DUALITY IN ALGEBRA AND TOPOLOGY
W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
1. Introduction
In this paper we take some classical ideas from commutative algebra,
mostly ideas involving duality, and apply them in algebraic topology.
To accomplish this we interpret properties of ordinary commutative
rings in such a way that they can be extended to the more general
rings that come up in homotopy theory. Amongst the rings we work
with are the differential graded ring of cochains on a space X, the dif-
ferential graded ring of chains on the loop space X, and various ring
spectra, e.g., the Spanier-Whitehead duals of finite spectra or chro-
matic localizations of the sphere spectrum.
Maybe the most important contribution of this paper is the concep-
tual framework, which allows us to view all of the following dualities
o Poincar'e duality for manifolds
o Gorenstein duality for commutative rings
o Benson-Carlson duality for cohomology rings of finite groups
o Poincar'e duality for groups
o Gross-Hopkins duality in chromatic stable homotopy theory
as examples of a single phenomenon. Beyond setting up this frame-
work, though, we prove some new results, both in algebra and topology,
and give new proofs of a number of old results. Some of the rings we
look at, such as C* X, are not commutative in any sense, and so im-
plicitly we extend the methods of commutative algebra to certain non-
commutative settings. We give a new formula for the dualizing module
of a Gorenstein ring; this formula involves differential graded algebras
(or ring spectra) in an essential way and is one instance of a general
construction that in another setting gives the Brown-Comenetz dual
of the sphere spectrum. We also prove the local cohomology theorem
for p-compact groups [16 ] and reprove it for compact Lie groups. The
existing proof for compact Lie groups [6] uses equivariant topology, but
our extension does not.
____________
Date: March 11, 2002.
1
2 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
1.1. Description of results. The objects we work with are fairly gen-
eral; briefly, we allow rings, differential graded algebras (DGAs), or ring
spectra; these are all covered under the general designation S-algebra
(see 1.3). We usually work in a derived category or in a homotopy
category of module spectra; most of the time we start with a homo-
morphism R ! k of S-algebras and let E denote the endomorphism
S-algebra End R (k). There are three main parts to the paper, which
deal with three different but related types of structures: regularity,
duality, and the Gorenstein condition.
Regularity. There are several different kinds of regularity which the ho-
momorphism R ! k might possess (2.12); the weakest and most flexi-
ble one is called proxy-regularity. Any surjection from a commutative
noetherian ring to a regular ring is proxy-regular (3.2). One property of
a proxy-regular homomorphism is particularly interesting to us. Given
an R-module M, there is an associated module Cellk(M), which is the
closest R-module approximation to M which can be cobbled together
from shifted copies of k by using sums and exact triangles. If R ! k
is proxy-regular, there is a canonical equivalence (2.10)
CellkM ~ Hom R(k, M) E k .
The notation Cellk(M) comes from topology [10 ], but if R is a commu-
tative ring and k = R=I for a finitely generated ideal I, then Cellk(M)
is the local (hyper)cohomology of M at I [13 , x6].
Duality. Given R ! k, we look for a notion of öP ntriagin duality"
over R which extends the notion of ordinary duality over k; in other
words, we look for an R-module I such that for any k-module X, there
is a natural identification
Hom R (X, I) ~ Hom k(X, k) .
The associated Pontriagin duality (or Matlis duality) for R-modules
sends M to Hom R(M, I). If R ! k is Z ! Fp, there is only one such
I, namely Z=p1 , and Hom Z(-, Z=p1 ) is ordinary p-local Pontriagin
duality for abelian groups. We find that in many circumstances, and
in particular if R ! k is proxy-regular, such dualizing modules I are
determined by right E-module structures on k; such a structure is a
new bit of information, since in its state of nature E acts on k from the
left. Given a suitable right action, the dualizing module I is given by
the formula
I ~ k E k ,
which mixes the exceptional right action of E on k with the canoni-
cal left action. This is a formula which in one setting constructs the
DUALITY 3
injective hull of the residue class field of a local ring (5.1), and in an-
other gives the p-primary component of the Brown-Comenetz dual of
the sphere spectrum (5.3). There are also other examples (x5).
The Gorenstein condition. The homomorphism R ! k is said to be
Gorenstein if Hom R(k, R) is equivalent to a shifted copy ak of k itself,
and the right action of E on k provided by this equivalence is suitable as
above for forming a dualizing module I. There are several consequences
of the Gorenstein condition. It is immediately clear (4.7) that there
are equivalences
I = k E k = -a Hom R(k, R) E k ~ -a CellkR .
In the commutative ring case this gives a connection between the du-
alizing module I and the local cohomology object Cellk R. In this
paper we head in a slightly different direction. Suppose that R is an
augmented k-algebra and R ! k is the augmentation; in this case
it is possible to compare the two right E-modules Hom R(k, R) and
Hom R(k, Hom k(R, k)). Given that R ! k is Gorenstein, the first is ab-
stractly equivalent to ak; the second, by an adjointness argument, is
always equivalent to k. If these two objects are the same as E-modules
after the appropriate shift, we obtain a formula
a CellkHom k(R, k) ~ CellkR ,
relating duality on the left to local cohomology on the right. In many
circumstances CellkHom k(R, k) is equivalent to Hom k(R, k) itself, and
in these cases the above formula becomes
a Hom k(R, k) ~ CellkR .
This leads to spectral sequences relating the local cohomology of a
ring to some kind of k-dual of the ring, for instance, if X is a suitable
space, relating the local cohomology of H*(X; k) to H*(X; k). We use
this approach to reprove the local cohomology theorem for compact Lie
groups and prove it for p-compact groups.
We intend to treat the two special cases of chromatic stable homo-
topy theory (Gross-Hopkins duality) and local algebra in papers [14 ]
and [15 ].
1.2. Organization of the paper. The three main themes, regularity,
duality, and the Gorenstein condition, are treated respectively in Sec-
tions 2, 4, and 6. Section 7 explains how to set up a local cohomology
spectral sequence for a suitable Gorenstein S-algebra. We spend a lot
of time dealing with examples; x3 has examples relating to regularity,
x5 examples related to duality, and x8 examples related to the Goren-
stein condition. In particular, Section 8 contains a proof of the local
4 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
cohomology theorem for p-compact groups (8.2) and for compact Lie
groups (8.3); following [6], this is one version of Benson-Carlson duality
[5]. The final section gathers together some technical material which
we refer to in the course of the paper.
1.3. Notation and terminology. In this paper we use the term S-
algebra to mean ring spectrum in the sense of [18 ] or [25 ]; the symbol
S stands for the sphere spectrum. If k is a commutative S-algebra,
we refer to algebra spectra over k as k-algebras. The sphere S is itself
a commutative ring spectrum, and, as the terminology "S-algebra"
suggests, any ring spectrum is an algebra spectrum over S.
Any ring R gives rise to an S-algebra (the corresponding Eilenberg-
MacLane ring spectrum), and we do not make a distinction in notation
between R and its associated spectrum. If R is commutative in the
usual sense it is also commutative as an S-algebra; the category of R-
algebras (in the way in which we use the term) is then equivalent to the
more familiar category of differential graded algebras (DGAs) over R.
For instance, Z-algebras are essentially DGAs; Q-algebras are DGAs
over the rationals. A module M over an S-algebra R is for us a module
spectrum over R; the category of these is denoted R Mod . If R is a
Z-algebra, this is essentially the same as the category of differential
graded modules over the corresponding DGA [38 ]. In particular, if R
is a ring, an R-module in our sense is essentially a chain complex of
ordinary R-modules; any ordinary module M gives rise to a module
in our sense (Eilenberg-MacLane module spectrum) by the analog of
treating M as a differential graded module concentrated in degree 0.
We will refer to such an M as a discrete module over R, and we will
not distinguish in notation between M and its associated Eilenberg-
MacLane spectrum. See [19 ] and [37 ] for details of the above.
Homotopy/homology. The homotopy groups of an S-algebra R and an
R-module M are denoted respectively ß*R and ß*M. The group ß0R is
always a ring, and a ring is distinguished among S-algebras by the fact
that ßiR ~=0 for i 6= 0. If R is a Z-algebra and M is an R-module, the
homotopy groups ß*R and ß*M amount to the homology groups of the
corresponding differential graded objects. A homomorphism R ! S of
S-algebras or M ! N of modules is an equivalence (weak equivalence,
quasi-isomorphism) if it induces an isomorphism on ß*. In this case
we write R ~ S or M ~ N. An S-algebra R is connective if ßiR = 0
for i < 0 and coconnective if ßiR = 0 for i > 0. An R-module M is
bounded below if ßiM = 0 for i << 0, and bounded above if ßiM = 0
for i >> 0.
DUALITY 5
Hom and tensor. Associated to two R-modules M and N is a spec-
trum Hom R(M, N) of homomorphisms; each R-module M also has an
endomorphism ring End R(M). These are derived objects; for instance,
in forming End R (M) we always tacitly assume that M has been re-
placed by an equivalent R-module which is cofibrant (projective) in
the appropriate sense. Note that unspecified modules are left modules.
If M and N are respectively right and left modules over R, there is
a derived smash product, which corresponds to tensor product of dif-
ferential graded modules, and which we write M R N. To fix ideas,
suppose that R is a ring, M is a discrete right module over R, and
N, K are discrete left modules. Then ßi(M R N) ~= Tor Ri(M, N),
while ßiHom R(K, N) ~= Ext-iR(K, N). In this situation we sometimes
write hom R (M, N) (with a lower-case "h") for the group Ext 0R(M, N)
of ordinary R-maps M ! N.
There are other contexts in which we follow the practice of tacitly
replacing one object by an equivalent one without changing the nota-
tion. For instance, suppose that R ! k is a map of S-algebras, and let
E = End R(k). The right action of k on itself commutes with the left
action of R, and so produces what we refer to as a öh momorphism
kop ! E", although in general this homomorphism can be realized as a
map of S-algebras only after adjusting k up to weak equivalence. The
issue is that in order to form End R(k), it is necessary to work with a
cofibrant (projective) surrogate for k as a left R-module, and the right
action of k on itself cannot in general be extended to an action of k on
such a surrogate without tweaking k to some extent. The reader might
want to consider the example R = Z, k = Fp from [13 , x3], where it
is clear that the ring Fp cannot map to the DGA representing E, al-
though a DGA weakly equivalent to Fp does map to E. In general we
silently pass over these adjustments and replacements in order to keep
the exposition within understandable bounds.
Derived category. The derived category D(R) = Ho (R Mod ) of an S-
algebra R is obtained from R Mod by formally inverting the weak equiv-
alences. A map between R-modules passes to an isomorphism in D(R)
if and only if it is a weak equivalence. Sometimes we have to consider a
homotopy category Ho (Mod R) involving right R-modules; since a right
R-module is the same as a left module over the opposite ring Rop,
we write Ho (Mod R) as D(Rop). If R is a ring, D(R) is categorically
equivalent to the usual derived category of R.
Augmentations. Many of the objects we work with are augmented.
An augmented k-algebra R is a k-algebra together with an augmenta-
tion homomorphism R ! k which splits the k-algebra structure map
6 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
k ! R. A map of augmented k-algebras is a map of k-algebras which
respects the augmentations. If R is an augmented k-algebra, we will
by default treat k as an R-module via the homomorphism R ! k.
Another path. The advantage of using the term S-algebra is that we
can refer to rings, DGAs, and ring spectra in one breath. The reader
can confidently take S = Z, read DGA for S-algebra, H* for ß*, and
work as in [13 ] in the algebraic context of [38 ]; only some examples
will be lost. Note however that all of the examples involving commuta-
tive objects (e.g. cochains on a space) will be put at risk, since under
the correspondence between Z-algebras and DGAs, the notion of com-
mutativity for Z-algebras does not carry over to the usual notion of
commutativity for DGAs (except in characteristic 0 [32 , App. C] [28 ]).
However, if R is a ring, then R is commutative as a Z-algebra if and
only if R is commutative in the usual sense.
1.4. Relationship to previous work. There is a substantial litera-
ture on Gorenstein rings. Our definition of a Gorenstein map R ! k of
S-algebras extends the definition of Avramov-Foxby [4] (see 6.4). F'elix,
Halperin, and Thomas have considered pretty much this same exten-
sion in the topological context of rational homotopy theory and DGAs
[20 ]; we generalize their work and have benefitted from it. Frankild
and Jorgensen [21 ] have also studied an extension of the Gorenstein
condition to DGAs, but their intentions are quite different from ours.
2. Smallness and regularity
In this section we describe the main setting that we work in; for
completeness, we work in slightly more detail than we will need later
on. We start with a pair (R, k), where R is an S-algebra and k is an
R-module. Later on it will usually be the case that k is an R-module
via an S-algebra homomorphism R ! k.
2.1. Cellular modules and complete modules. A map U ! V
of R-modules is a k-equivalence if the induced map Hom R(k, U) !
Hom R(k, V ) is an equivalence. An R-module M is said to be k-cellular
or k-torsion ([13 , x4], [10 ]) if any such k-equivalence induces an equiv-
alence Hom R(M, U) ! Hom R(M, V ). This turns out to be the same as
requiring that M be built from k, in the sense that M belongs to the
smallest class of R-modules which contains ik, i 2 Z, and is closed
under coproducts, cofibration sequences (triangles), retracts, and weak
equivalences. A k-equivalence between k-cellular objects is necessar-
ily an equivalence. We let Cell(R, k) denote the full subcategory of
R Mod containing the k-cellular objects, and DCell(R, k) the corre-
sponding subcategory of the derived category D(R). For any R-module
DUALITY 7
X there is a k-cellular object Cellk(X) together with a k-equivalence
Cellk(X) ! X; such an object is unique up to a canonical equiva-
lence and is called the k-cellular approximation to X. If we want to
emphasize the role of R we write CellRk(X).
Dually, an R-module M is k-complete if any k-equivalence U ! V
induces an equivalence Hom R(V, M) ! Hom R(U, M). A k-equivalence
between k-complete objects is necessarily an equivalence. The category
Comp (R, k) is the full subcategory of R Mod containing the k-complete
objects, and DComp (R, k) the corresponding subcategory of D(R). If
X is an R-module, a k-completion of X is a k-complete module Y
together with a k-equivalence X ! Y ; such a k-completion, if it exists,
is unique up to a canonical equivalence.
2.2. Smallness. We say that M is a finite k-cellular complex, or M
is finitely built from k if M 2 Cell(R, k) and M can be constructed in
finitely many steps from k and its shifts by cofibration sequences and
retracts. There are three special cases to consider.
2.3. Definition. The R-module k is small if k is finitely built from
R, and cosmall if R is finitely built from k. Finally, k is proxy-small
if there exists an R-module K, finitely built from R and also finitely
built from k, such that Cell(R, k) = Cell(R, K). The object K is then
called a Koszul complex associated to k (cf. 3.2).
2.4. Remark. The R-module k is small if and only if Hom R(k, -) com-
mutes with arbitrary coproducts; if R is a ring this is equivalent to
requiring that k be a perfect complex, i.e., isomorphic in D(R) to a
chain complex of finite length whose constituents are finitely generated
projective R-modules.
2.5. Remark. The condition Cell(R, k) = Cell(R, K) in 2.3 amounts
to the requirement that k and K can be built from one another; this
implies that k-equivalences are the same as K-equivalences, and hence
that Comp (R, k) = Comp (R, K). If k is either small or cosmall it
is also proxy-small; in the former case take K = k and in the latter
K = R.
One of the main results of [13 ] is the following; although in [13 ] it is
phrased for DGAs, the proof for general S-algebras is the same.
2.6. Theorem. Suppose that k is a small R-module. Let E = End R(k),
and let E be the functor which assigns to an R-module M the right E-
module Hom R(k, M). Then E restricts to give categorical equivalences
DCell(R, k) ! D(Eop) and DComp (R, k) ! D(Eop).
8 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
2.7. Remark. The inverse functors are given by
T :D(Eop) ! DCell(R, k) T (X) = X E k
C :D(Eop) ! DComp (R, k) C(X) = Hom Eop(k# , X)
Here k# is the ordinary R-dual Hom R(k, R) of k. The functor T is
always a left adjoint to E; under the assumption that k is small, C is a
right adjoint. If k is small, then for any R-module M, T E(M) ! M is
a k-cellular approximation map, and M ! CE(M) is a k-completion
map.
There is a generalization of this to the proxy-small case.
2.8. Theorem. Suppose that k is a proxy-small R-module with Koszul
complex K. Let E = End R(k), J = Hom R(k, K), and EK = End R(K).
Then the five categories
DCell(R, k), DComp (R, k), DCell(Eop, J), DComp (Eop, J), D(EopK)
are all equivalent to one another.
2.9. Remark. We leave it to the reader to work out the functors that
induce the various equivalences.
Proof of 2.8. We will show that J is a small Eop-module, and that the
natural map EK ! End Eop(J) is an equivalence. The theorem is then
proved by applying 2.6 serially to the pairs (EopK, J) and (R, K) whilst
keeping 2.5 in mind. For the smallness, observe that since K is finitely
built from k as an R-module, J = Hom R(k, K) is finitely built from
E = Hom R(k, k) as a right E-module. Next, consider all R-modules X
with the property that for any R-module M the natural map
Hom R (X, M) ! Hom Eop(Hom R(k, X), Hom R (k, M))
is an equivalence. The class includes X = k by inspection, and hence
by triangle arguments any X finitely built from k, in particular X =
K.
2.10. Proposition. Suppose that k is a proxy-small R-module, and let
E = End R(k). Then for any R-module M the natural map
Hom R (k, M) E k ! M
is a k-cellular approximation. In particular, the map is a k-equivalence,
and an equivalence if M is k-cellular.
Proof. Let K be a Koszul complex for k, and EK = End R (K). By
2.7, the natural map Hom R(K, M) EK K ! M is a K-cellular ap-
proximation, and hence (2.5) a k-cellular approximation. We wish to
analyze the domain of the map. Let J = Hom R(k, K). As in the
proof of 2.8, Hom R(K, M) is equivalent to Hom Eop(J, Hom R (k, M)),
which, because J is small as a right E-module, is itself equivalent to
DUALITY 9
Hom R(k, M) E Hom Eop(J, E). Since E ~ Hom R(k, k), the second fac-
tor of the tensor product is (again as in the above proof) equivalent to
Hom R(K, k). We conclude that the natural map
Hom R (k, M) E (Hom R(K, k) EK K) ! M
is a k-cellular approximation. But the factor Hom R(K, k) EK K is
equivalent to k, since by 2.6 the map Hom R(K, k) EK K ! k is a K-
cellular approximation and hence also (2.5) a k-cellular approximation.
Of course, k itself is already k-cellular.
2.11. Regularity conditions. Now we identify certain S-algebra ho-
momorphisms which are particularly convenient to work with. See 3.2
for the main motivating example.
2.12. Definition. An S-algebra homomorphism R ! k is regular if k
is small as an R-module, coregular if k is cosmall, and proxy-regular if
k is proxy-small.
2.13. Remark. As in 2.5, if R ! k is either regular or coregular it is
also proxy-regular. These are three very different conditions to put on
the map R ! k, with proxy-regularity being by far the weakest one
(see 3.2).
When it comes to rings, our terminology differs in some instances
from the usage in commutative algebra. Recall that a commutative ring
R is regular (in the absolute sense) if every finitely-generated discrete
R-module M is small, i.e., has a finite length resolution by finitely
generated projectives. Suppose that f : R ! k is a surjection of
commutative noetherian rings. If f is regular as a map of rings it
is regular as a map of S-algebras, but the converse holds in general
only if k is a regular ring; the point is that for f to be regular in
the ring-theoretic sense certain additional conditions must be satisfied
by the fibres of R ! k. Perhaps this terminological discrepancy will
eventually be cleared up by a better understanding of the algebraic
geometry of S-algebras.
2.14. Relationships between types of regularity. Suppose that k is an R-
module and that E = End R(k). The double centralizer of R is the ring
^R= End E(k). Left multiplication gives a ring homomorphism R ! ^R,
and the pair (R, k) is said to be dc-complete if the homomorphism
R ! R^ is an equivalence. Note that if R ! k is a surjective map of
noetherian commutative rings with kernel I R, then, as long as k is
a regular ring, (R, k) is dc-complete if and only if R is isomorphic to
its I-adic completion (9.18).
If R is an augmented k-algebra, then E = End R(k) is also an aug-
mented k-algebra. The augmentation is provided by the natural map
10 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
End R(k) ! End k(k) ~ k induced by the k-algebra structure homo-
morphism k ! R.
2.15. Proposition. Suppose that R is an augmented k-algebra, and
let E = End R (k). Assume that the pair (R, k) is dc-complete. Then
R ! k is regular if and only if E ! k is coregular. Similarly, R ! k
is proxy-regular if and only if E ! k is proxy-regular.
Proof. If k is finitely built from R as an R-module, then by applying
Hom R(-, k) to the construction process, we see that E = Hom R(k, k)
is finitely built from k = Hom R(R, k) as an E-module. Conversely, if E
is finitely built from k as an E-module, it follows that k = Hom E(E, k)
is finitely built from R^ ~ Hom E(k, k) as an R-module. If R ~ ^R, this
implies that k is finitely built from R.
For the rest, it is enough by symmetry to show that if R ! k is
proxy-regular, then so is E ! k. Suppose then that k is proxy-small
over R with Koszul complex K. Let L = Hom R(K, k). Arguments
as above show that L is finitely built both from Hom R(R, k) ~ k and
from Hom R(k, k) ~ E as an E-module. This means that L will serve as
a Koszul complex for k over E, as long as L builds k over E. Let EK =
End R(K). By 2.10, the natural map L EK K ! k is an equivalence;
it is evidently a map of E-modules. Since EK builds K over EK , L ~
L EK EK builds k over E.
2.16. Proposition. Suppose that S ! R and R ! k are homomor-
phisms of commutative S-algebras, and let Q = R S k. Note that Q
is a commutative S-algebra and that there is a natural homomorphism
Q ! k which extends R ! k. Assume that one of the following holds:
(1) S ! k is proxy-regular and Q ! k is coregular, or
(2) S ! k is regular and Q ! k is proxy-regular.
Then R ! k is proxy-regular.
Proof. In case 1, suppose that K is a Koszul complex for k over S. We
will show that R S K is a Koszul complex for k over R. Since K is
small over S, R S K is small over R. Since k finitely builds K over S,
R S k = Q finitely builds R S K over R. But k finitely builds Q over
Q, and hence over R; it follows that k finitely builds R S K over R.
Finally, K builds k over S, and so R S K builds Q over R; however,
Q clearly builds k as a Q-module, and so a fortiori builds k over R.
In case 2, let K be a Koszul complex for k over Q. We will show that
K is also a Koszul complex for k over R. Note that S ! k is regular,
so that k is small over S and hence Q = R S k is small over R. But
K is finitely built from Q over Q and hence over R; it follows that K
DUALITY 11
is small over R. Since k finitely builds K over Q, it does so over R; for
a similar reason K builds k over R.
3. Examples of regularity
In this section we look at some sample cases in which the regular-
ity conditions of x2 are or are not satisfied. Several of the examples
are topological, so before proceeding we recall some topological back-
ground.
3.1. Topological background. Suppose that X is a connected pointed
topological space, and that k is a commutative S-algebra. For any Y
let 1 Y denote the unpointed suspension spectrum of Y , in other
words, the ordinary suspension spectrum of Y+ , where Y+ is Y with a
disjoint basepoint added. We will consider two k-algebras associated
to the pair (X, k): the chain algebra C*( X; k) = k S 1 ( X) and
the cochain algebra C*(X; k) = Map S( 1 X, k). Here X is the loop
space on X, and C*( X; k) is an S-algebra because X can be con-
structed as a topological or simplicial group; C*( X; k) is essentially
the group ring k[ X]. The multiplication on C*(X; k) is cup product
coming from the diagonal map on X, and so C*(X; k) is a commuta-
tive k-algebra. Both of these objects are augmented, one by the map
C*( X; k) ! k induced by the map X ! pt, the other by the map
C*(X; k) ! k induced by the basepoint inclusion pt ! X. If k is a
ring, then ßiC*( X; k) ~=Hi( X; k) and ßiC*(X; k) ~=H-i(X; k).
The Rothenberg-Steenrod construction [36 ] shows that for any X
and k there is an equivalence C*(X; k) ~ End C*( X;k)(k). We will
say that the pair (X, k) is of Eilenberg-Moore type if k is a field, each
homology group Hi(X; k) is finite dimensional over k, and either
(1) X is simply connected, or
(2) k is of characteristic p and ß1X is a finite p-group.
If (X, k) is of Eilenberg-Moore type, then by the Eilenberg-Moore spec-
tral sequence construction ([17 ], [11 ], [32 , Appendix C]), C*( X; k) ~
End C*(X;k)(k) and both of the pairs (C*( X; k), k) and (C*(X; k), k)
are dc-complete (2.14).
3.2. Commutative rings. If R is a commutative Noetherian ring and
I R is an ideal such that the quotient R=I = k is a regular ring (2.13),
then R ! k is proxy-regular [13 , x6]; the complex K can be chosen to
be the Koszul complex associated to any finite set of generators for I.
The construction of the Koszul complex is sketched below in the proof
of 7.3. The pair (R, k) is dc-complete if and only if R is complete and
Hausdorff with respect to the I-adic topology (9.18).
12 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
For example, if R is a noetherian local ring with residue field k, then
the map R ! k is proxy-regular; this map is regular if and only if R is
regular (Serre's Theorem) and coregular if and only if R is artinian.
3.3. The sphere spectrum. Consider the map S ! Fp of commu-
tative S-algebras; here as usual S is the sphere spectrum and the ring
Fp is identified with the associated Eilenberg-MacLane spectrum. This
map is not proxy-regular. A Koszul complex K for S ! Fp would
be a stable finite complex with nontrivial mod p homology (because
K would build Fp), and only a finite number of non-trivial homotopy
groups, each one a finite p-group (because Fp would finitely build K).
We leave it to the reader to show that no such K exists, for instance
because of Lin's theorem [31 ] that Map S(Fp, S) ~ 0.
Let Sp denote the p-completion of the sphere spectrum. The map
S ! Fp is not dc-complete, but Sp ! Fp is; this can be interepreted
in terms of the convergence of the classical mod p Adams spectral
sequence.
3.4. Cochains. Suppose that X is a pointed connected topological
space and that R is the augmented k-algebra C*(X; k).
(1) The map R ! k is coregular if X is a finite complex (9.16).
(2) If k is a field, then R ! k is coregular if and only if H*(X; k)
is finite-dimensional (9.14).
(3) If (X, k) is of Eilenberg-Moore type, then R ! k is regular if
and only if H*( X; k) is finite-dimensional (3.4, 9.14).
3.5. Chains. Suppose that X is a pointed connected topological space
and that R is the augmented k-algebra C*( X; k).
(1) The map R ! k is regular if X is a finite complex (9.12).
(2) If (X, k) is of Eilenberg-Moore type, then R ! k is regular if
and only if H*(X; k) is finite-dimensional (2.15, 3.4).
(3) If (X, k) is of Eilenberg-Moore type, then R ! k is coregular if
and only if H*( X; k) is finite-dimensional (9.14).
If (X, k) is of Eilenberg-Moore type, the parallels between 3.4 and 3.5
are explained by 2.15.
3.6. Completed classifying spaces. Suppose that G is a compact
Lie group (e.g., a finite group), that k = Fp, and that X is the p-
completion of the classifying space BG in the sense of Bousfield-Kan
[8]. Let R = C*(X; k) and E = C*( X; k). We will show in the
following paragraph that R ! k and E ! k are both proxy-regular,
and that the pair (X, k) is of Eilenberg-Moore type. There are many
G for which neither H*( X; k) nor H*(X; k) is finite dimensional [30 ];
by 3.4 and 3.5, in such cases the maps R ! k and E ! k are neither
DUALITY 13
regular nor coregular. We are interested in these examples for the sake
of local cohomology theorems (8.3).
By elementary representation theory there is a faithful embedding
æ : G ! SU(n) for some n, where SU(n) is the special unitary group of
n x n Hermitian matrices of determinant one. Consider the associated
fibration sequence
(3.7) M = SU(n)=G ! BG ! BSU(n) .
The fibre M is a finite complex. Recall that R = C*(BG; k); write
S = C*(BSU(n); k) and Q = C*(M; k). Since BSU(n) is simply-
connected, the Eilenberg-Moore spectral sequence of 3.7 converges and
Q ~ k S R (cf. [32 , 5.2]). The map S ! k is regular by 3.5 and Q ! k
is coregular by 3.4; it follows from 2.16 that R ! k is proxy-regular.
Since ß1BG = ß0G is finite, BG is Fp-good (i.e., C*(X; k) ~ R), and
ß1X is a finite p-group [8, VII.5]. In particular, (X, k) is of Eilenberg-
Moore type. Since E = C*( X; k) is thus equivalent to End R (k), we
conclude from 2.15 that E ! k is also proxy-regular.
3.8. Group rings. If G is a finite group and k is a commutative ring,
then the augmentation map k[G] ! k is proxy-regular. We will prove
this by producing a Koszul complex K for Z over Z[G]; it is then easy
to argue that k Z K is a Koszul complex for k over k[G]. Embed
G as above into a unitary group SU(n) and let K = C*(SU(n); Z).
The space SU(n) with the induced left G-action is a compact manifold
on which G acts smoothly and freely, and so by transformation group
theory [26 ] can be constructed from a finite number of G-cells of the
form (G x Di, G x Si-1). This implies that K is small over Z[G], since,
up to equivalence over Z[G], K can be identified with the G-cellular
chains on SU(n). Note that G acts trivially on ß*K = H*(SU(n); Z)
(because SU(n) is connected) and that, since H*(SU(n); Z) is torsion
free, each group ßiK is isomorphic over G to a finite direct sum of
copies of the augmentation module Z. The Postnikov argument in the
proof of 9.14 thus shows that K is finitely built from Z over Z[G].
Finally, K itself is an S-algebra, the action of Z[G] on K is induced by
a homomorphism Z[G] ! K, and the augmentation Z[G] ! Z extends
to an augmentation K ! Z. Since K builds Z over K, it certainly
builds Z over Z[G].
4. Matlis lifts
Suppose that R is a commutative noetherian local ring, and that
R ! k is reduction modulo the maximal ideal. Let I(k) be the injective
hull of k as an R-module. The starting point of this section is the
14 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
isomorphism
(4.1) Hom k(X, k) ~ Hom R(X, I(k)) ,
which holds for any k-module X. We think of I(k) as a lift of k to an
R-module, not the obvious lift obtained by using the homomorphism
R ! k, but a more mysterious construction that allows for 4.1. The
Pontriagin dual of an R-module M is defined to be Hom R (M, I(k)). By
4.1, Pontriagin duality is a construction for R-modules which extends
ordinary k-duality for k-modules.
We want to generalize this. Given a map R ! k of S-algebras and a
k-module N, we look for R-modules I(N) with the property that
(4.2) Hom k(X, N) ~ Hom R(X, I(N))
for any k-module N. To avoid delay, we will give the construction
right away and discuss it later in 4.8. Let E = End R(k). Observe that
the right multiplication action of k on itself gives a homomorphism
kop ! E, or equivalently k ! Eop, so it makes sense to look at right
E-actions on N which extend the left k-action.
4.3. Definition. Suppose that R ! k is a map of S-algebras, and that
N is a k-module. Let E = End R(k). An E-lift of N is a right E-module
structure on N which extends the left k-action. An E-lift of N is said
to be of Matlis type if the natural map
(4.4) N ~ N E Hom R(k, k) ! Hom R(k, N E k)
is an equivalence; in this case the R-module N E k is said to be a
Matlis lift of N. (Note that the action of R on N E k is obtained from
the left action of R on k.)
4.5. Remark. In general, a right E-module N is said to be of Matlis
type if the map 4.4 is an equivalence.
4.6. Proposition. In the situation of 4.3, suppose that I = N E k is a
Matlis lift of N. Then for any k-module X, Hom R(X, I) is equivalent
to Hom k(X, N).
Proof. By adjointness, there is an equivalence
Hom R(X, I) ~ Hom k(X, Hom R (k, I)) .
The proposition follows from the fact that Hom R(k, I) is by assumption
equivalent to N as a left k-module.
The following observation is useful for recognizing Matlis lifts.
DUALITY 15
4.7. Proposition. Suppose that R ! k is a map of S-algebras, that
E = End R(k), and that M is an R-module. Then the right E-module
Hom R(k, M) is of Matlis type if and only if the evaluation map
Hom R (k, M) E k ! M
is a k-cellular approximation.
Proof. Let N = Hom R(k, M). Since N is Eop-cellular over Eop, N E k
is k-cellular over R. This implies that the evaluation map ffl is a k-
cellular approximation if and only if it is a k-equivalence. Consider the
chain
Hom R(k,ffl)
N E Hom R(k, k) ! Hom R(k, N E k) --- - - - !N .
It is easy to check that the composite is the obvious equivalence, so the
left hand map is an equivalence (N is of Matlis type) if and only if the
right-hand map is an equivalence (ffl is a k-equivalence).
4.8. Remark. The reader may wonder about the source of 4.3, since it is
probably not clear how to get from 4.2 to 4.3. Suppose that I = I(N)
is an R-module for which 4.2 holds. First of all, to tighten things up a
bit we may as well replace I by CellRk(I), since Hom R(X, CellRk(I)) ~
Hom R(X, I) for all R-modules X which are built from k, and in par-
ticular for k-modules X. Secondly, the case X = k of 4.2 gives
Hom R(k, I) ~ N; this provides a right E-action on N that (given a
little naturality in 4.2) extends the left k-action and is hence an E-lift.
There is an induced evaluation map
(4.9) N E k ~ Hom R(k, I) E k ! I .
If the E-lift is of Matlis type, this map is a k-cellular approximation
(4.7) and therefore an equivalence, since I is k-cellular. The question
then becomes whether or not it is reasonable to expect an E-lift of N
to be of Matlis type. There are some examples below (4.11), but for
now note that if R ! k is proxy-regular, e.g., if R ! k is a surjection
of commutative noetherian rings with a regular quotient ring k (3.2),
then 4.9 is always a k-equivalence (2.10). In this case, at least, any
R-module I(N) satisfying 4.2 must be a Matlis lift in the sense of 4.3.
4.10. Matlis duality. In the situation of 4.3, let N = k and let I =
k E k be a Matlis lift of k. The Pontriagin dual or Matlis dual of
an R-module M (with respect to I) is defined to be Hom R(M, I).
By 4.6, Matlis duality is a construction for R-modules which extends
ordinary k-duality for k-modules. Note, however, that in the absence
of additional structure (e.g., commutativity of R) it is not clear that
16 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
Hom R(M, I) is a right R-module. We will come up with one way to
remedy this later on (6.3).
4.11. Existence of Matlis lifts. We give four conditions under which
a right E-module is of Matlis type, and so gives rise to a Matlis lift of
the underlying k-module. The first two conditions are of an algebraic
nature; the second two may seem technical, but they apply to many ring
spectra, chain algebras, and cochain algebras. In all of the statements
below, R ! k is a map of S-algebras, E = End R(k), and N is a right
E-module.
4.12. Proposition. If R ! k is regular, then any N is of Matlis type.
Proof. Calculate
Hom R (k, N E k) ~ N E Hom R(k, k) ~ N E E ~ N
where the first weak equivalence comes from the fact that k is small as
an R-module.
4.13. Proposition. If R ! k is proxy-regular, then N is of Matlis type
if and only if there exists an R-module M such that N is equivalent to
Hom R(k, M) as a right E-module.
Proof. If N is of Matlis type, then M = N E k will do. Given M, the
fact that Hom R(k, M) is of Matlis type follows from 4.7 and 2.10.
4.14. Definition. Suppose that X and Y are R-modules and {Aff} is
a collection of R-modules. Then Y is obtained from X by attaching
copies of the modules Affif there is a cofibration sequence U ! X ! Y
in which U is a equivalent to a coproduct of modules from the collec-
tion {Aff}. More generally, Y is obtained from X by iteratively attach-
ing copies of {Aff} if Y is the homotopy colimit of a directed system
{X!}!2 , indexed by an ordinal , such that
o X0 = X,
o X!+1 is obtained from X! by attaching copies of the Aff, and
o for a limit ordinal ! 2 , X! ~ hocolim !0 A.
ßiHom R(k, N E k) -! ßiHom R(X, N E k)
Now N is of downward type as a right E-module, so if we choose A
small enough we can guarantee that the map
ßi(N E Hom R(k, k)) ! ßi(N E Hom R(X, k))
is an isomorphism for i > B. By reducing A if necessary (which of
course affects the choice of X), we can assume A B. Now consider
the commutative diagram
N E Hom R(k, k) --- ! Hom R(k, N E k)
? ?
(4.18) ?y ?y
N E Hom R(X, k) --- ! Hom R (X, N E k)
The lower arrow is an equivalence, because X is finitely built from R,
and the vertical arrows are isomorphisms on ßi for i > B. Since B is
arbitrary, it follows that the upper arrow is an equivalence.
Proof of 4.16. This is very similar to the proof above, but with the
inequalities reversed. Observe that since k and N are bounded below,
and N is of upward type as an E-module, N E k is also bounded below.
Pick an integer B, and let A be another integer. Since k is of downward
18 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
finite type as an R-module and both k and N E k are bounded below,
there exists an X finitely built from R such that the maps in 4.17 are
isomorphisms for i < A. Now N is of upward type as a right E-module,
so if we choose A large enough we can guarantee that the map
ßi(N E Hom R(k, k)) ! ßi(N E Hom R(X, k))
is an isomorphism for i < B. By making A larger if necessary, we can
assume A > B. The proof is now completed by using the commutative
diagram 4.18.
5. Examples of Matlis lifting
In this section we look at particular examples of Matlis lifting (x4).
In each case we start with a morphism R ! k of rings, and look for
Matlis lifts of k. As usual, E denotes End R(k).
5.1. Local rings. Suppose that R is a commutative Noetherian local
ring with maximal ideal I and residue field R=I = k, and that R ! k
is the quotient map. Let I = I(k) be the injective hull of k (as an
R-module). Then I is a Matlis lift of k.
To see this, first note that I is k-cellular, or equivalently [13 , 6.12],
that each element of I is annihilated by some power of I. Pick an
element x 2 I; by Krull's Theorem [2, 10.20] the intersection \jIjx is
trivial. But each submodule Ijx of I is either trivial itself or contains
k I [33 , p. 281]. The conclusion is that Ijx = 0 for j >> 0. Since
Hom R(k, I) ~ k (again, for instance, by [33 ]), I provides an E-lift of
k (cf. 4.8), and the induced map k E k ~ Hom R(k, I) E k ! I is
an equivalence by 3.2 and 2.10. Up to equivalence there is exactly one
E-lift of k (9.2), and so in fact I(k) is the only Matlis lift of k.
For instance, if R ! k is Z(p)! Fp, then I ~ k E k is Z=p1 (cf.
[13 , x3]), and Matlis duality (4.10) for R-modules is Pontriagin duality
for p-local abelian groups.
5.2. k-algebras. Suppose that R is an augmented k-algebra, and let
M be the R-module Hom k(R, k). The left R-action on M is induced
by the right R-action of R on itself. By an adjointness calculation,
Hom R(k, M) is equivalent to k, and so in this way M provides an E-lift
of k. If this E-lift is of Matlis type, then the R-module k E k, which
by 4.7 is equivalent to CellkHom k(R, k), is a Matlis lift of k. There are
equivalences
Hom k(k E k, k) ~ Hom E(k, Hom k(k, k)) ~ Hom E(k, k) ~ ^R,
so that if (R, k) is dc-complete, the Matlis lift k E k is pre-dual to
R. Note that this calculation does not depend on assuming that R is
DUALITY 19
small in any sense as a k-module; there is an interesting example below
in 5.6.
5.3. The sphere spectrum. Let R ! k be the unit map S ! Fp. (Re-
call that we are willing to identify Fp with the corresponding Eilenberg-
MacLane ring spectrum.) The endomorphism S-algebra E is the Steen-
rod algebra spectrum, with ß-iE isomorphic to the degree i homoge-
neous component of the Steenrod algebra. Since k has a unique E-lift
(9.2) and the conditions of 4.15 are satisfied (9.8, 9.9), k has a unique
Matlis lift given by k E k. Let J be the Brown-Comenetz dual of S [9]
and Jp its p-primary summand. We argue below that Jp is k-cellular; by
the basic property of Brown-Comenetz duality, Hom R(k, Jp) ~ k. By
4.7 the evaluation map k E k ! Jp is a k-cellular approximation and
hence, because J is k-cellular, an equivalence. Matlis duality amounts
to the p-primary part of Brown-Comenetz duality. Arguments parallel
to those in the proof of 4.15 show that if X is a connective spectrum
of finite type then the natural map
k E Hom R(X, k) ! Hom R(X, k E k)
is an equivalence. Suppose that X* is an Adams resolution of the
sphere. Taking the Brown-Comenetz dual Hom R(X*, k E k) gives a
spectral sequence which is the Fp-dual of the mod p Adams spectral
sequence. On the other hand, computing ß* Hom R (X*, k) amounts to
taking the cohomology of X* and so gives a free resolution of k over the
Steenrod Algebra; the spectral sequence associated to k EHom R(X*, k)
is then the Kunneth spectral sequence
Tor i*E*(ß*k, ß*k) ) ß*(k E k) ~=ß*Jp .
It follows that these two spectral sequences are isomorphic.
To see that Jp is k-cellular, write Jp = hocolim Jp(-i), where Jp(-i)
is the (-i)-connective cover of Jp. Each Jp(-i) has only a finite number
of homotopy groups, each of which is a finite p-primary torsion group,
and it follows immediately that Jp can be finitely built from k. Thus
Jp, as a homotopy colimit of k-cellular objects, is itself k-cellular.
5.4. Cochains. Suppose that X is a pointed connected space and k
is a field. Let R = C*(X; k) and E = End R (k), and suppose that
some E-lift of k is given. By 9.8, k is of upward type over Eop. If
(X, k) is of Eilenberg-Moore type (3.1), then k is of downward finite
type over R (9.10), the conditions of 4.16 are satisfied, and I = k E k
is a Matlis lift of k. If X is 1-connected then ß0E ~= k and there is
only one E-lift of k (9.2); more generally, there is only one E-lift of k
if (X, k) is of Eilenberg-Moore type. (This last statement follows from
the fact that if k is a field of characteristic p and G is a finite p-group,
20 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
any homomorphism G ! kx is trivial.) In these cases the Matlis lift
I = k E k is equivalent by the Rothenberg-Steenrod construction
to C*(X; k) = Hom k(R, k). Observe in particular that Hom k(R, k) is
k-cellular as an R-module; this also follows from 9.15.
5.5. Chains. Let X be a pointed space, k a field, and R the chain
algebra C*( X; k), so that E ~ C*(X; k). By 9.2 there is only one
E-lift of k, necessarily given by the augmentation action of E on k.
Suppose that k has upward finite type as an R-module, for instance,
suppose that the conditions of 9.8 hold, or that X has finite skeleta
(9.12). Then, by 9.9 and 4.15, k has a unique Matlis lift, given by
k E k, or alternatively (5.2) by CellkHom k(R, k) ~ CellkC*( X; k).
We have not assumed that (X, k) is of Eilenberg-Moore type, and so
the identification
k E k ~ CellkC*( X; k)
gives an interpretation of the abutment of the cohomology Eilenberg-
Moore spectral sequence associated to the path fibration over X; this
is in some sense dual to the interpretation of the abutment of the
corresponding homology spectral as a suitable completion of C*( X)
[12 ].
5.6. Suspension spectra of loop spaces. Suppose that X is a pointed
finite complex, let k = S, and let R be the augmented k-algebra
C*( X; k). Then E is equivalent to C*(X; k), i.e., to the Spanier-
Whitehead dual of X (3.1). Since X is finite, k is small as an R-module
(9.12). It follows from 4.12 that Matlis lifts of k correspond bijectively
to E-lifts of k. Note that since the augmentation action of E on k fac-
tors through E ! k, and k is commutative, this augmentation action
amounts in itself to an E-lift. (It is possible to show that this is the
only E-lift of k, but we will not do that here.) By inspection, this
augmentation E-lift of k is the same as the E-lift obtained by letting E
act in the natural way on Hom R(k, Hom k(R, k)) ~ k as in 5.2. By 4.7,
the corresponding Matlis lift k E k is CellkHom k(R, k).
Suppose in addition that X is 1-connected, and write k E k as the
realization of the ordinary simplicial bar construction
k S k ( k S E S k W k S E S E S k . ...
The spectrum Hom S(k S k, S) is then the total complex of the corre-
sponding cosimplicial object
Hom S(k S k, S) ) Hom S(k S E S k, S) V . ...
This is the cosimplicial object obtained by applying the unpointed sus-
pension spectrum functor to the cobar construction on X, and by a
theorem of Bousfield [7] its total complex is the suspension spectrum
DUALITY 21
of X, i.e., R. Equivalently, Bousfield's theorem shows that in this
case (R, k) is dc-complete. In this way if X is 1-connected the Matlis
lift of k is a Spanier-Whitehead pre-dual of R (cf. 5.2). This object
has come up in a different way in work of N. Kuhn [29 ].
6.Gorenstein S-algebras
If R is a commutative Noetherian local ring with maximal ideal I and
residue field R=I = k, one says that R is Gorenstein if Ext *R(k, R) is
concentrated in a single degree, and is isomorphic to k there. We give
a similar definition for S-algebras, with an extra technical condition
added on.
6.1. Definition. Suppose that R ! k is a map of S-algebras, and let
E = End R(k). Then R ! k is Gorenstein of shift a if the following two
conditions hold:
(1) as a left k-module, Hom R(k, R) is equivalent to ak, and
(2) as a right E-module, Hom R(k, R) is of Matlis type (4.5).
6.2. Remark. Suppose that R ! k is Gorenstein of shift a, and give
ak the right E-module structure from 6.1(1). Then by 4.7, Cellk(R)
is equivalent to ak E k.
6.3. Remark. Definition 6.1 does not exhaust all of the structure in
Hom R(k, R); in fact, the right action of R on itself gives a right R-
action on Hom R(k, R) which commutes with the right E-action (since
E acts through k). This implies that if R ! k is Gorenstein and k
is given the right E-action obtained from k ~ -a Hom R(k, R), then
the Matlis lift I = k E k of k inherits a right R-action. In this case
the Matlis dual Hom R(M, I) of a left R-module is naturally a right
R-module.
In the proxy-regular case it is possible to simplify definition 6.1. We
record the following, which is a consequence of 4.13.
6.4. Proposition. Suppose that the map R ! k of S-algebras is proxy-
regular. Then R ! k is Gorenstein of shift a if and only if Hom R(k, R)
is equivalent to ak as a left k-module.
The rest of the section provides techniques for recognizing Gorenstein
homomorphisms R ! k.
6.5. Proposition. Suppose that R is an augmented k-algebra, and let
E = End R (k). Assume that (R, k) is dc-complete, and that R ! k
is proxy-regular. Then R ! k is Gorenstein if and only if E ! k is
Gorenstein.
See [20 , 2.1] for a differential graded version of this.
22 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
Proof. Compute
Hom R(k, R)~ Hom R(k, Hom E(k, k)) ~ Hom R kE(k k k, k)
Hom E(k, E)~ Hom E(k, Hom R (k, k)) ~ Hom E kR(k k k, k) .
There's a subtlety here: k k k is certainly equivalent to k, but not
necessarily in a way which relates the tensor product action of R kE on
k kk to the action of R kE on k given by E = End R(k). Nevertheless,
it is clear that Hom R(k, R) is equivalent to a shift of k if and only if
Hom E(k, E) is. If E is Gorenstein, the R is Gorenstein by 6.4. If R is
Gorenstein, E is Gorenstein by 2.15 and 6.4.
6.6. Proposition. Suppose that S ! R is a map of augmented com-
mutative k-algebras such that R is small as an S-module. Let Q be the
augmented k-algebra k S R. Then there is an equivalence of k-modules
Hom R(k, R) ~ Hom Q(k, Hom S (k, S) k Q),
where the action of Q on Q k Hom S(k, S) is induced by the usual
action of Q on itself.
There is a rational version in [20 , 4.3]. The argument below depends
on the following general lemma, whose proof we leave to the reader.
6.7. Lemma. Suppose that R is a k-algebra, that A is a right R-module,
and that B and C are left R-modules. Then there are natural equiva-
lences
Hom R(B, C) ~ Hom R kRop(R, Hom k(B, C))
A R B ~ R R kRop (A k B) .
Proof of 6.6. Since R is commutative, we do not distinguish in notation
between R and Rop. First note that
Hom R(k, R) ~ Hom R SR (R, Hom S (k, R))
as in 6.7. Now observe that R is small over S, so that
(6.8) Hom S(k, R) ~=Hom S(k, S) S R .
Under this equivalence, the left action of R on Hom S(k, S) S R is
induced by the left action of R on itself, and the right action of R by
the left action of R on k. Now since S is commutative, the right and
left actions of S on Hom S(k, S) are the same. In particular, the right
action (which is used in forming Hom S(k, S) S R) factors through the
homomorphism S ! k, and we obtain an equivalence
(6.9) Hom S (k, S) S R ~ Hom S(k, S) k(k S R) ~ Hom S(k, S) kQ .
Let M = Hom S(k, S) k Q. Under 6.8 and 6.9 the left action of R on
M is induced by the left action of R on Q, while the right action of
R is induced by the left action of R on k. In particular, the action of
DUALITY 23
R S R on M factors through an action of k S R ~ Q on M, and so
by adjointness we have
Hom R SR (R, M) ~ Hom Q(Q R SR R, M)
~ Hom Q(k, M) ,
where the last equivalence depends on the calculation (6.7)
(k S R) R SR R ~ k R R ~ k .
The action of Q on this object is the obvious one that factors through
Q ! k. Combining the above gives the desired statement.
6.10. Proposition. Let S ! R be a homomorphism of commutative
augmented k-algebras, and set Q = k S R. Suppose that R is small
as an S-module, and that R ! k is proxy-regular. Then if the maps
S ! k and Q ! k are Gorenstein, so is R ! k.
Proof. By 6.6, Hom R(k, R) ~ ak. It follows from 6.4 that R ! k is
Gorenstein.
6.11. Poincar'e Duality. A k-algebra R is said to satisfy Poincar'e
duality of dimension a if there is an R-module equivalence aR !
Hom k(R, k); note that here we give Hom k(R, k) the left R-module
structure induced by the right action of R on itself. The algebra
R satisfies this condition if and only if there is an orientation class
! 2 ßa Hom k(R, k) with the property that ß* Hom k(R, k) is a free
module of rank one over ß*R with generator !. If k is a field, then
ß* Hom k(R, k) = hom k(ß*R, k), and R satisfies Poincar'e duality if and
only if ß*R satisfies Poincar'e duality in the simplest algebraic sense.
6.12. Proposition. Suppose that R is an augmented k algebra such
that the map R ! k is proxy-regular. If R satisfies Poincar'e duality of
dimension a, then R is Gorenstein of shift -a.
Proof. As in 5.2, compute
Hom R(k, R) ~ Hom R(k, -a Hom k(R, k)) ~ -a Hom k(R R k, k) ~ -ak .
The fact that R ! k is Gorenstein follows from 6.4.
We now give a version of the result from commutative ring theory
that "regular implies Gorenstein".
6.13. Proposition. Suppose that k is a field, R is a connective com-
mutative S-algebra, and R ! k is a regular homomorphism which is
surjective on ß0. Assume that the pair (R, k) is dc-complete. Then
R ! k is Gorenstein.
24 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
6.14. Remark. It is possible to omit the dc-completeness hypothesis
from 6.13 in the commutative ring case. Suppose that R is a commu-
tative noetherian ring, I R is a maximal ideal, k = R=I is the residue
field, and R ! k is regular. We show that R ! k is also Gorenstein.
To see this, let S = lims R=Is be the I-adic completion of R. As in
the proof of 9.18, S is flat over R and TorR0(S, k) ~=k; in addition, the
map R ! S is a k-equivalence (of R-modules). This gives a chain of
equivalences
Hom R(k, R) ~ Hom R(k, S) ~ Hom S(S R k, S) ~ Hom S(k, S) .
The flatness easily implies that S ! k is also regular, and so R ! k
is Gorenstein if and only if S ! k is Gorenstein. But it follows from
9.18 that the pair (S, k) is dc-complete, and so S ! k is Gorenstein
by 6.13.
6.15. Lemma. Suppose that k is a field, R is a connective commutative
S-algebra, and R ! k is a regular homomorphism which is surjective
on ß0. Assume that k is of upward finite type over R. Then ß* End R(k)
is in a natural way a cocommutative Hopf algebra over k.
Proof. The diagram chasing necessary to prove this is described in de-
tail in [1, pp. 56-76], with a focus at the end on the case in which
R = S, k = Fp, and ß* End R(k) is the mod p Steenrod algebra. Let
E = End R(k). The key idea is that ß*E is the k-dual of the commuta-
tive k-algebra ß*(k R k): as in 5.2 there are equivalences
Hom k(k R k, k) ~ Hom R(k, Hom k(k, k)) ~ End R(k) .
The k-dual of the multiplication on ß*(k R k) then provides the comul-
tiplication on ß* End R(k). The fact that k is of upward finite type over
R guarantees that the groups ßi(k E k) are finite-dimensional over k.
There is a technicality: k R k is a bimodule over k, not an algebra
over k. However k R k is an algebra over R, so that the surjection
ß0R ! k guarantees that the left and right action of k on ß*(k R k)
agree. For the same reason, the left and right actions of k on ß* End R(k)
agree, and this graded ring becomes a Hopf algebra over k.
Proof of 6.13. Let E = End R(k). The connectivity assumptions on R
imply that ß0E ~= k and that E is coconnective; by 6.15, E is a Hopf
algebra over k. In fact, E is finitely built from k (2.15), and so ß*E is
a finite dimensional Hopf algebra over k. Sweedler has remarked that
a connected finite-dimensional Hopf algebra over k with commutative
comultiplication and involution satisfies algebraic Poincar'e duality [35 ];
a somewhat more general result can be derived from [39 , 5.1.6]. The
map E ! k is thus Gorenstein by 6.12, and R ! k by 6.5.
DUALITY 25
6.16. Remark. The above arguments are related to those of Avramov
and Golod [3], who show that a noetherian local ring R is Gorenstein if
and only if the homology of the associated Koszul complex is a Poincar'e
duality algebra.
7. A local cohomology theorem
One of the attractions of the Gorenstein condition on a S-algebra
R is that it has structural implications for ß*R, which can sometimes
be thought of as duality properties. To illustrate this, we look at the
special case in which R ! k is a Gorenstein map of augmented k-
algebras, where k is a field. Let E = End R(k). By 6.2, the Gorenstein
condition gives
ak E k ~ CellkR .
We next assume that the right E-structure on ak given by k ~
Hom R(k, R) is equivalent to the right E-structure given by
ak ~ Hom R(k, a Hom k(R, k)) .
By 4.7 this gives an equivalence
ak E k ~ a CellkHom k(R, k) .
Assume in addition that Hom k(R, k) is itself k-cellular as an R-module.
Combining the above then gives
(7.1) a Hom k(R, k) ~ CellkR .
Now in some reasonable circumstances we might expect a spectral se-
quence
(7.2) E2i,j= ßiCelli*Rk(ß*R)j ) ßi+jCellRk(R)
which in the special situation we are considering would give
E2i,j= ßiCelli*Rk(ß*R)j ) ßi+j-aHom k(R, k) .
(The subscript j refers to the j'th homogeneous component of an ap-
propriate grading on ßiCelli*Rk(ß*R).) This is what we mean by a
duality property for ß*R: a spectral sequence starting from some co-
variant algebraic data associated to ß*R and abutting to the dual ob-
ject ß* Hom k(R, k) ~= Hom k(ß*R, k). If R is k-cellular as a module
over itself, then 7.1 gives a Hom k(R, k) ~ R, and we obtain ordinary
Poincar'e duality.
The problematic point here is the existence of the spectral sequence
7.2. Rather than trying to construct this spectral sequence in general
and study its convergence properties, we concentrate on a special case
in which it is possible to identify CellRk(R) explicitly. To connect the
following statement with 7.2, recall [13 , x6] that if S is a commutative
26 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
ring and I S a finitely generated ideal with quotient ring k = S=I,
then for any discrete S-module M the local cohomology group HiI(M)
can be identified with ß-i CellSk(M).
7.3. Proposition. Suppose that k is a field, and that R is a coconnec-
tive commutative augmented k-algebra. Assume that ß*R is noether-
ian, and that the augmentation map induces an isomorphism ß0R ~=k.
Then for any R-module M there is a spectral sequence
E2i,j= H-iI(M)j ) ßi+jCellRk(M) .
Given the above discussion, this leads to the following local coho-
mology theorem.
7.4. Proposition. In the situation of 7.3, assume in addition that
R ! k is Gorenstein of shift a, that k has a unique E-lift (where
E = End R(k)), and that Hom k(R, k) is k-cellular as an R-module.
Then there is a spectral sequence
E2i,j= H-iI(ß*R)j ) ßi+j-aHom k(R, k) .
7.5. Remark. The structural implications of this spectral sequence for
the geometry of the ring ß*R are investigated in [23 ]. For examples in
which Hom k(R, k) is k-cellular over R, see 5.4 or 9.15.
Proof of 7.3. We first copy some constructions from [13 , x6]. For any
x 2 ß*R we can form an R-module R[1=x] by taking the homotopy
colimit of the sequence
R x-!R x-!R x-!. ...
(Actually, R[1=x] can also be given the structure of a commutative S-
algebra, in such a way that R ! R[1=x] is a homomorphism.) Write
Km (x) for the fibre of xm : R ! R, and K1 (x) for the fibre of the map
R ! R[1=x]. Now choose a finite sequence x1, . .,.xn of generators for
I ß*R, and let
Km = Km (x1) R . . .RKm (xn)
K1 = K1 (x1) R . . .RK1 (xn) .
Recall that R is commutative, so that right and left R-module struc-
tures are interchangeable, and tensoring two R-modules over R pro-
duces a third R-module. Write K = K1. It is easy to see that ß*K is
finitely built from k as a module over ß*R, and hence (9.14) that K is
finitely built from k as a module over R. An inductive argument (us-
ing cofibration sequences Km (xi) ! Km+1 (xi) ! K1(xi)) shows that
K builds Km and hence also builds K1 ~ hocolim Km (cf. [13 , 6.6]).
It is easy to see that the evident map K1 ! R gives equivalences
(7.6) k R K1 ~ k K R K1 ~ K .
DUALITY 27
See [13 , proof of 6.9]; the second equivalence follows from the first
because K is built from k. The first equivalence implies that K1
builds k and this in turn shows that K build k. Since K is small
over R, we see that R ! k is proxy-regular with Koszul complex K.
In particular, a map A ! B of R-modules is a k-equivalence if and
only if it is a K-equivalence, or (since Hom R(K(xi), R) ~ K(xi) and
hence Hom R(K, R) ~ nK)) if and only if it induces an equivalence
K R A ! K R B. Since K1 is built from k as an R-module, so
is K1 R M. The right hand equivalence in 7.6 implies that the map
K1 R M ! M induces an equivalence
K R K1 R M ! K R M ,
and it follows that K1 R M is CellRk(M). Each module K1 (xi) lies
in a cofibration sequence
-1R[1=xi] ! K1 (xi) ! R
which can be interpreted as a one-step increasing filtration of K1 (xi).
Tensoring these together gives an n-step filtration of K1 ,
0 = Fn+1 ! Fn ! Fn-1 ! . .!.F0 = K1 ,
with the property that there are equivalences
M
Fs=Fs+1 ~ R[1=xi1] R . . .RR[1=xis] .
{i1,...,is}
The sums here are indexed over subsets of cardinality s from {1, . .,.n}.
Tensoring this filtration with M gives a finite filtration of CellRk(M),
and the spectral sequence of the proposition is the homotopy spectral
sequence associated to the filtration. The identification of the E2-page
as local cohomology is standard [13 , x6] [24 ]; the main point here is
to notice that since ß*R[1=xi] is flat over ß*R, there are isomorphisms
ß*(R[1=xi] R M) ~=ß*(R[1=xi]) i*Rß*M ~= (ß*R)[1=xi] i*Rß*M.
8. Gorenstein examples
We give several examples of S-algebras which are Gorenstein, and at
least one example of an S-algebra which is not.
8.1. Regular chains. Suppose that X is a pointed connected topolog-
ical space and that k is a field such that the pair (X, k) is of Eilenberg-
Moore type (3.1). Let R = C*(X; k), E = C*( X; k) ~ End R(k), and
assume that H*(X; k) ~= ß-*R is finite dimensional. Then R ! k is
coregular and E ! k is regular (3.4). If H*(X; k) satisfies Poincar'e
duality of dimension a, (e.g., if X is a closed orientable manifold of
dimension a), then R ! k is Gorenstein of shift -a (6.12) and E ! k
28 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
is also Gorenstein with the same shift (6.5). The ring R has a local
cohomology spectral sequence (7.4), but this collapses to a restatement
of Poincar'e duality:
E2 = ß*R ~=Celli*Rk(ß*R) ~=ß* Hom k(R, k) .
In the absence of the hypothetical spectral sequence 7.2, there is noth-
ing like a local cohomology theorem for the noncommutative S-algebra
E.
8.2. Regular cochains. Suppose that k is a field and G is a topologi-
cal group such that H*(G; k) is finite dimensional. Let R = C*(BG; k)
and E = C*(G; k). Assume in addition that (BG, k) is of Eilenberg-
Moore type; this covers the cases in which k = Fp, and G is a finite
p-group, a compact Lie group with ß0G a finite p-group, or a p-compact
group. The map E ! k is coregular (9.14), and hence R ! k is reg-
ular (3.5). The graded ring H*(G; k) is a finite dimensional group-like
Hopf algebra over k, and so by Sweedler (cf. [39 , 5.1.6]) satisfies alge-
braic Poincar'e duality of some dimension, say a. (If G is a connected
compact Lie group, then a = - dim G; the üf ndamental class" ! lies
in H-a (G; k) = ßaR.) By 6.12, E ! k is Gorenstein of shift a, and
so R ! k is also Gorenstein with the same shift (6.5). The graded
ring H*(BG; k) = ß*R is noetherian. If k is of characteristic zero, this
follows from the fact that the ring is a finitely generated polynomial
algebra over k; see [34 , 7.20]. If k = Fp and G is a compact Lie group,
the finite generation statement is a classical theorem of Golod [22 ] and
Venkov [40 ]; in the general case it amounts to the main result of [16 ].
By 9.15 and 7.4 there is a local cohomology theorem for R.
8.3. Compact Lie groups. Suppose that G is a general compact Lie
group, e.g., a finite group, and that k = Fp. We continue the discussion
in 3.6, with the same notation. Recall that X is the p-completion of
BG, R = C*(X; k) ~ C*(BG; k), and E = C*( X; k); the space X
plays the role of G above in 8.2, but we do not have that H*( X; k)
is finite dimensional. The fibre M in 3.7 is a compact manifold; it
is orientable because its tangent bundle is the bundle associated to
the conjugation action of G on the Lie algebra of SU(n), and, since
SU(n) is connected, this conjugation action preserves orientation. As
in 8.1, Q = C*(M; k) is coregular and Gorenstein. Similarly, S =
C*(BSU(n); k) is regular and Gorenstein by 8.2. It follows from 9.11
that R is small as a module over S. By 2.16 and 6.10, R ! k is
proxy-regular and Gorenstein, as is E ! k (6.5). Since Hom k(R, k) is
k-cellular over R (9.15), there is a local cohomology spectral sequence
for R (7.4).
DUALITY 29
8.4. Finite complexes. Suppose that X is a pointed connected finite
complex which is a Poincar'e duality complex over k of formal dimension
a; in other words, assume that X satisfies possibly unoriented Poincar'e
duality with arbitrary (twisted) k-module coefficients. To be specific,
assume that k is a finite field, the field Q, or the ring Z of integers. Let
R denote the augmented k-algebra C*( X; k), so that ß0R ~=k[ß1X].
Note that R ! k is regular (9.12). Any module M over k[ß1X] gives
a module over R, and (by a version of the Rothenberg-Steenrod con-
struction) there are isomorphisms
Hi(X; M) ~=ßi(k R M) Hi(X; M) ~=ß-i Hom R (k, M) .
The duality condition on X can be expressed by saying that there is
a module ~ over k[ß1X] whose underlying k-module is isomorphic to
k itself, and an orientation class ! 2 ßa(~ R k), such that, for any
k[ß1X]-module M, evaluation on ! gives an equivalence
(8.5) Hom R(k, M) ! -a~ R M .
By 9.2, it follows that 8.5 is an equivalence for any R-module M which
has only one nonvanishing homotopy group. By triangle arguments (cf.
9.4) it is easy to conclude that 8.5 is an equivalence for all M which
have only a finite number of nonvanishing homotopy groups, and by
passing to a limit (cf. proof of 4.15) that 8.5 is actually an equivalence
for all R-modules M. Note that this passage to the limit depends on
the fact that k is small over R. The case M = R of 8.5 gives
Hom R (k, R) ~ -a~ R R ~ ~ ~ k ,
and so by 6.4, R ! k is Gorenstein of shift -a. Let E = End R (k).
The pair (R, k) is not necessarily dc-complete, and so E ! k is not
necessarily Gorenstein; for example, it is clear that ß*E ~= H*(X; k)
need not satisfy algebraic Poincar'e duality in the nonorientable case.
The equivalence Hom R(k, R) ~ ~ is an R-module equivalence as long
as Hom R(k, R) is given the right R-module structure obtained from
the right action of R on itself. In this way the orientation character
of the Poincar'e complex X is derived from the one bit of structure on
Hom R(k, R) that does not play a role in the definition of what it means
for R ! k to be Gorenstein (6.3).
8.6. Suspension spectra of loop spaces. We continue the discussion
from 5.6: X is a pointed connected finite complex, k = S, R is the
augmented k-algebra C*( X; k), and E is C*(X; k) ~ End R (k). The
map R ! k is regular; if X is simply connected, then R ! k is
dc-complete. Note that S = Z S R ~ C*( X; Z) (3.1). Suppose
that X is a Poincar'e duality complex of formal dimension a. We wish
30 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
to show that R ! k is Gorenstein of shift -a, or equivalently, that
Hom R(k, R) ~ -ak. The spectrum Y = -ak is characterized by
a combination of the homotopical property that Y is bounded below,
and the homological property that Z k Y ~ -aZ. The spectrum
Hom R(k, R) is bounded below because R is bounded below and k is
small over R. Similarly, the fact that k is small over R implies that
Z k Hom R (k, R) ~ Hom R(k, Z k R). Now compute
Hom R(k, Z k R) ~ Hom Z kR(Z, Z k R) ~ -aZ ,
where the first equivalence comes from adjointness, and the second from
8.4. It follows that R ! k is Gorenstein. If X is simply connected,
then E ! k is coregular and Gorenstein.
The stable homotopy orientation character of X is given by the action
of R on k ~ S obtained via -ak ~ Hom R(k, R) from the right action
of k on itself; see 8.5 for the homological version of this. It is not too
far off to interpret this character as a homomorphism X ! Sx ; in
any case it determines a stable spherical fibration over X which can
be identified with the Spivak normal bundle. (To see this, note that
the Thom complex of this spherical fibration is Hom R(k, R) R k ~
Hom R(k, k) = E, and the top cell has a spherical reduction given by
the unit homomorphism S ! E.) For some more details see [27 ].
8.7. The sphere spectrum. Let R = S and k = Fp. The map R ! k
is not Gorenstein; in fact, by Lin's theorem [31 ], Hom R(k, R) is trivial.
9. Basic constructions
This section looks into some properties of S-algebras and modules
which we refer to in the rest of the paper.
9.1. Uniqueness of module structures. We first aim for the follow-
ing elementary uniqueness result.
9.2. Proposition. Suppose that R is connective or that R is coconnec-
tive with ß0R a field, and that M and N are R-modules with nonva-
nishing homotopy only in a single dimension n. Then M and N are
equivalent as R-modules if and only if ßnM and ßnN are isomorphic
over ß0R.
9.3. Remark. It follows easily from the proof below that if R is as in
9.2, A is a discrete module over ß0R, and n is an integer, then there
exists up to equivalence a unique R-module K(M, n) with ßnK(A, n) ~=
A (over ß0R) and ßiK(A, n) ~= 0 for i 6= n. If R is connective the
construction of K(A, n) can be made functorially in A, otherwise in
general not. If A and B are two discrete ß0R-modules, the natural
DUALITY 31
map ß0 Hom R (K(A, n), K(B, n)) ! hom i0R(A, B) is an isomorphism
if R is connective but only a surjection in general if R is coconnective.
9.4. Lemma. Suppose that R is connective, that M is an R-module,
and that n is an integer. Then there is a natural R-module PnM with
ßi(PnM) ~=0 for i > n and ßi(PnM) ~=ßiM for i n, together with a
natural map M ! Pn inducing isomorphisms on ßi for i n.
Proof. Form PnM by iteratively attaching copies of iR, i > n to M
(4.14) to kill off the higher homotopy of M. The construction can be
made functorial by repeatedly doing the attachments in all possible
ways.
9.5. Lemma. Suppose that R is coconnective with ß0R a field, that M
is an R-module, and that n is an integer. Then there is an R-module
QnM with ßi(QnM) = 0 for i < n and ßi(QnM) ~=ßiM for i n. The
R-module QnM is obtained by iteratively attaching copies of iR, i < n
to M (4.14), and there is a map M ! QnM inducing isomorphisms
on ßi for i n.
9.6. Remark. The construction of QnM cannot be made functorial
in any reasonable sense. Consider the DGA E of [13 , x3]; E is co-
connective and ß0E ~=Fp. Then ß0 Hom E(E, E) ~= ß0E ~=Fp, while
ß0 Hom E(Q0E, Q0E) ~ Zp. Since there is no additive map Fp ! Zp,
there can be no way to form Q0E functorially from E.
Proof of 9.5. Attach copies of iR, i < n, to kill off the lower homo-
topy of M, as in the proof of 9.9 below. The fact that ß0R is a field
guarantees that the attachments can be done in such a way as not
to introduce new homotopy in dimensions n. But the attachments
have to be done minimally, and it is this requirement that prevents the
construction from being functorial.
Proof of 9.2. One way to prove this is to construct a suitable spec-
tral sequence converging to ß* Hom R (M, N); under the connectivity
assumptions on R, hom i0R(ßnM, ßnN) will appear in one corner of the
E2-page and subsequently remain undisturbed for positional reasons.
This implies that any map ßnM ! ßnN of ß0R-modules, in particular
any isomorphism, can be realized by an R-map M ! N. We will take
a more elementary approach. Assume without loss of generality that
n = 0 and suppose that there are isomorphisms ß0M ~= ß0N ~= A over
ß0R. First we treat the case in which R is connective. Find a free
presentation
OE1 ! OE0 ! A ! 0
of A over ß0R and construct a map F1 ! F0 of R-modules such that
each Fi is a sum of copies of R, and such that ß0F1 ! ß0F0 is OE1 ! OE0.
32 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
Let C be the cofibre of F1 ! F0. By inspection ß0C ~= A and there are
isomorphisms ß0 Hom R (C, M) ~= hom i0R(A, A) and ß0 Hom R (C, N) ~=
hom i0R(A, A). Choose maps C ! M and C ! N which induce iso-
morphisms on ß0, and apply the functor P0 (9.4) to conclude M ~ N.
Now suppose that R is coconnective, and that ß0R is a field. Write
A ~= ffß0R over ß0R, let F = ffR, and construct maps F ! M
and F ! N inducing isomorphisms on ß0. Consider Q0F (9.5). Since
Q0F is obtained from F by attaching copies of iR, i < 0, there
are surjections (not necessarily isomorphisms) ß0 Hom R (Q0F, M) i
hom i0R(A, A) and ß0 Hom R (Q0F, N) i hom i0R(A, A). Clearly, then,
there are equivalences Q0F ! M and Q0F ! N.
9.7. Finite type and smallness. We look for conditions under which
an R-module has upward (finite) type, downward (finite) type, or is
small. See 2.3 and the discussion preceding 4.15 for definitions of these
concepts. The first proof we leave to the reader.
9.8. Proposition. Suppose that R is a connective S-algebra, and that
M is a module over R which is bounded below. Then M is of upward
type. If in addition ß0R is noetherian, and the groups ßiR and ßiM (i 2
Z) are individually finitely generated over ß0R, then M is of upward
finite type.
9.9. Proposition. Suppose that R is a coconnective S-algebra such
that ß0R is a field, and that M is an R-module which is bounded above.
Then M is of downward type. If in addition ß-1R = 0 and the groups
ßiR and ßiM (i 2 Z) are individually finitely generated over ß0R, then
M is of downward finite type.
Proof. Given an R-module X and an integer m, choose a basis for ßm X
over ß0R, and let Vm X be a sum of copies of m R, one for each basis
element. There is a map Vm X ! X which induces an isomorphism on
ßm , and we let Wm X be the cofibre of this map. Now suppose that
M is nontrivial and bounded above, let n be the greatest integer such
that ßnM 6= 0, and let WnM be the cofibre of the map VnM ! M.
Iteration gives a sequence of maps M ! WnM ! Wn2M ! . .,.and we
let Wn1M = hocolim kWnkM. Then ßnWn1M = colimk ßnWnkM = 0.
Define modules Ui inductively by U0 = M, Ui+1 = Wn1-iUi. There
are maps Ui ! Ui+1 and it is clear that hocolim Ui ~ 0. Let Fi be
the homotopy fibre of M ! Ui. Then hocolim Fi is equivalent to
M, and Fi+1 is obtained from Fi by repeatedly attaching copies of
n-i-1R. This shows that M is of downward type. If ß-1R = 0, then
ßn-iWn-iUi ~=0, so that Wn1-iUi ~ Wn-iUi. Under the stated finiteness
hypotheses, one sees by an inductive argument that the groups ßjUi,
j 2 Z, are finite dimensional over k, and so Fi+1 is obtained from Fi
DUALITY 33
by attaching a finite number of copies of n-i-1R. This shows that M
is of downward finite type.
The next two propositions are two sides of the same coin; they
roughly correspond under taking double centralizers, but we don't as-
sume that the augmented k-algebras involved are dc-complete.
9.10. Proposition. Suppose that k is a field and that R is an aug-
mented connective k-algebra with the property that the augmentation
ideal I = ker(ß0R ! k) is contained in the Jacobson radical of ß0R.
Let M be an R-module which is bounded below, and let E = End R(k).
Then Hom R(M, k) is of downward finite type over E if the groups ßiM
(i 2 Z) are individually finite dimensional over k, and small over E if
ß*M is finite dimensional over k.
Proof. Let Mi denote the Postnikov stage PiM (9.4), so that the mod-
ule M is equivalent holim Mi. We claim that Hom R(M, k) is equiva-
lent to hocolim Hom R(Mi, k). This follows from the fact that Mn is
obtained from M by attaching copies of iR for i > n, and so the
natural map Hom R(Mn, k) ! Hom R(M, k) induces isomorphisms on
ßi for i -n. Now observe that the fibre of F of Mn ! Mn-1 has
only one nonzero homotopy group, ßnF = ßnM; this group is finite
dimensional over k and hence, since the augmentation ideal of ß0R is
contained in the Jacobson radical, has a finite composition series over
ß0R whose associated graded module consists of copies of the augmen-
tation module k. The triangle Mn ! Mn-1 ! K(ßnR, n + 1) (9.3) du-
alizes to Hom R(K(ßnR, n + 1), k) ! Hom R(Mn-1, k) ! Hom R(Mn, k),
which, in light of 9.2 and the above remarks, shows that Hom R(Mn, k)
is obtained from Hom R(Mn-1, k) by successively attaching copies of
Hom R( n+1k, k) ~ -(n+1)E. Since Mi ~ 0 for i << 0, the proposition
follows.
9.11. Proposition. Suppose that k is a field and that R is a cocon-
nective augmented k-algebra such that ß-1R ~=0 and the augmentation
gives an isomorphism ß0R ~= k. Let M be a module over R which
is bounded above. Then M is of downward finite type over R if and
only if the groups ßi(k R M) are individually finite dimensional over
k, and small as a module over R if and only if ß*(k R M) is finite
dimensional over k.
Proof. We will prove the smallness statement; the statement involving
downward finite type is handled similarly. If M is small as a module
over R then k R M is small as a module over k, and so has finite
dimensional homotopy. Suppose conversely that k R M has finite
dimensional homotopy. Observe that if N is any R-module with the
34 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
property that ßiN ~= 0 for i > n, then the natural map ßnN ! ßn(k R
N) is an isomorphism; this depends on the fact that ß-1R ~=0, and is
proved by inspecting the technique for building N from free modules
used in the proof of 9.9. Now we prove by induction on the number
(M) of integers i such that ßi(k R M) 6= 0 that M is small over R.
If (M) = 0 there is nothing to prove. Otherwise, let n be the greatest
integer such that ßnM 6= 0 and let VnM and W M be as in the proof of
9.9. The map VnM ! M induces an isomorphism ßn(VnM) ! ßnM
and hence an isomorphism ßn(k R VnM) ! ßnM. Since ßi(k R VnM)
vanishes for i 6= n (recall that VnM is a sum of copies of nR), it follows
from the cofibration sequence
k R VnM ! k R M ! k R W M
that ßnW M ~= 0 and that the map ßi(k R M) ! ßi(W M) is an
isomorphism for i 6= n. By induction, W M is small over R; since M
is obtained from VnM by attaching -1W M, M itself is small over
R.
9.12. Proposition. Suppose that X is a pointed connected space of
finite type (i.e., with a finite number of cells of each dimension), that
k is a commutative S-algebra, and that R is the augmented k-algebra
C*( X; k). Then k is of upward finite type as an R-module. If X is a
finite complex, then k is small as an R-module.
Proof. Let E be the total space of the universal principal bundle over
X with fibre X, so that E is contractible and M = C*(E; k) ~ k.
The action of X on E induces an action of R on M which amounts
to the augmentation action of R on k. Let Ei be the inverse image in
E of the i-skeleton of X, and let Mi be the R-module C*(Ei; k). Then
Mi=Mi-1 is equivalent to a finite sum iR indexed by the i-cells of
X. Since k ~ M = hocolim Mi, it follows that M is of upward finite
type. If X is finite then Mi = Mi-1 for i >> 0, and it follows that M
is small.
9.13. Coregularity. Finally, given an augmented k-algebra R, we look
at the problem of building an R-module M, or R itself, from k.
9.14. Proposition. Suppose that k is a field, that R is an augmented
k-algebra, and that M is an R-module. Assume either that R is con-
nective and the kernel of the augmentation ß0R ! k is nilpotent, or
that R is coconnective and ß0R ~= k. Then M is finitely built from k
over R if and only if ß*M is finite dimensional over k.
9.15. Remark. A similar argument that if R is coconnective and ß0R ~
k, then any R-module M which is bounded below is built from k over
DUALITY 35
R. It is only necessary to note that the fibre Fn of M ! QnM is built
from k (it has only a finite number of nontrivial homotopy groups) and
that M ~ hocolim Fn. Along the same lines, if R is connective and
ß0R is as in 9.14, then any R-module M which is bounded above is
built from k over R.
Proof. It is clear that if M is finitely built from k then ß*M is finite
dimensional. Suppose then that ß*M is finite-dimensional, so that in
particular ßiM vanishes for all but a finite number of i. By using
the Postnikov constructions P* (9.4) or Q* (9.5), we can find a finite
filtration of M such that the associated graded objects are of the form
K(ßnM, n) (9.3). It is enough to show that if A is a discrete module
over ß0R which is finite-dimensional over k, then K(A, n) is finitely
built from k over R. But this follows from 9.3 and that fact that under
the given assumptions, A has a finite filtration by ß0R-submodules such
that the successive quotients are isomorphic to k.
9.16. Proposition. Suppose that X is a pointed connected finite com-
plex, that k is a commutative S-algebra, and that R is the augmented
k-algebra C*(X; k). Then R ! k is coregular.
Proof. Let E be the augmented k-algebra C*( X; k) so that by 9.12,
E ! k is regular. Since R ~ End E(k) (3.1), the argument in the proof
of 2.15 (with R and E switched) shows that R ! k is coregular.
9.17. Completeness. Section 2 describes various notions of complete-
ness. We show that for commutative noetherian rings the notions usu-
ally agree. Recall that a ring k is said to be regular if every finitely
generated discrete module over k has a finite projective resolution, i.e.,
if every finitely generated discrete module over k is small over k.
9.18. Proposition. Suppose that R ! k is a surjection of commutative
noetherian rings with kernel ideal I R. Assume that k is a regular
ring. Then the following are equivalent:
(1) (R, k) is dc-complete (2.14),
(2) R is I-adically complete, i.e., R ~=lim sR=Is,
(3) R is k-complete in the sense of 2.1.
9.19. Remark. The proof below shows that under the conditions of 9.18,
the double centralizer map R ! R^ (2.14) can be identified with the
I-adic completion map R ! limsR=Is.
9.20. Lemma. [13 , 6.4] Let R ! k be a surjection of commutative
rings, and assume that the kernel I R is finitely generated as an
ideal. Then a map M ! N of R-modules is a k-equivalence (2.1) if
and only if it induces an equivalence k R M ~ k R N.
36 W. G. DWYER, J. P. C. GREENLEES, AND S. IYENGAR
Proof of 9.18. Let E = End R (k), and R^ = End E (k), so that there
is a natural homomorphism R ! R^ which is an equivalence if and
only if (R, k) is dc-complete. We will show that R^ is equivalent to
R^I= limsR=Is, and then show that the map R ! R^Iis a k-completion
map. The conclusion is that (R, k) is dc-complete if and only if the
map R ! R^Iis an isomorphism, and that this last occurs if and only
if R is k-complete.
Consider the class of all R-modules X with the property that the
natural map
(9.21) X ! Hom E(Hom R(X, k), k)
is an equivalence. The class includes k, and hence all R-modules finitely
built from k. Each quotient Is=Is+1 is finitely generated over k, hence
small over k, and hence finitely built from k over R. It follows from
an inductive argument that the modules R=Is are finitely built from
k over R, and consequently that 9.21 is an equivalence for X = R=Is.
By a theorem of Grothendieck [24 , 2.8], there are isomorphisms
(
k i = 0
colim sExt iR(R=Is, k) ~=
0 i > 0
which (1.3) assemble into an equivalence
hocolim sHom R(R=Is, k) ~ Hom R(R, k) ~ k .
This allows for the calculation
R^ ~ Hom E(k, k)~ Hom E(hocolim sHom R(R=Is, k), k)
~ holims Hom E(Hom R(R=Is, k), k)
~ holims R=Is ~ R^I.
It is easy to check that under this chain of equivalences the map R ! ^R
corresponds to the completion map R ! R^I.
It remains to show that R ! R^Iis a k-completion map. We first
show that k itself is k-complete (2.1). Suppose that f : M ! N
is a k-equivalence of R-modules. By 9.20, f induces an equivalence
k R M ! k R N and hence an equivalence
Hom R(N, k) ~ Hom k(k R N, k) ! Hom k(k R M, k) ~ Hom R(M, k) .
This is exactly what is needed in order for k to be k-complete. It follows
that any R-module which is finitely built out of k is k-complete. In
particular, as above, R=Is is k-complete, and hence the homotopy limit
R^I~ holims R=Is is k-complete. By [2, 10.14, 10.15], the natural map
DUALITY 37
k = k R R ! k R R^Iis an equivalence. Given 9.20, this implies that
R ! R^Iis a k-equivalence and that R ! R^Iis k-completion.
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Department of Mathematics, University of Notre Dame, Notre Dame,
IN 46556. USA
Department of Pure Mathematics, Hick Building, Sheffield S3 7RH.
UK
DUALITY 39
202 Mathematical Sciences Building, University of Missouri, Columbia,
MO 65211. USA
E-mail address: dwyer.1@nd.edu
E-mail address: j.greenlees@sheffield.ac.uk
E-mail address: iyengar@math.missouri.edu