DERIVED CATEGORIES AND PROJECTIVE CLASSES



                            J. DANIEL CHRISTENSEN


       Abstract.An important example of a model category is the category of unb*
 *ounded chain com-
       plexes of R-modules, which has as its homotopy category the derived cate*
 *gory of the ring R. This
       example shows that traditional homological algebra is encompassed by Qui*
 *llen's homotopical alge-
       bra. The goal of this paper is to show that more general forms of homolo*
 *gical algebra also fit into
       Quillen's framework. Specifically, any set of objects in a complete and *
 *cocomplete abelian category
       A generates a projective class on A, which is exactly the information ne*
 *eded to do homological al-
       gebra in A. The main result is that if the generating objects are "small*
 *" in an appropriate sense, then
       the category of chain complexes of objects of A has a model category str*
 *ucture which reflects the
       homological algebra of the projective class. The motivation for this wor*
 *k is the construction of the
       üp re derived categoryö  f a ring R. Finally, we explain how the categor*
 *y of simplicial objects in
       a possibly non-abelian category can be equipped with a model category st*
 *ructure reflecting a given
       projective class.


                                Contents

   Introduction                                                            1
   1.  Projective classes                                                     3
   2.  Cofibrantly generated model categories                                  *
 *   4
   3.  Derived categories                                                    5
   4.  The pure and categorical derived categories                             *
 *    11
   5.  Simplicial objects and the bounded below derived category               *
 *       12
   References                                                            13

                              Introduction

   An important example of a model category is the category Ch of unbounded cha*
 *in complexes

of R-modules, which has as its homotopy category the derived category D of the *
 *ring R. The

formation of a projective resolution is an example of cofibrant replacement, an*
 *d traditional derived

functors are examples of derived functors in the model category sense. This exa*
 *mple shows that

traditional homological algebra is encompassed by Quillen's homotopical algebra*
 *, and indeed this

unification was one of the main points of Quillen's work [14].

   The goal of this paper is to illustrate that more general forms of homologic*
 *al algebra also fit into

Quillen's framework. In any abelian category A there is a natural notion of "pr*
 *ojective objectä  nd

"exact sequence." However, it is sometimes useful to impose different definitio*
 *ns of these terms.
___________
  Date: November 17, 1998.
  1991 Mathematics Subject Classification. Primary 18E30; Secondary 18G35, 55U3*
 *5, 18G25.
  Key words and phrases. Derived category, chain complex, relative homological *
 *algebra, projective class, pure homo-
logical algebra.


                                    1




2                            J. DANIEL CHRISTENSEN


If this is done in a way that satisfies some natural axioms, what is obtained i*
 *s a "projective class,"

which is exactly the information needed to do homological algebra in A. Our mai*
 *n result is that

given any projective class satisfying a set-theoretic hypothesis, the category *
 *of chain complexes of

objects of A has a model category structure which reflects the homological alge*
 *bra of the projective

class. The motivation for this work is the construction of the üp re derived ca*
 *tegoryö  f a ring R.

Pure homological algebra has applications to phantom maps in the stable homotop*
 *y category and

in the (usual) derived category of a ring, and these connections will be descri*
 *bed.

   When A has enough projectives, the projective objects and exact sequences fo*
 *rm a projective

class. Therefore the results of this paper apply to traditional homological alg*
 *ebra as well. Even

in this special case, it is not a trivial fact that the category of unbounded c*
 *hain complexes can be

given a model category structure, and indeed Quillen restricted himself to the *
 *bounded below case.

I know of three other written proofs that the category of unbounded chain compl*
 *exes is a model

category [7, 8, 10], which do the case of R-modules, but this was probably know*
 *n to others as well.

   An important corollary of the fact that a derived category D is the homotopy*
 * category of a model

category is that D(X, Y ) is a set (as opposed to a proper class) for any two c*
 *hain complexes X and

Y . This is not the case in general, and much work on derived categories ignore*
 *s this possibility. It

is not just a pedantic point; if one uses the morphisms in the derived category*
 * to index constructions

in other categories or to define cohomology groups, one needs to know that the *
 *indexing class is

actually a set. Recently, the unbounded case has been handled under various ass*
 *umptions on A.

(See Weibel [16] Remark 10.4.5, which credits Gabber, and Exercise 10.4.5, whic*
 *h credits Lewis,

May and Steinberger [13]. See also Kriz and May [12, Part III].) The assumption*
 *s that appear in the

present paper are different from those that have appeared before and the proof *
 *is somewhat easier

(because of our use of the theory of cofibrantly generated model categories), s*
 *o this paper may be

of some interest even in this special case.

   Another consequence of the fact that Ch is a model category is the existence*
 * of resolutions

coming from cofibrant and fibrant approximations, and the related derived funct*
 *ors. Some of these

are discussed in [1] and [15]. We do not discuss these topics here, but just me*
 *ntion that the model

category point of view can provide a framework for some of this material.

   We also briefly discuss the category of non-negatively graded chain complexe*
 *s. In this case

we describe a model category structure that works for an arbitrary projective c*
 *lass, without any

set-theoretic hypotheses. More generally, we show that under appropriate hypoth*
 *eses a projective

class on a possibly non-abelian category A determines a model category structur*
 *e on the category

of simplicial objects in A. As an example, we deduce that the category of equiv*
 *ariant simplicial

sets has various model category structures.

   We now briefly outline the paper. In Section 1 we give the axioms for a proj*
 *ective class and

mention a few examples. In Section 3 we describe the model category structure o*
 *n the category of

chain complexes which takes into account a given projective class. Then we stat*
 *e the set-theoretic

assumption needed and prove our main result, using the recognition lemma for co*
 *fibrantly gener-

ated categories, which is recalled in Section 2. In Section 4 we give two examp*
 *les, the traditional

derived category of R-modules and the pure derived category, and we describe wh*
 *y the pure derived

category is interesting. In the final section we discuss the bounded below case*
 *, which works for

any projective class, and describe a result for simplicial objects in a possibl*
 *y non-abelian category.




                     DERIVED CATEGORIES AND PROJECTIVE CLASSES                 *
 *   3


   I thank Haynes Miller for asking the question which led to this paper and Ma*
 *rk Hovey, Haynes

Miller and John Palmieri for fruitful and enjoyable discussions.


                           1.Projective classes

   In this section we explain the notion of a projective class, which is the in*
 *formation necessary

in order to do homological algebra.  Intuitively, a projective class is a choic*
 *e of which sort of

"elements" we wish to think about.

   The elements of a set X correspond bijectively to the maps from a singleton *
 *to X, and the

elements of an abelian group A correspond bijectively to the maps from Z to A. *
 *Motivated by this,

we call a map P ! A in any category a P -element of A. If we don't wish to ment*
 *ion P , we call

such a map a generalized element of A. A map A ! B in any category is determine*
 *d by what it

does on generalized elements. If P is a collection of objects, then a P-element*
 * means a P -element

for some P in P.

   Let A be a pointed category, i.e., a category in which initial and terminal *
 *objects exist and

agree. In a pointed category, any initial (equivalently, terminal) object is ca*
 *lled a zero object. Let

P be an object of A. A sequence

                              A -! B -! C

is said to be P -exact if the composite A ! C is the zero map (the unique map w*
 *hich factors

through a zero object) and

                       A(P, A) -! A(P, B) -! A(P, C)

is an exact sequence of pointed sets (the base points being the zero maps).  Th*
 *e latter can be

rephrased as the condition that A ! B ! C induces an exact sequence of P -eleme*
 *nts. A P-exact

sequence is one which is P -exact for all P in P. A map A ! B is P -epic (resp.*
 * P-epic) if it

induces a surjection of P -elements (resp. P-elements).

   A projective class on A is a collection P of objects of A and a collection E*
 * of sequences

A ! B ! C in A such that

  (i)E is precisely the collection of all P-exact sequences;

 (ii)P is precisely the collection of all objects P such that each sequence in *
 *E is P -exact;

 (iii)any map A ! B can be extended to a sequence P ! A ! B in E with P in P.

When a collection P is part of a projective class (P, E), the projective class *
 *is unique, and so we

say that P determines a projective class or even that P is a projective class. *
 *An object of P is called

a P-projective. Condition (iii) says that there are "enough P-projectives."


Example 1.1. Let A be the category of pointed sets, let P be the collection of *
 *all pointed sets and

let E be the collection of all exact sequences of pointed sets. Then E is preci*
 *sely the collection of

P-exact sequences, and P is a projective class.


Example 1.2. For an associative ring R, let A be the category of left R-modules*
 *, let P be the

collection of all summands of free R-modules and let E be the collection of all*
 * exact sequences of

R-modules. Then E is precisely the collection of P-exact sequences, and P is a *
 *projective class.




4                            J. DANIEL CHRISTENSEN


   The above two examples are äc tegorical" projective classes in the sense tha*
 *t the P-epimor-

phisms are just the epimorphisms and the P-projectives are the categorical proj*
 *ectives, i.e., those

objects P such that maps from P lift through epimorphisms.

   Here are two examples of non-categorical projective classes.


Example 1.3. Let A be the category of left R-modules, as in Example 1.2. Let P *
 *consist of all

summands of sums of finitely presented modules and define E to consist of all P*
 *-exact sequences.

Then P is a projective class. A short exact sequence is P-exact iff it remains *
 *exact after tensoring

with every right module.


Example 1.4. Let S be the homotopy category of spectra and let P consist of all*
 * retracts of wedges

of finite spectra. Then P determines a projective class, and a map is a P-epimo*
 *rphism iff its cofibre

is a phantom map.


   Examples 1.2 and 1.3 will be discussed further in Section 4. Example 1.4 is *
 *studied in [2], along

with similar examples.

   We note that all of the examples mentioned above are determined by a set in *
 *the sense of the

following lemma, whose proof is easy.


Lemma 1.5. Suppose F is any set of objects in a pointed category with coproduct*
 *s. Let E be the

collection of F-exact sequences and let P be the collection of all objects P su*
 *ch that every sequence

in E is P -exact. Then P is the collection of retracts of coproducts of objects*
 * of F and (P, E) is a

projective class.


   A projective class is precisely the information needed to form projective re*
 *solutions and define

derived functors. For further information, we refer the reader to [6] for the g*
 *eneral theory and to [2]

in the special case of a triangulated category.


                2. Cofibrantly generated model categories

   In this section we briefly recall the basics of cofibrantly generated model *
 *categories. This ma-

terial will be used in the next section to prove our main result. For more deta*
 *ils, see the books by

Dwyer, Hirschhorn and Kan [5] and Hirschhorn [9]. We assume knowledge of the ba*
 *sics of model

categories, for which [4] is an excellent reference.

   We will always assume our model categories to be complete and cocomplete.

   In the following, a cardinal number is thought of as the first ordinal with *
 *that cardinality.


Definition 2.1. Given an ordinal ~, a ~-sequence in a category M is a diagram

                   X0 -! X1 -! . .-.! X~ -! X~+1 -! . . .

indexed by the ordinals less than ~, such that for each limit ordinal fl less t*
 *han ~ the natural map

colim~<flX~ ! Xflis an isomorphism.

   A cardinal fl is said to be regular if any sum of fewer than fl cardinals ea*
 *ch less than fl is less

than fl. The first few regular cardinals are 0, 1, 2 and @0.

   An object P is said to be small relative to a subcategory B if there exists *
 *a cardinal ~ such that

for any regular cardinal fl   ~ and any fl-sequence X in M taking values in B, *
 *the natural map




                     DERIVED CATEGORIES AND PROJECTIVE CLASSES                 *
 *   5


colim~<flM(P, X~) ! M(P, colim~<flX~) is an isomorphism, where the last colimit*
 * is taken in

M.

Definition 2.2. Let I be a class of maps in a cocomplete category. A map is sai*
 *d to be I-injective

if it has the right lifting property with respect to each map in I, and we writ*
 *e I-injfor the category

containing these maps. A map is said to be an I-cofibration if it has the left *
 *lifting property with

respect to each map in I-inj, and we write I-coffor the category containing the*
 *se maps. A map is

said to be I-cellular if it is a transfinite composite of pushouts of coproduct*
 *s of maps in I, and we

write I-cellfor the category containing these maps.

   Note that I-cellis a subcategory of I-cof.

Definition 2.3. A cofibrantly generated model category is a model category M fo*
 *r which there exist

sets I and J of morphisms with domains which are small relative to I-cofand J-c*
 *of, respectively,

such that I-cofis the category of cofibrations and J-cofis the category of acyc*
 *lic cofibrations. It

follows that I-injis the category of acyclic fibrations and that J-injis the ca*
 *tegory of fibrations.

   For example, take M to be the category of spaces and take I = {Sn ! Bn+1} an*
 *d J =

{Bn x 0 ! Bn x [0, 1]}.

Proposition 2.4. (Recognition Lemma.) Let M be a category which is complete and*
 * cocomplete, let

W be a class of maps which is closed under retracts and satisfies the two-out-o*
 *f-three axiom, and

let I and J be sets of maps with domains which are small relative to I-celland *
 *J-cell, respectively,

such that

  (i)J-cell  I-cof\ W and I-inj  J-inj\ W , and

 (ii)J-cof  I-cell\ W or I-inj  J-inj\ W .

Then M is cofibrantly generated by I and J, and W is the subcategory of weak eq*
 *uivalences.

Moreover, a map is in I-cofif and only if it is a retract of a map in I-cell, a*
 *nd similarly for J-cof

and J-cell.

   The proof, which uses the small object argument, can be found in [5] and [9].


                          3. Derived categories

   Let A be a complete and cocomplete abelian category equipped with a projecti*
 *ve class P. Our

goal is to construct a model category structure on the category of chain comple*
 *xes of objects of A

in a way that takes into account the projective class P. At a certain point we *
 *will need a set-theoretic

assumption, but we proceed as far as possible without it.

   We write Ch for the category of unbounded chain complexes of objects of A an*
 *d degree zero

chain maps. To fix notation, assume that the differentials lower degree. For an*
 * object X of Ch,

define ZnX := ker(d : Xn ! Xn-1) and BnX := im(d : Xn+1 ! Xn), and write HnX fo*
 *r the

quotient. Given an object P of A and a chain complex X, we write A(P, X) for th*
 *e chain complex

which has the abelian group A(P, Xn) in degree n. This is the chain complex of *
 *P -elements of

X. (See Section 1 for a discussion of P -elements.) The cycles, boundaries and *
 *homology classes

in A(P, X) are called P -cycles, P -boundaries and P -homology classes in X. No*
 *te that a P -

cycle is the same as a P -element of ZnX, but that not every P -element of BnX *
 *is necessarily a

P -boundary (unless P is a categorical projective).




6                            J. DANIEL CHRISTENSEN


   Using this terminology, we can say which maps in our category should be weak*
 * equivalences,

fibrations and cofibrations.

Definition 3.1. Consider a chain map f : X ! Y . The map f is said to be a quas*
 *i-isomorphism if

it induces an isomorphism HnX ! HnY for each n. The map f is said to be a weak *
 *equivalence

if for each P-projective P the chain map A(P, X) ! A(P, Y ) is a quasi-isomorph*
 *ism, i.e., if f

induces an isomorphism of P -homology groups. The map f is said to be a fibrati*
 *on if for each P-

projective P the chain map A(P, X) ! A(P, Y ) is degreewise epic, i.e., if f in*
 *duces a surjection

of P -elements. And f is said to be a cofibration if each fn : Xn ! Yn is split*
 *-monic and the

complex of cokernels is öc fibrant": a complex is cofibrant if for some ordinal*
 * ~ it is a retract of a

complex which is a colimit of a ~-sequence

                        0 = C0 -! C1 -! C2 -! . . .

such that each map is degreewise split-monic and the successive quotients are c*
 *omplexes of P-

projective objects with zero differential. As usual, a map is said to be an acy*
 *clic fibration (resp.

acyclic cofibration) if it is both a fibration (resp. cofibration) and a weak e*
 *quivalence.

   It will be useful to have the following notation. We write   for the usual a*
 *utomorphism of Ch,

which sends a complex X to the complex  X with ( X)n = Xn-1and d X = -dX . The *
 *functor

  is defined on morphisms by ( f)n = fn-1. Given an object A in A, we consider *
 *A as a chain

complex concentrated in degree 0. Thus we write  nA for the complex with A conc*
 *entrated in

degree n. The maps from  nA to a complex X biject with the A-elements of ZnX. S*
 *imilarly, we

write DnA for the complex which has A in degrees n-1 and n connected by the ide*
 *ntity map, and

which is zero in other degrees, and we find that the maps from DnA to a complex*
 * X biject with the

A-elements of Xn. There is a natural map  n-1A ! DnA.

   The following lemma will be used to prove our main result.

Lemma 3.2. Consider an object A 2 A and a map f : X ! Y of chain complexes.

  (i)The map f has the RLP with respect to each of the maps 0 ! DnA, n 2 Z, if *
 *and only if the

    induced map of A-elements is a surjection.

 (ii)The map f has the RLP with respect to each of the maps  n-1A ! DnA, n 2 Z,*
 * if and only

    if the induced map of A-elements is a surjection and a quasi-isomorphism.

Proof.We begin with (i): The map f has the RLP with respect to the map 0 ! DnA *
 *if and only if

each map DnA ! Y factors through f, i.e., if and only if each A-element of Yn i*
 *s in the image of

fn.

   Now (ii): The map f has the RLP with respect to the map  n-1A ! DnA if and o*
 *nly if for each

A-element y of Yn whose boundary is the image of an A-cycle x of Xn-1, there is*
 * an A-element x0

of Xn which hits y under f and x under the differential. (In other words, if an*
 *d only if the natural

map Xn ! Zn-1X xZn-1YYn induces a surjection of A-elements.)

   So suppose that f has the RLP with respect to each map  n-1A ! DnA. As a pre*
 *liminary

result, we prove that f induces a surjection of A-cycles. Suppose we are given *
 *an A-cycle y of Y .

Its boundary is zero and is thus in the image of the A-cycle 0 of X. Therefore *
 *y is the image of an

A-cycle x0.

   It follows immediately that f induces a surjection in A-homology.




                     DERIVED CATEGORIES AND PROJECTIVE CLASSES                 *
 *   7


   Now we prove that f induces a surjection of A-elements. Suppose we are given*
 * an A-element y

of Y . By the above argument, its boundary, which is an A-cycle, is the image o*
 *f an A-cycle x of

X. Thus, by the characterization of maps f having the RLP, we see that there is*
 * an A-element x0

which hits y.

   Finally, we prove that f induces an injection in A-homology. Suppose x is an*
 * A-cycle of X

whose image under f is the boundary of an A-element y. Then, using the characte*
 *rization of the

RLP, we see that there is an A-element x0whose boundary is x. This shows that f*
 * induces an

injection in A-homology.

   We have proved that if f has the RLP with respect to the maps  n-1A ! DnA, t*
 *hen f induces

an isomorphism in A-homology and a surjection of A-elements.  The proof of the *
 *converse is

similar.


Corollary 3.3. A map f : X ! Y is a fibration (resp. acyclic fibration) if and *
 *only if it has the

RLP with respect to the map 0 ! DnP (resp.  n-1P ! DnP ) for each P-projective *
 *P and each

n 2 Z.


   We want to claim that the above definitions lead to a model category structu*
 *re on the category

Ch . In order to prove this, we need to assume that there is a set S of P-proje*
 *ctives such that a map

f : A ! B is P-epic if and only if f induces a surjection of P -elements for ea*
 *ch P in S. This

implies that for any B there is a P-epimorphism P ! B with P a coproduct of obj*
 *ects from S, and

that every P-projective object is a retract of such a coproduct. We also need t*
 *o assume that each

P in S is small relative to the subcategory K := {(1, 0) : A ! A   Q | A 2 A an*
 *dQ 2 P}.

(See the previous section for the definition of "small.") When these conditions*
 * hold we say that P

is determined by a set of small objects.


Theorem 3.4. Assume that P is determined by a set S of small objects. Then the *
 *category Chis a

cofibrantly generated model category with the following generating sets:

                     I := { n-1P ! DnP | P 2 S, n 2 Z},

                       J := {0 ! DnP | P 2 S, n 2 Z}.

Moreover, the weak equivalences, fibrations, cofibrations and cofibrant objects*
 * are as described in

Definition 3.1, and every object is fibrant.


Proof.In this proof we use the terms weak equivalence, (acyclic) fibration, (ac*
 *yclic) cofibration,

and cofibrant as defined in Definition 3.1.

   We check the hypotheses of the Recognition Lemma from the previous section. *
 * Since A is

complete and cocomplete, so is Ch; limits and colimits are taken degreewise. Th*
 *e class of weak

equivalences is easily seen to be closed under retracts and to satisfy the two-*
 *out-of-three condition.

The zero chain complex is certainly small relative to J-cell. We assumed that e*
 *ach P in S is small

relative to K   A, and it follows that each  kP is small relative to I-cell  Ch*
 *. In more detail: A

map is in I-cellif and only if it is a degreewise split monomorphism whose coke*
 *rnel is a transfinite

colimit of degreewise split monomorphisms whose cokernels are complexes of P-pr*
 *ojectives with

zero differential. In particular, every map in I-cellis a degreewise split mono*
 *morphism whose

cokernel is a complex of P-projectives. Let P be an object of S. For large enou*
 *gh regular cardinals




8                            J. DANIEL CHRISTENSEN


~, P is small with respect to ~-sequences of maps in K. Suppose that for such a*
 * ~ we have a ~-

sequence X0 ! X1 ! . .o.f maps in I-cellwith colimit X. A chain map from  kP to*
 * X is just

a map f : P ! Xk such that df = 0. Given such an f, it factors through some g :*
 * P ! Xffk, since

P is small relative to K, and the sequence X0k! X1k! . .i.s in K. Moreover, sin*
 *ce df = 0, it

follows that dg = 0, at least after passing to Xfikfor some fi   ff, again usin*
 *g smallness. Thus the

map colimCh( kP, X~) ! Ch( kP, X) is surjective. An easier argument shows that *
 *it is monic,

and therefore  kP is small relative to I-cell.

   Since the projectives in S may be used to test whether a map is a fibration *
 *or weak equivalence,

Corollary 3.3 tells us that I-injis the collection of acyclic fibrations and th*
 *at J-injis the collection

of fibrations. Thus we have an equality I-inj= J-inj\ W , giving us two of the *
 *inclusions required

by the Recognition Lemma.

   We now prove that J-cell  I-cof\ W . Since I-inj  J-inj, it is clear that J-*
 *cof  I-cof, so in

particular J-cell  I-cof. We must prove that J-cell  W , i.e., that each map wh*
 *ich is a transfinite

composite of pushouts of coproducts of maps in J is a weak equivalence. A map i*
 *n J is of the

form 0 ! DnP for some P 2 S. Thus a coproduct of maps in J is of the form 0 ! C*
 * for some

contractible complex C. A pushout of such a coproduct is of the form X ! X   C,*
 * again with C

a contractible complex. And a transfinite composite of such maps is of the same*
 * form as well. In

particular, a transfinite composite of pushouts of coproducts of maps from J is*
 * a weak equivalence.

   We can now apply the Recognition Lemma and conclude that Chis a model catego*
 *ry with weak

equivalences as described in Definition 3.1, cofibrations given by the maps in *
 *I-cofand fibrations

given by the maps in J-inj. We saw above that the maps in J-injare precisely th*
 *ose we called

fibrations in Definition 3.1, and it is clear from this that every object is fi*
 *brant.

   It remains to check that the maps in I-cofare as described in Definition 3.1*
 *. We continue use the

words öc fibrationä  nd öc fibrantä  s defined in Definition 3.1. Suppose that *
 *i is in I-cof. By 2.4,

i is a retract of a map j which is a transfinite composite of pushouts of copro*
 *ducts of maps in I.

As above, one can show that j is degreewise split-monic and that the complex C *
 *of cokernels is

cofibrant. It follows that i is degreewise split-monic and that the complex of *
 *cokernels is a retract

of C and is therefore cofibrant. That is, i is a cofibration. And now it is cle*
 *ar that if the map 0 ! X

is in I-cof, then X is cofibrant.

   Using that I-cofis closed under transfinite compositions and retracts, one c*
 *an show that every

cofibration is in I-cofand in particular that if X is cofibrant then the map 0 *
 *! X is in I-cof.

   For the rest of this section, assume that the projective class P is determin*
 *ed by a set of small

objects, so that Ch is a model category. The homotopy category of Ch, formed by*
 * inverting the

weak equivalences, is called the derived category of A (with respect to P). It *
 *is denoted D, and

a fundamental result in model category theory asserts that D(X, Y ) is a set fo*
 *r each X and Y . In

Exercise 10.4.5 of [16], Weibel outlines an argument which proves that D(X, Y )*
 * is a set when there

are enough (categorical) projectives, P is the categorical projective class, an*
 *d A satisfies AB5. A

connection between Weibel's hypotheses and ours is that if A has enough project*
 *ives which are

small with respect to all filtered diagrams in A, then AB5 holds. Of course, th*
 *e smallness condition

needed for our theorem is weaker than this in that smallness is only required f*
 *or diagrams indexed

by large cardinals and taking values in the subcategory K.




                     DERIVED CATEGORIES AND PROJECTIVE CLASSES                 *
 *   9


   We now show that the notion of homotopy determined from the model category s*
 *tructure (see

[4] or [14]) corresponds to the usual notion of chain homotopy.


Definition 3.5. If M is an object in a model category C, a good cylinder object*
 * for M is an object
                            `    i        p
M x I and a factorization M   M -! M x I -! M of the codiagonal map, with i a c*
 *ofibration

and p a weak equivalence. (Despite the notation, M x I is not in general a prod*
 *uct of M with an

object I.) A left homotopy between`maps`f, g : M ! N is a map H : M x I ! N suc*
 *h that the

composite Hi is equal to f   g : M   M ! N, for some good cylinder object M x I.


   The notion of good path object NI for N is dual to that of good cylinder obj*
 *ect and leads to

the notion of right homotopy. The following standard result can be found in [4,*
 * Section 4], for

example.


Lemma 3.6. For M cofibrant and N fibrant, two maps f, g : M ! N are left homoto*
 *pic if and

only if they are right homotopic, and both of these relations are equivalence r*
 *elations and respect

composition. Moreover, if M x I is a fixed good cylinder object for M, then f a*
 *nd g are left

homotopic if and only if they are left homotopic using M x I; similarly for a f*
 *ixed good path

object.


   Because of the lemma, for M cofibrant and N fibrant we have a well-defined r*
 *elation of homo-

topy on maps M ! N. One can show that the homotopy category of C, which is by d*
 *efinition

the category of fractions formed by inverting the weak equivalences, is equival*
 *ent to the category

consisting of objects which are both fibrant and cofibrant with morphisms being*
 * homotopy classes

of morphisms.

   Now we return to the study of the model category Ch.


Lemma 3.7. Let M and N be objects of Chwith M cofibrant. Two maps M ! N are hom*
 *otopic

if and only if they are chain homotopic.


Proof.We construct a factorization N ! NI ! N x N of the diagonal map   : N ! N*
 * x N

in the following way. Let NI be the chain complex which has Nn   Nn+1   Nn in d*
 *egree n.

We describe the differential by saying that it sends a generalized element (n, *
 *~n, n0) in (NI)n to

(dn, n - n0- d~n, dn0). Let ff : N ! NI be the map which sends n to (n, 0, n) a*
 *nd let fi : NI !

N x N be the map which sends (n, ~n, n0) to (n, n0). One can check easily that *
 *NI is a chain

complex and that ff and fi are chain maps whose composite is  . The map ff is a*
 * chain homotopy

equivalence with chain homotopy inverse sending (n, ~n, n0) to n; this implies *
 *that it induces a

chain homotopy equivalence of generalized elements and is thus a weak equivalen*
 *ce. The map fi is

degreewise split-epi; this implies that it induces a split epimorphism of gener*
 *alized elements and is

thus a fibration. Therefore NI is a good path object for N.

   It is easy to see that a chain homotopy between two maps M ! N is the same a*
 *s a right

homotopy using the path object NI. By Lemma 3.6, two maps are homotopic if and *
 *only if they

are right homotopic using NI. Thus the model category notion of homotopy is the*
 * same as the

notion of chain homotopy when the source is cofibrant.

   There is a dual proof which proceeds by constructing a specific good cylinde*
 *r object M x I for

M such that a left homotopy using M x I is the same as a chain homotopy.




10                           J. DANIEL CHRISTENSEN


Corollary 3.8. Let A and B be objects of A considered as chain complexes concen*
 *trated in degree

0. Then D(A,  nB) ~=ExtnP(A, B).


   Here we denote by ExtnP(-, B) the right derived functors of A(-, B) with res*
 *pect to the projec-

tive class P. The group ExtnP(A, B) can be calculated by forming a P-projective*
 * P-exact resolution

of A, applying A(-, B), and taking cohomology. One can show that ExtnP(A, B) cl*
 *assifies equiv-

alence classes of P-exact sequences 0 ! B ! C1 ! . .!.Cn ! A ! 0.

Proof.The group D(A,  nB) may be calculated by choosing a cofibrant replacement*
 * A0for A and

computing the homotopy classes of maps from A0to  nB. (Recall that all objects *
 *are fibrant, so

there is no need to take a fibrant replacement for  nB.) A P-projective P-exact*
 * resolution P of A

serves as a cofibrant replacement for A, and by Lemma 3.7 the homotopy relation*
 * on Ch(P,  nB)

is chain homotopy, so it follows that D(A,  nB) is isomorphic to ExtnP(A, B).

   More generally, a similar argument shows that the derived functors of a func*
 *tor F can be ex-

pressed as the cohomology of the derived functor of F in the model category sen*
 *se. To make the

story complete, we next show that the shift functor   corresponds to the notion*
 * of suspension that

the category D obtains as the homotopy category of a pointed model category.


Definition 3.9. Let C be a pointed model`category. If M is cofibrant, we define*
 * the suspension

 M of M to be the cofibre of the map M   M ! M`x I for some good cylinder objec*
 *t M x I.

(The cofibre of a map X ! Y is the pushout *  X Y , where * is the zero object.*
 *)  M is cofibrant

and well-defined up to homotopy equivalence.


   The loop object  N of a fibrant object N is defined dually. These operations*
 * induce adjoint

functors on the homotopy category. A straightforward argument based on the path*
 * object described

above (and a dual cylinder object) proves the following lemma.


Lemma 3.10. In the model category Ch, the functor   defined in Definition 3.9 c*
 *an be taken to be

the usual suspension, so that ( X)n = Xn-1 and d X = -dX . Similarly,  X can be*
 * taken to be

the complex  -1X. That is, ( X)n = Xn+1 and d X = -dX .


   In particular,   and   are inverse functors. Hovey [10] has shown that this *
 *implies that cofibre

sequences and fibre sequences agree (up to the usual sign) and that   and the c*
 *ofibre sequences give

rise to a triangulation of the homotopy category. (See [14, Section I.3] for th*
 *e definition of cofibre

and fibre sequences in any pointed model category.) Applying Hovey's result to *
 *our situation we

can be more explicit. Given a map f : L ! M of chain complexes, the standard tr*
 *iangle on f is

the sequence L ! M ! N !  L, where Nn = Mn   Ln-1, dN (m, l) = (dM m + fl, -dLl*
 *) on

generalized elements, and the maps are the inclusion and projection.


Corollary 3.11. The category D is triangulated. A sequence L0! M0! N0!  L0is a *
 *triangle

if and only if it is isomorphic in D to a standard triangle.


Proof.That D is triangulated follows from Hovey's result, since we have shown t*
 *hat   is a self-

equivalence. The identification of the triangles uses the definition of fibre s*
 *equences from [14] and

the construction of the path object from the proof of Lemma 3.7. There is also *
 *a dual proof using

cofibre sequences and cylinder objects.




                     DERIVED CATEGORIES AND PROJECTIVE CLASSES                 *
 *   11


   The results of this section probably apply with A replaced by a complete and*
 * cocomplete additive

category.


               4.The pure and categorical derived categories

   In this section we let R be an associative ring with unit and we take for A *
 *the category of left

R-modules. We are concerned with two projective classes on the category A. The *
 *first is the cate-

gorical projective class C whose projectives are summands of free modules, whos*
 *e exact sequences

are the usual exact sequences, and whose epimorphisms are the surjections. The *
 *second is the pure

projective class P whose projectives are summands of sums of finitely presented*
 * modules. A short

exact sequence is P-exact iff it remains exact after tensoring with any right m*
 *odule. A map is a

P-epimorphism iff it appears in a P-exact short exact sequence. We say pure pro*
 *jective instead

of P-projective, and similarly for pure exact and pure epimorphism. We assume t*
 *hat the reader

has some familiarity with these projective classes. A brief summary with furthe*
 *r references may

be found in [2, Section 9]. As usual, we write Ext*(-, B) (resp. PExt*(-, B)) f*
 *or the derived

functors of A(-, B) with respect to the categorical (resp. pure) projective cla*
 *ss.

   Both of these projective classes are determined by sets of small objects: C *
 *is determined by {R}

and P is determined by any set of finitely presented modules containing a repre*
 *sentative from each

isomorphism class. Thus we get two model category structures on Chand two deriv*
 *ed categories,

the categorical derived category DC and the pure derived category DP. We refer *
 *to the pure weak

equivalences in Chas pure quasi-isomorphisms, and as usual call the categorical*
 * weak equivalences

simply quasi-isomorphisms. Similarly, we talk of pure fibrations and fibrations*
 *, pure cofibrations

and cofibrations, etc.

   Pure homological algebra is of interest in stable homotopy theory because of*
 * the following result.


Theorem 4.1. [3] Phantom maps from a spectrum X to an Eilenberg-Mac Lane spectr*
 *um HG are

given by PExt1Z(H-1X, G), that is, by maps of degree one from H-1X to G in the *
 *pure derived

category of abelian groups.


   The two derived categories are connected in various ways. For example, since*
 * every categorical

projective is pure projective, it follows that every pure quasi-isomorphism is *
 *a quasi-isomorphism.

This implies that there is a unique functor R : DP ! DC commuting with the func*
 *tors from Ch.

This functor has a left adjoint L. We can see this in the following way: the id*
 *entity functor on

Ch is adjoint to itself. It is easy to see that every pure fibration is a fibra*
 *tion and that every pure

acyclic fibration is an acyclic fibration. Therefore, by [4, Theorem 9.7], ther*
 *e is an induced pair

of adjoint functors between the pure derived category and the categorical deriv*
 *ed category, and the

right adjoint is the functor R mentioned above. This right adjoint is the ident*
 *ity on objects, since

everything is fibrant, and induces the natural map PExt*(A, B) ! Ext*(A, B) for*
 * R-modules A

and B. The left adjoint sends a complex X in DC to a (categorical) cofibrant re*
 *placement ~Xfor X.

Similar adjoint functors exist whenever one has two projective classes on a cat*
 *egory, one containing

the other.

   In addition to the connection between phantom maps and pure homological alge*
 *bra, the author

is interested in the pure derived category as a tool for connecting the global *
 *pure dimension of a ring




12                           J. DANIEL CHRISTENSEN


R to the behaviour of phantom maps in DC and DP under composition. This work wi*
 *ll hopefully

appear in a future paper.


        5.Simplicial objects and the bounded below derived category

   In this section, let A be a complete and cocomplete pointed category with a *
 *projective class P.

We do not assume that A is abelian. Write sAfor the category of simplicial obje*
 *cts in A. Given a

simplicial object X and an object P , write A(P, X) for the simplicial set whic*
 *h has A(P, Xn) in

degree n. Consider the following conditions:

    (*)For each X in sAand each P in P, A(P, X) is a fibrant simplicial set.

   (**)P is determined by a set S of small objects. Here we require that each P*
 * in S be small with

       respect to all monomorphisms in A, not just the split monomorphisms with*
 * P-projective

       cokernels.


Theorem 5.1. Let A be a complete and cocomplete pointed category with a project*
 *ive class P. If

A and P satisfy (*) or (**), then sA is a simplicial model category. The weak e*
 *quivalences (resp.

fibrations) are the maps f such that A(P, f) is a weak equivalence (resp. fibra*
 *tion) of simplicial

sets for each P in S. (Here we write S for the set of small objects which deter*
 *mines P when (**)

holds, and we take S to be equal to P when (*) holds.) In general the cofibrati*
 *ons are determined

by the lifting property. When (**) holds, sAis cofibrantly generated by the sets

                  I := {P   `[n] ! P    [n] | P 2 S, n   0},

              J := {P   V [n, k] ! P    [n] | P 2 S, 0 < n   k   0}.


   In the above,  [n] denotes the standard n-simplex simplicial set; `[n] denot*
 *es its boundary;

and V [n, k] denotes the subcomplex with the n-cell and its kth face removed. F*
 *or an object P of A

and a set K, P   K denotes the coproduct of copies of P indexed by K. When K is*
 * a simplicial

set, P   K denotes the simplicial object in A which has P   Kn in degree n.

   If A is abelian, then (*) holds.  Moreover, in this case there is an equival*
 *ence of categories

given by the normalization functor sA! Ch+, where Ch+denotes the category of no*
 *n-negatively

graded chain complexes of objects of A. Thus we can deduce:


Corollary 5.2. Let A be a complete and cocomplete abelian category with a proje*
 *ctive class P.

Then Ch+ is a model category. A map f is a weak equivalence iff A(P, f) is a qu*
 *asi-isomorphism

for each P in P.  A map f is a fibration iff A(P, f) is surjective in positive *
 *degrees (but not

necessarily in degree 0) for each P in P. A map is a cofibration iff it is degr*
 *eewise split-monic with

degreewise P-projective cokernel. Every complex is fibrant, and a complex is co*
 *fibrant iff it is a

complex of P-projectives.


   Note that no conditions on the projective class are required, and that the d*
 *escription of the

cofibrations is simpler than in the unbounded case.

   As another example of the theorem, let G be a group and consider G-simplicia*
 *l sets, or equiv-

alently, simplicial objects in the category A of (say, left) G-sets. Let F be a*
 * family of subgroups

of G, and consider the set S = {G=H | H 2 F} of homogeneous spaces. These are s*
 *mall, and

determine a projective class P. Thus, using case (**) of the above theorem, we *
 *can deduce:




                     DERIVED CATEGORIES AND PROJECTIVE CLASSES                 *
 *   13


Corollary 5.3. Let G be a group and let F be a family of subgroups of G. The ca*
 *tegory of G-

equivariant simplicial sets has a model category structure in which the weak eq*
 *uivalences are

precisely the maps which induce a weak equivalence on H-fixed points for each H*
 * in F.


   We omit the proof of Theorem 5.1, and simply note that it follows the argume*
 *nt in [14, Section

II.4] fairly closely. It is a bit simpler, in that Quillen spends part of the t*
 *ime (specifically Proposition

2) proving that effective epimorphisms give rise to a projective class, althoug*
 *h he doesn't use this

terminology. It is also a bit more complicated, in the (**) case, in that a tra*
 *nsfinite version of Kan's

Ex 1 functor [11] is required, since we make a weaker smallness assumption. Qui*
 *llen's argument

in this case can be interpreted as a verification of the hypotheses of the reco*
 *gnition lemma for

cofibrantly generated model categories (Proposition 2.4).


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   Department of Mathematics, Johns Hopkins University, 3400 N. Charles St., Ba*
 *ltimore, MD
21218-2686
   E-mail address: jdc@math.jhu.edu