smodules are triangulated equivalent:
Kb(projR) ' Kb(projS).
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(projR)
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(projR) and Kb(projS). Let S[0] be the free Smodule on one generator,
viewed as a complex in ChS concentrated in dimension zero; let T be its im
age in Kb(projR). We have DR (T, T ) ~=DS(S[0], S[0]) ~=S. Since S[0] gener
ates Kb(projS), T generates Kb(projR). Since R[0] is a generator of DR and
R[0] 2 Kb(projR), the only localizing subcategory of DR containing Kb(projR) *
* __
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(projR) 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 Gtheory
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(projR), etc. The notation K+,hb(projR) means the intersection of
K+ (projR) and Khb(projR). It is an easy exercise to check that Dh+(Mod R) =
K+ (ProjR) 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(projR), K+ (ProjR) = Dh+(Mod R), Dh(Mod R),
Dhb(Mod R) = Db(Mod R), K+ (projR), K+,hb(projR).
Proof.The result [BN , 6.4] identifies Kb(projR) 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+ (projR) is harder, but was proven by Rickard_see the first
paragraph in the proof of [R1 , 8.1]. Finally, K+,hb(projR) is just the_inters*
*ection_
of K+ (projR) and Dhb(Mod R). __
K+,hb(projR) is the full subcategory of DR consisting of complexes which are
quasiisomorphic to a boundedbelow complex of finitelygenerated projectives,
and which also have bounded homology. So one has the inclusions Dc(R)
K+,hb(projR) DR . Note that K+,hb(projR) is the image in DR of the Wald
hausen subcategory UG (R) ChR . It is an easy exercise to check that when R is
rightNoetherian one has K+,hb(projR) = Db(mod R), where the latter denotes
the full subcategory of DR consisting of the bounded complexes of finitelygene*
*rated
modules.
Theorem C follows immediately from the following more comprehensive state
ment:
Theorem 5.2. Let R and S be rightNoetherian.
(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(projR) and K+,hb(projS)
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(projR) for
what we call K+,hb(projR), 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(projR) and K+,hb(projS). 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(projR), 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 cocomplete 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 Ynk
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 ModA.
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 leftliftingproperty 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 rightadjoint 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 chainhomotopyclasses 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 quasiisomorphic 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 quasiisomorphisms 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 quasiisomorphism A ! B of differential graded algebras
induces a Quillen equivalence ModA ! Mod B.
16 DANIEL DUGGER AND BROOKE SHIPLEY
Proof.Any map f :A ! B induces a Quillen adjoint pair between ModA 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 Smodules (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 dgmodules 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
quasiisomorphism 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 dgmodules 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 Ktheory
groups are nonisomorphic 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 Abcategor*
*y (a
category enriched over abelian groups); this terminology makes sense because an
Abcategory 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 cocomplete 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 Abcategories with Chcategories (categories enriched over Ch = ChZ.) S*
*ince
a Chcategory with one object is a differential graded algebra, one may think o*
*f a
small Chcategory as a DGA with many objects. Given a small Chcategory R,
a (right) Rmodule M is a contravariant Chfunctor 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 Chcategories, 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 Chcategory 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 quasiisomorphisms 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 finitelygenerated 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 finitelygenerated 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 manyobjects 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. (Manyobjects tilting theorem) Theorem 4.2 holds when the
rings R and S are replaced by ringswithmanyobjects 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 ringswithmanyobjects 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 ringwithmanyobjects.
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 cocomplete 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(ChModR ).
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(projR*
*_)
and Kb(projS_). For s any object in S_, consider the module FsS_as a complex
concentrated in dimension zero; let Ts be its image in Kb(projR_). Since the o*
*bjects
in {Fs}s2S_generate Kb(projS_), the objects in T = {Ts}s2S_generate Kb(projR_*
*).
But Kb(projR_) 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 quasiisomorphic 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 fibrantreplacement functor in
N and let Q be a cofibrantreplacement 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 qXn1 Yn1 ! *
*Yn
are cofibrations. In particular, an object Xo is cofibrant if and only if the *
*maps
Xn1 ! 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(Un1): 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
wUn1 _____//_wVn1
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 . __
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Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
Email address: ddugger@math.uoregon.edu
Department of Mathematics, Purdue University, W. Lafayette, IN 47907, USA
Email address: bshipley@math.purdue.edu