DERIVED CATEGORIES IN ALGEBRA AND TOPOLOGY
J. P. MAY
I will give a philosophical overview of some joint work with Igor Kriz (in al*
*gebra),
with Tony Elmendorf and Kriz (in topology), and with John Greenlees (in equivar*
*iant
topology). I will begin with a description of some foundational issues before s*
*aying
anything about the applications. This is not the best way to motivate people, b*
*ut I
must explain the issues involved in order to describe what we have done. Let me*
* just
say that the emphasis I shall give to an analogy between algebra and topology i*
*s not
just an expository device. The algebraic work that I will describe both illumin*
*ates
the deeper topological theory and has applications to algebraic geometry.
We begin by displaying an analogy that is familiar to topologists. It is the *
*starting
point of_our_work._____________________________________________________
| ALGEBRA | TOPOLOGY |
|_______________________________|___________________________|____|||
|_______a_commutative_ring_k_______|___the_sphere_spectrum_S_____||||
|____differential_graded_k-modules____|_______spectra____________||||
|__________tensor_product__________|_______smash_product_________||||
|______internal_hom_Hom_(X;_Y_)______|function_spectrum_F_(X;_Y_)_||||
|_______dual_DX_=_Hom_(X;_k)_______|____dual_DX_=_F_(X;_S)______|_|||
|________projective_k-module________|______CW_spectrum_________|_|||
|____finitely_generated_projective____|_finite_CW_spectrum_______||
~ Hom (X; Y E) | F (X; Y ) ^ E ' F (X; Y ^ E) |
||Hom (X; Y ) ~E = || | |
|| DX E = Hom (X; E) | | DX ^ E ' F (X; E) | |
|____(X_or_E_fin._gen._projective)_(X_|or_E_a_finite_CW_spectrum)_||||
|__________hyperhomology__________|_____Eq(X)__ssq(X_^_E)______|_||q|
|_________hypercohomology_________|_____E_(X)__ss-qF_(X;_E)______|
Note that the right column already encodes the important topological theory of
Spanier-Whitehead duality: if X is a finite CW spectrum, then
E*(DX) ~=E-*(X):
Provisionally, we regard the columns as providing ground categories in which to*
* study
homotopical algebra. The usual starting point in algebra is a commutative diffe*
*rential
graded k-algebra, or DGA, A. The usual starting point in topology is a commutat*
*ive
ring spectrum R. This is an algebraic structure defined not in the ground categ*
*ory
___________
The author thanks the organizers of the conference for their generosity and h*
*ospitality; he also
acknowledges support from the NSF.
1
2 J. P. MAY
of spectra but rather in its "derived category", which is called the stable hom*
*otopy
category of spectra and denoted by hS . A map of spectra is called a weak equiv*
*alence
if it induces an isomorphism on homotopy groups, and hS is constructed from the
homotopy category of spectra by formally inverting the weak equivalences. Thus a
ring spectrum is a spectrum R together with a product OE : R ^ R -! R and unit
j : S -! R such that the following diagrams commute in hS :
j^1 // o1^jo 1^OE//
S ^ RK____R_^KR _____R ^ S and R ^ R ^ R _____R ^ R
KKK | ssss | |
'KKKK OE|ss's OE^1| |OE
%%|fflfflyyss |fflfflOE |fflffl
R R ^ R _________R//
The unlabelled equivalences are canonical isomorphisms in hS that give the un*
*ital
property, and we have suppressed such an associativity isomorphism in the second
diagram. Intuitively, these diagrams commute only up to homotopy. Similarly, th*
*ere
is a transposition isomorphism o : E ^ F - ! F ^ E in hS , and R is commutative
if the following diagram commutes in hS :
R ^ RGG_____o____/R/^ R
GG www
OEGGG##--wOEwww
R
An R-module is a spectrum M together with a map : R ^ M -! M such that
the following diagrams commute in hS :
j^1// 1^ //
S ^ ML_____RL^ M and R ^ R ^ M _____R ^ M
LLL | | |
' LLLL | OE^1| |
&&|fflffl |fflffl |fflffl
M R ^ M _________M//
The analog in algebra is a (differential graded) module M over a DGA A. A
map of A-modules is a quasi-isomorphism if it induces an isomorphism on homolog*
*y.
The derived category DA = hMA of A-modules is constructed from the homotopy
category of A-modules by formally inverting the quasi-isomorphisms.
The categories MA of A-modules and S of spectra admit suspension functors
and cofiber, or mapping cone, constructions that lead to long exact sequences of
homological or homotopical invariants. For a map f :M -! N , the sequence
f
M -! N -! Cf -! M
is called an "exact triangle" and leads to a "triangulation" of the derived cat*
*egory.
Analogously, in topology, we would like to have a triangulated category of R-mo*
*dules
for a ring spectrum R. However, this fails hopelessly with the definitions just*
* given.
The cofiber of a map f :M -! N of R-modules need not be an R-module. We can
DERIVED CATEGORIES IN ALGEBRA AND TOPOLOGY 3
find a map R ^ Cf -! Cf such that the following diagram commutes in hS , but
the map depends on a choice of homotopy making the left square commute, and the
associativity diagram that we required of R-modules generally fails to commute *
*for
Cf.
R ^ M ____R_^/N/ ____R_^/Cf/ ____R_^/M/
| | | |
| | | |
|fflffl |fflffl |fflffl |fflffl
M ________N_//_______Cf_//_______M_//
More deeply, when R is commutative, we would like to construct a smash product
M ^R N analogous to the tensor product M A N of A-modules in algebra. It is
far from clear how to begin. The algebraic constructions are easy because of t*
*he
good properties of the concrete underlying category of k-modules. Specifically,*
* Mk is
symmetric monoidal in the sense that its tensor product is associative, commuta*
*tive,
and unital up to coherent natural isomorphism. The smash product in the category
S of spectra is not associative, commutative, or unital. It only becomes so on *
*passage
to the derived category hS , which is symmetric monoidal. It is this limitation*
* on the
smash product that forced the homotopical definitions of ring and module spectra
that we gave above.
There is a significant difference in paradigm: algebraic topologists are ent*
*irely
comfortable working with fuzzy objects in the stable homotopy category, with no
point-set level models in mind. Algebraic geometers work with more concrete obj*
*ects,
and they wouldn't dream of taking a "ring in the derived category" seriously, as
topologists routinely make use of ring spectra.
The theory that I will describe gives both a new point-set level topological *
*theory
of rings and modules and a new algebraic theory of algebras and modules up to h*
*o-
motopy. These allow a far more precise analogy than the one displayed above. The
new topological theory allows the wholesale importation of techniques of commut*
*a-
tive algebra into stable homotopy theory. Applications include:
o A homotopical replacement for the Baas-Sullivan theory of manifolds with sing*
*u-
larities as a tool for the construction of new spectra from cobordism spectra.
o New generalized universal coefficient and K"unneth spectral sequences.
o New constructions of topological Hochschild homology and topological cyclic h*
*o-
mology.
o The construction of equivariant versions of such module spectra over the comp*
*lex
cobordism spectrum MU as the Brown-Peterson and Morava K-theory spectra.
o A completion theorem analogous to the Atiyah-Segal completion theorem in K-
theory that applies to module spectra over MU.
The new algebraic theory leads to the construction of a sensible site in whic*
*h to
define "integral mixed Tate motives" in algebraic geometry, realizing a program*
* that
was proposed by Deligne.
4 J. P. MAY
These applications are described in our announcements [1] and [2]. However, t*
*hose
notes say nothing about the actual constructions, and my purpose here is to giv*
*e an
intuitive introduction to the foundations that lead to these applications.
I shall sketch some topological definitions to give substance to the discussi*
*on. This
is a distillation of an introduction to the stable homotopy category and is int*
*ended
to give some feeling for the issues involved. Recall that the smash product X *
*^ Y
of based spaces X and Y is the quotient X x Y=X _ Y ; we write F (X; Y ) for t*
*he
function space of based maps X -! Y .
A "universe" U is a countably infinite dimensional real inner product space. *
* It
suffices to think of U = R 1. If V and W are finite dimensional sub inner prod*
*uct
spaces of U and V W , we let W - V denote the orthogonal complement of V in
W . For a based space X, we let V X = X ^ SV and V X = F (SV ; X), where SV
is the one-point compactification of V .
A "prespectrum" T is a collection of based spaces T V and based maps
oe :W-V T V - ! T W
that satisfy an evident transitivity condition. We write
"oe:T V-! W-V T W
for the adjoint of oe. A prespectrum is a "spectrum" if each map "oeis a homeom*
*or-
phism. (We generally write E for a spectrum and T for a prespectrum.) A map
of prespectra is a collection of maps f :T V -! T 0V that are strictly compatib*
*le
with the structure maps oe; a map E -! E0 of spectra is a weak equivalence if *
*each
f :EV -! E0V is a weak equivalence of spaces.
Let PU and S U denote the categories of prespectra and of spectra indexed on
U. There is a "spectrification" functor L: PU - ! S U that is left adjoint t*
*o the
forgetful functor `: S U - ! PU . This is analogous to sheafification from pre*
*sheaves
to sheaves. Constructions made on prespectra are transported to spectra via (L;*
* `).
For example, the smash product of a prespectrum T and a based space X is given
by (T ^ X)(V ) = (T V ) ^ X. The smash product of a spectrum E and a based space
X is then E ^ X = L(`E ^ X). Typically, this procedure is necessary for functors
that are left adjoints, whereas functors that are right adjoints preserve spect*
*ra. For
example, F (X; T )(V ) = F (X; T V ) gives the function prespectrum of a based *
*space
and a prespectrum; if T is a spectrum, then so is F (X; T ).
Since we have based cylinders E^I+ , where the plus denotes adjunction of a d*
*isjoint
basepoint, we have the notion of a homotopy between maps of spectra. There resu*
*lts
a homotopy category hS U of spectra indexed on U, and we obtain hS U by adjoini*
*ng
inverses to the weak equivalences; we abbreviate hS U to hS when U is understoo*
*d.
The suspension and loop functors, E = E ^ S1 and E = F (S1; E), become inverse
equivalences of categories on hS . It is in that sense that hS is a "stable cat*
*egory".
DERIVED CATEGORIES IN ALGEBRA AND TOPOLOGY 5
There is a functor 1 from based spaces to spectra specified by 1 X = {QV X},
where QY = [V V Y . It is left adjoint to the zeroth space functor 1 E = E(0).
We think of 1 X as the stabilization of the space X. Spaces of the form E(0) are
called infinite loop spaces.
There is a theory of CW-spectra that is analogous to the theory of CW complex*
*es.
The only twist is that we have negative dimensional spheres and our cells must
be allowed to take positive and negative dimensions. The stable category hS is
equivalent to the homotopy category of CW spectra and cellular maps.
We now come to the crux of the matter: the construction of smash products of
prespectra and spectra. The obvious definitions would seem to be
(T ^ T 0)(V V 0) = T V ^ T 0V 0 and E ^ E0= L(`E ^ `E0):
This makes sense and works, provided that it is interpreted in the right way. I*
*n fact,
this constructs ^ as a functor S U x S U0 -! S (U U0) for a pair of universes U
and U0. Similarly, we can define explicit function spectra F (E0; E00), where *
*F is a
functor S U0 x S U00- ! S U. The changes of universe are essential. We refer *
*to
this operation as an "external" smash product; it is associative and commutativ*
*e.
To internalize, we choose a linear isometry f : U U -! U and construct a func*
*tor
f* : S (U U) -! S U. We then define an internal smash product by ^ = f* O ^.
Two choices of f give equivalent functors on passage to stable categories, and *
*this
independence of the choice leads to the proof that the stable category level sm*
*ash
product is associative and commutative. It is this internal smash product that *
*was
relevant to the product R ^ R -! R in our original definition of a ring spectr*
*um.
We can collect all choices of f into a single parameter space for smash produ*
*cts
and so eliminate the apparent dependence on f. More generally, we can construct
such parameter spaces for j-fold smash products. Thus let L (j) = I (Uj; U) be *
*the
space of linear isometries Uj ! U. This is a contractible space with a free act*
*ion of
the symmetric group j. We have a system of maps
L (k) x L (j1) x . .x.L (jk) -! L (j1 + . .+.jk);
(g; f1; : :;:fk) -! g O (f1 . . .fk):
These maps are suitably associative, unital, and equivariant. These laws are co*
*dified
in the general notion of an operad, and operads whose jth space is j-free and
contractible for each j are called E1 operads.
We can construct "twisted half-smash product" functors L (j) n E(j), where E(*
*j)
denotes the j-fold external smash power of E. These are functors S U - ! S U.
They are the spectrum level analogs of the evident functors
L (j) n X(j) L (j)+ ^ X(j)
on based spaces X, and we have
1 (L (j) n X(j)) ~=L (j) n (1 X)(j):
6 J. P. MAY
We can now give the fundamental definitions [4, 3]: an E1 ring spectrum is a
spectrum R together with maps
j : L (j) n R(j)-! R
that are suitably associative, unital, and equivariant on the point set level. *
* This is
as close to a commutative and associative ring spectrum as one can hope to get.*
* A
module (or E1 module) over R is a spectrum M together with maps
j : L (j) n (R(j-1)^ M) -! M
that, with the j, are suitably associative, unital, and equivariant.
Most of the important cohomology theories in algebraic topology are represent*
*ed by
E1 ring spectra. Examples include the sphere spectrum S; the Eilenberg-MacLane
spectra HA for discrete commutative rings A; the Thom spectra MO, MU, and
MSp; the connective K-theory spectra kO and kU; and the algebraic K-theory
spectra KA of discrete commutative rings. Many other examples are constructed
via multiplicative infinite loop space theory [5]; that theory allows one to co*
*nstruct
E1 ring spectra from E1 ring spaces, which in turn arise from suitable categor*
*ies
with and . Recent work of Hopkins, Miller, McClure, Kriz, Elmendorf, Vogt,
Schw"anzl, and others has given still more examples. A very rich theory of E1 *
*rings,
including "cell theory" (Hopkins) and "Postnikov systems" (Kriz), is now emergi*
*ng.
Much of it depends on the theory I am about to describe.
The definitions just given are the right ones, but they are rather hard to wo*
*rk with.
Our recent breakthrough recasts these notions in a far more conceptual and work*
*able
form. To explain the idea, we return to our analogy and consider its algebraic *
*side.
Operads of (differential graded) k-modules are defined by replacing Cartesian*
* prod-
ucts of spaces with tensor products of k-modules. They can be obtained, for exa*
*mple,
by applying the normalized singular k-chain functor to an operad of spaces. An *
*op-
erad C of k-modules is an E1 operad if C (j) is a free k[j]-resolution of k fo*
*r each
j; the chain operad of an E1 operad of spaces is an example. To fix ideas, we *
*agree
to let C denote the chain operad so obtained from our topological E1 operad L .
We define an E1 k-algebra to be a k-module A together with maps
j : C (j) A(j)-! A;
where A(j)denotes the j-fold tensor power of A; these maps must satisfy associa*
*tiv-
ity, unity, and equivariance relations exactly like those in the definition of *
*E1 ring
spectra. Modules over such algebras are defined similarly in terms of maps
j : C (j) (A(j-1) M) -! M:
These definitions are forced by examples from algebraic geometry. Deligne, se*
*eking
foundations for an integral theory of mixed Tate motives, asked me if Bloch's C*
*how
complex of an algebraic variety, which is a simplicial abelian group with a par*
*tially
DERIVED CATEGORIES IN ALGEBRA AND TOPOLOGY 7
defined product, might give rise to a quasi-isomorphic E1 algebra, and, if so,*
* if there
might then be a good derived category of modules over an E1 algebra. Kriz and I
gave positive answers to these questions. Unless k is a field of characteristi*
*c zero,
one cannot hope to replace the resulting E1 algebras by quasi-isomorphic genui*
*ne
DGA's. The present level of generality is essential. As an aside, the topolog*
*ical
theory applied to the algebraic K-theory spectra of fields gives an alternative*
* site for
a possible theory of mixed Tate motives.
As another digression, operads and their actions are now playing a serious ro*
*le in
differential geometry and mathematical string theory. Here En operads, related*
* to
n-fold loop spaces, play a fundamental role. Different types of discrete operad*
*s define
different types of algebras, such as Lie algebras, and Lie algebras up to homot*
*opy
characterized by actions by appropropriate Lie-like chain operads are also play*
*ing an
important role. Recall that a module over a Lie algebra is the same thing as a *
*mod-
ule over its universal enveloping algebra, which is an associative algebra. Pr*
*ecisely
mimicking the proof, one can show that, for any operad C and C -algebra A, there
is an associative universal enveloping DGA U(A) such that an A-module is the sa*
*me
thing as a U(A)-module.
Now return to our particular operad C . The ground ring k is a C -algebra via
augmentations, and its universal enveloping algebra turns out to be U(k) = C (1*
*).
That is, as one can easily check directly from the formal definitions, if we re*
*gard k as
an E1 k-algebra, we find that an E1 k-module is the same thing as a C (1)-mod*
*ule.
Analogously, in topology, S is an E1 ring spectrum, and an E1 S-module is the
same thing as a spectrum with an "action of the monoid L (1)" defined in terms *
*of
an associative and unital action map L (1) n M - ! M .
A fundamental idea at this point is to switch ground categories from k-modules
and spectra to E1 k-modules and S-modules. Here a miracle occurs. We define a
new tensor product of E1 k-modules in algebra or smash product of E1 S-modules
in topology. In algebra, the definition is:
M N C (2) C(1)C(1) M N:
In more detail, instances of the operad structure maps
C (k) x C (j1) x . .x.C (jk) -! C (j1 + . .+.jk)
give a left action of C (1) and a right action of C (1)C (1) on C (2). The latt*
*er action
is used to make sense of the displayed tensor product, and the former action gi*
*ves
the new tensor product a structure of C (1)-module. This is already remarkable:*
* the
algebra C (1) is not commutative, so it is rather surprising to have an interna*
*l tensor
product on its modules. This much would be true for any operad. The real miracle
is that, with our particular choice of operad C , this tensor product turns out*
* to be
associative and commutative, with a natural unit equivalence : k M - ! M .
8 J. P. MAY
We can now define an E1 k-algebra A to be a C (1)-module together with a pro*
*d-
uct OE : A A - ! A and unit j : k - ! A such that the evident associativity,
commutativity, and unit diagrams (like those displayed at the start) are commut*
*a-
tive. Similarly, we define an A-module M to be a C (1)-module together with an
action : A M -! M such that the evident associativity and unit diagrams
commute. Moreover, we can define the tensor product of A-modules M and N with
actions and to be the coequalizer (or difference cokernel) displayed in the d*
*iagram
__Id______//
M A N __Id_____M_/N/ ____M_A/N:/
There is a concomitant internal hom functor HomA(M; N); it is defined as an app*
*ro-
priate equalizer.
Actually, one can be more categorically precise about this. For C (1)-modules*
* M
with a given "unit map" j :k -! M , we can define a variant, say, of the prod*
*uct
which is not only associative and commutative, but also unital with unit k. The
modified product is defined by the pushout diagram
k N [kk M k [____M/[k/N
jId+ Idj || ||
|fflffl |fflffl
M N __________M//N:
An E1 k-algebra is precisely the same thing as a commutative monoid in the sym-
metric monoidal category of unital C (1)-modules. There is a similar way to be *
*more
precise about the notion of an A-module.
The topological theory works just the same way. For (E1 ) S-modules M and N,
we can make sense of the definition
M ^S N L (2) nL (1)xL (1)M ^ N:
This is again an S-module, and this smash product over S is an associative and *
*com-
mutative operation with a natural unit equivalence : S ^S M - ! M . We redefi*
*ne
an E1 ring spectrum to be an S-module with a product OE : R ^S R -! R and unit
j : S -! R such that, with ^ replaced by ^S, the diagrams that we gave at the s*
*tart
commute. The point is that the commutation now makes sense and is required on t*
*he
point set level, that is, in the category of S-modules. We define R-modules sim*
*ilarly
in terms of action maps : R ^S M -! M, and we define the smash product over
R of R-modules M and N to be the coequalizer
___^Id____//
M ^S R ^S N __________M/^S/N _____M/^R/N:
Id^
There is a concomitant right adjoint function R-module FR(M; N).
Again, there is a variant of the smash product over S, ?S say, that is define*
*d on
S-modules M with unit maps j :S -! M and that is associative, commutative, and
DERIVED CATEGORIES IN ALGEBRA AND TOPOLOGY 9
unital. It is defined by a pushout diagram just like that defining . An E1 r*
*ing
spectrum is precisely a commutative monoid in the symmetric monoidal category of
unital S-module spectra. There is a similar way to be more precise about R-modu*
*les.
Here another miracle occurs: these simple conceptual definitions turn out to be
equivalent to the pre-existing definitions in terms of actions by the linear is*
*ometries
operad L . This allows use of the older theory to supply examples, which can th*
*en
be studied algebraically by means of our new theory.
In particular, we can mimic the theory of cell spectra to develop a theory of*
* cell
R-modules. A map of R-module spectra is said to be a weak equivalence if it is*
* a
weak equivalence as a map of spectra. The derived stable homotopy category of R-
modules, hMR, is constructed from the homotopy category of R-modules by formally
inverting the weak equivalences, and it is equivalent to the homotopy category *
*of cell
R-modules. It is a triangulated category, and it is symmetric monoidal under t*
*he
derived smash product of R-modules. It provides the starting point for the vari*
*ous
applications that we listed at the start.
We think of the sphere spectrum S as a universal ground ring. For any E1 ring
spectrum R, R-modules are S-modules by neglect of structure. The stable homotopy
category hMS provides an improved substitute for the stable homotopy category hS
that we started with.
Theorem 1. The forgetful functor MS -! S induces an equivalence of categories
hMS -! hS . For S-modules M and N, there are natural isomorphisms in hS
M ^ N ' M ^S N and F (M; N) ' FS(M; N):
The topology now feeds back into algebra in a most amusing fashion. The stand*
*ard
treatment of tensor products and internal hom functors in the derived category *
*of
differential graded modules M over a DGA A entails the use of suitable projecti*
*ve
resolutions of such modules. These are awkward to deal with for general, unboun*
*ded,
modules. These difficulties disappear if one mimics the topologists' treatment*
* of
smash products and function spectra in the stable homotopy category. There is a*
* very
simple theory of cell A-modules which provides a substitute for projective reso*
*lutions.
Here free A-modules on one generator substitute for sphere spectra as the domai*
*ns
of attaching maps of cells. Topological results such as Whitehead's theorem and
Brown's representability theorem transcribe directly into algebra. Every A-mod*
*ule
M is quasi-isomorphic to a cell A-module, and the derived category DA is equiva*
*lent
to the homotopy category of cell A-modules. Moreover, this treatment works equa*
*lly
well in the more general context of modules over E1 k-algebras A. Here again, *
*the
derived category DA of A-modules is triangulated, and it is symmetric monoidal
under the derived tensor product. It is just such a generalized derived categor*
*y that
is needed to realize Deligne's program for defining a good category of integral*
* mixed
Tate motives.
10 J. P. MAY
To summarize, we display the more sophisticated and precise analogy between
algebra and topology that emerges from our discussion. For a (differential gra*
*ded)
k-module M, write
Mq = Hq(M) = M-q:
For a spectrum M, write
Mq = ssq(M) = M-q:
______________________________________________________________________|||
|______________ALGEBRA______________|___________TOPOLOGY__________|___|||
|_____E1_k_modules_=_C_(1)-modules_____|E1_S-modules_=_L_(1)-modules__||||
|__________"tensor_product"___________|_____smash_product_over_S______||||
|_______internal_hom_Hom___(X;_Y_)_______|function_spectrum_FS(X;_Y_)_||||
|| E1 k-algebra A | | E1 ring spectrum R ||
|| = commutative monoid in the | =|commutative monoid in the | |
|____category_of_unital_C_(1)-modules__ca|tegory_of_unital_L_(1)-modules_||||
|_______________A-module_______________|_________R-module____________|_|||
|___________quasi-isomorphism___________|_____weak_equivalence________ |
| derived category D s|table homotopy category hM |
|____________________________A__________|__________________________R__||||
|_____________cell_A-module_____________|_______cell_R-module_________||||
|__________finite_cell_A-module__________|___finite_cell_R-module_____||||
|_________tensor_product_over_A_________|___smash_product_over_R______||||
|_______internal_hom_Hom_A_(M;_N)_______|function_R-module_FR(M;_N)___||||
|______dual_DA(M)_=_Hom_A(M;_A)______|____dual_DR(M)_=_FR(M;_R)_____|_|||
||Hom A (X; Y ) A E ' Hom A(X; Y E)|| FR(X; Y ) ^R E ' FR(X; Y ^R E)||
|| DAX A E ' Hom A(X; E) | |DRX ^R E ' FR(X; E) | |
|_____(X_or_E_a_finite_cell_A-module)___(X|or_E_a_finite_cell_R-module)_||||
| spectral sequence | spectral sequence |
| TorA*(M ; N ) =) (M N) | TorR*(M ; N ) =) (M ^ N) |
|____________*___*________A___*_____|__________*___*_________R___*__|_|||
|| spectral*sequence * | | spectral*sequence * | |
|___Ext_A*(M*;_N_)_=)_HomA(M;_N)____|___ExtR*(M*;_N_)_=)_FR(M;_N)___|_
In one important case, the analogy reduces to an equivalence of derived categ*
*ories
in algebra and topology.
Theorem 2. Let A be a commutative ring. Then A-modules M can be realized
functorially by Eilenberg-Mac Lane spectra HM that are modules over the E1 ring
spectrum HA, and
Tor A*(M; N) ~=(HM ^HA HN)* and Ext*A(M; N) ~=FHA (HM; HN)*
as A-modules. Further, the stable homotopy category of HA-modules is equivalent
to the derived category of A-modules.
DERIVED CATEGORIES IN ALGEBRA AND TOPOLOGY 11
The essential point is that HM ^HA HN and FHA (HM; HN) are equivalent to
derived tensor product and Hom functors in the category of chain complexes of A-
modules. The spectral sequences at the end of our displayed analogy are the app*
*ro-
priate generalizations to E1 algrabras and E1 ring spectra of the isomorphism*
*s of
the theorem. In topology, they specialize to give generalized K"unneth and univ*
*ersal
coefficients spectral sequences.
References
1. A. Elmendorf, J. P. C. Greenlees, I. Kriz, and J. P. May. Commutative algebr*
*a in stable homo-
topy theory and a completion theorem. Mathematical Research Letters. To appe*
*ar.
2. I. Kriz and J. P. May. Derived categories and motives. Mathematical Research*
* Letters. To
appear.
3. L. G. Lewis, Jr., J. P. May, and M. Steinberger (with contributions by J. E.*
* McClure). Equiv-
ariant stable homotopy theory. Springer Lecture Notes Vol 1213. 1986.
4. J. P. May (with contributions by F. Quinn, N. Ray, and J. T"ornehave). E1 ri*
*ng spaces and
E1 ring spectra. Springer Lecture Notes Vol 577. 1977.
5. J. P. May. Multiplicative infinite loop space theory. J. Pure and Applied Al*
*gebra 26(1982), 1-69.
University of Chicago, Chicago Il 60637
E-mail address: may@math.uchicago.edu