RINGS WITH A LOCAL COHOMOLOGY THEOREM AND APPLICATIONS
TO COHOMOLOGY RINGS OF GROUPS.
J.P.C.GREENLEES AND G.LYUBEZNIK
Abstract. Cohomology rings of various classes of groups have curious dual*
*ity properties ex-
pressed in terms of their local cohomology [2, 3, 12, 4, 5, 6]. We formul*
*ate a purely algebraic
form of this duality, and investigate its consequences. It is obvious tha*
*t a Cohen-Macaulay ring
of this sort is automatically Gorenstein, and that its Hilbert series the*
*refore satisfies a functional
equation, and our main result is a generalization of this to rings with d*
*epth one less than their
dimension: this proves a conjecture of Benson and Greenlees [4]
1. Introduction
It has recently emerged that the rings of coefficients of equivariant cohomol*
*ogy theories very
often have remarkable duality properties. It is the purpose of the present pape*
*r to formulate the
duality purely algebraically in a particularly favourable case, and to investig*
*ate its ring theoretic
implications. We give a little background in Appendix A, but readers finding th*
*e definition of
interest as commutative algebra in its own right may ignore the topology.
Before we describe the duality properties, we need some terminology. Rings a*
*rising as co-
efficients of cohomology theories are Z-graded, and in this paper all elements *
*and ideals are
required to be homogeneous. These rings are also graded-commutative in the sens*
*e that rs =
(-1)deg(r)deg(s)sr for all elements r; s. Graded-commutative rings are very clo*
*se to being commu-
tative, and we want to apply the techniques of commutative algebra to them. Sin*
*ce left and right
ideals coincide, the notions of a Noetherian ring and a prime ideal behaves as *
*in the commu-
tative case. The formula mr := (-1)deg(r)deg(m)rm allows one to consider left m*
*odules as right
modules, and we henceforth restrict ourselves to left modules. If R is of chara*
*cteristic 2 then R
is itself commutative; in general the subring Revof even degree elements is com*
*mutative, and the
inclusion induces a bijection of primes. Readers uncomfortable with graded-comm*
*utative rings
should note that our constructions may be made using only the structure of a mo*
*dule over the
commutative subring Rev.
For the rest of the paper, R will be a Noetherian graded-commutative local ri*
*ng of dimension
r, with maximal ideal m and residue field k. We also suppose R is connected in*
* the sense that
it is zero in negative degrees and R0 = k.
To state the duality property we use Grothendieck's local cohomology functors*
* H*m(.). Since
R is Noetherian, these calculate the right derived functors of the m-power tors*
*ion functor on
graded R-modules M 7-! mM = {x 2 M | msx = 0 fors >> 0}:
H*m(M) = R*mM:
______________
This work began whilst the second author was visiting Sheffield, supported by*
* a Visiting Fellowship from the
EPSRC.
1
2 J.P.C.GREENLEES AND G.LYUBEZNIK
Notice that since M is graded, the local cohomology module Him(M) is a graded m*
*odule, and
we write Hi;jm(M) for the degree j part. Readers uncomfortable with graded-comm*
*utative rings
may interpret m as the maximal ideal of Revthroughout.
Local cohomology detects depth in the sense that depth(M) = min{i | Him(M) 6=*
* 0}, so that
R is Cohen-Macaulay if and only if H*m(R) is concentrated in degree r; in this *
*case R is then
Gorenstein if and only if Hrm(R) is isomorphic to DR = Hom k(R; k) up to a shif*
*t of degrees.
Remark 1.1. (Grading conventions). Since we are much concerned with modules n*
*on-zero
in both positive and negative degrees, it is essential to be clear about gradin*
*g conventions.
All grading will be cohomological (upper indexing), and R is concentrated in de*
*grees 0 and
above. Other constructions are graded in the standard way. Thus (DR)n = Hom n*
*k(R; k) =
HomQk(R-n; k), and DR is concentrated in degrees 0 and below. In general Hom n*
*k(M; N) =
i i+n
iHom k(M ; N ).
We also use topological notation to denote shifts in degrees. Thus (M)n = Mn-*
*1, and we
refer to M as the (cohomological) suspension of M; the alternative notation M =*
* M(1) is
also widely used.
We are now equipped to define the class of rings we wish to study.
Definition 1.2.We say that R has a local cohomology theorem with shift v (or th*
*at R is an
LCT v ring) if there is a spectral sequence
Es;t2= Hs;tm(R) =) vDR
with differentials
du : Es;tu-! Es+u;t-u+1u;
and so that du : Es;*u-! Es+u;*uis a map of R modules.
The interest in LCT v rings arises from a number of examples supplied by the *
*cohomology of
groups.
Example 1.3. (a) If R = H*(G; k) for a finite group G, then R admits a local c*
*ohomology
theorem with shift 0 [12].
(b) If R = H*(BG; k) for a compact Lie group of dimension w, and if the adjoint*
* representation
is orientable over k then it admits a local cohomology theorem with shift -w [4*
*].
(c) If R = H*(G; k) for a k-orientable virtual Poincare duality group G of virt*
*ual dimension v,
then it admits a local cohomology theorem with shift v [5].
(d) If R = H*(G; k) for a p-adic Lie group G of dimension v then it admits a lo*
*cal cohomology
theorem with shift v [6].
The existence of an LCT v structure is a form of duality, since the E2 term i*
*s covariant in R,
whilst the spectral sequence converges to DR, which is contravariant. For examp*
*le, if R is an
LCT v ring and Cohen-Macaulay, then the spectral sequence collapses to give an *
*isomorphism
Hrm(R) = v-rDR, showing that R is also Gorenstein. Conversely, it is immediate*
* that any
Gorenstein ring is an LCT v ring for some v.
Our main result, Theorem 5.4, describes the consequences of an LCT v structur*
*e on a ring
which is almost Cohen-Macaulay in the sense that its depth is one less than its*
* dimension. In
particular we prove a conjecture of Benson and the first author for the rings o*
*f Example 1.3 (b)
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
* 3
above, by giving a pair of functional equations 6.2 for the Hilbert series of a*
*ny almost Cohen-
Macaulay LCT v ring. Benson and Carlson [3] had previously proved the analogous*
* result for
cohomology rings of finite groups as in Example 1.3 (a), by using the particula*
*r features of the
definition of the cohomology ring.
We show in Section 7 that Grothendieck's method of dual localization shows th*
*at the local-
ization of an LCT v ring at a prime of dimension d is an LCT v-dring. According*
*ly, our results
for r-dimensional LCT v rings of depth r - 1 or r allow us to deduce 7.4 that a*
*ny LCT v ring R
is generically Gorenstein, and also very well behaved in codimension 1.
Readers particularly interested in the applications to cohomology rings may n*
*ot be familiar
with standard methods of local cohomology, so we have therefore explained vario*
*us well-known
methods from commutative algebra at some length, particularly in Section 2.
Contents
1. Introduction *
* 1
2. Implications of local duality *
* 3
3. The LCT approximation map. *
* 5
4. When is R quasi-Gorenstein? *
* 6
5. Interaction with Grothendieck's spectral sequence. *
* 7
6. Hilbert functions. *
* 9
7. Localization of the spectral sequence. *
* 11
8. Minimal associated primes of dual local cohomology. *
* 13
Appendix A. Topological background *
* 14
References *
* 16
2. Implications of local duality
In this section we record a number of consequences of local duality that are *
*central to the
analysis. These are all well known [18], but some readers may appreciate the si*
*mplicity of the
proofs in our case.
Let R be an r-dimensional connected graded-commutative local ring with maxima*
*l ideal m
and residue field k. Throughout this section M will be a finitely generated R-m*
*odule, although
our main interest is in the case M = R.
By Noether normalization we may choose a polynomial subring "R R over which R*
* is finite;
for definiteness we suppose the generators of R" are in even degrees a1; a2; : *
*:;:ar, and we let
a = a1 + a2 + . .+.ar.
Remark 2.1. In what follows the primary objects are R-modules. However, it is*
* sometimes
convenient to establish various properties by considering the underlying "R-mod*
*ule, and we pause
to clarify this.
(1) The Matlis duality functor is defined on an R-module M by DM = Hom R (M; *
*E(k)).
Note that E(k) = Hom R(R; k), and hence we have DM = Hom R(M; E(k)) = Hom k(M; *
*k) =
Hom R"(M; "E(k)):
(2) Throughout we shallpbe_discussing local cohomology relative to the maxima*
*l ideal of the
ambient ring. Since m"R= m, where "mis the maximal ideal of "R, for every R-m*
*odule M we
4 J.P.C.GREENLEES AND G.LYUBEZNIK
have Hi"m(M) = Him(M); it should therefore cause no confusion if we always writ*
*e H*mfor local
cohomology functors for the maximal ideal.
(3) It is now clear that the depth of an R-module, the dimension of its suppo*
*rt and the
dimension of its associated primes can be established by considering it as an "*
*R-module.
(4) If M is a finitely generated R-module, it is a finitely generated "R-modu*
*le, so Him(M) is an
*
* __
Artinian "R-module. *
* |__|
The principal ingredient in our analysis is local duality [17]. We shall use *
*the graded version
(suitable forms are immediate from [13, 3.8] or [7, 3.6.19]), which states that*
* for any finitely
generated R-module M,
DHim(M) = aExtr-i"R(M; "R):
Working with the dual of local cohomology allows us to measure the significance*
* of local coho-
mology modules by their dimension. Since DHim(M) is finitely generated, its dim*
*ension is equal
to that of its support.
Lemma 2.2.
dim(DHim(M)) i
Proof: For a prime " of dimension d let ""be the inverse image of " in "R, a*
*lso of dimension
d. Since "R"is a regular local ring of dimension r - d we see
Extr-i"R(M; "R)""= Extr-i"R"(M"; "R") = 0
*
* __
for r - i > r - d. *
* |__|
The top local cohomology module is particularly well behaved and plays a spec*
*ial role: its
dual is the canonical module = DHrm(R) of R.
Lemma 2.3. All associated primes of are of dimension r.
Proof: There is an exact sequence 0 -! "F-! R -! Q -! 0 of "R-modules where "Fi*
*s a free "R-
module of finite rank and Q has non-zero annihilator. Accordingly = DHrm(R) = *
*Hom "R(R; "R)
*
* __
is a submodule of Hom "R(F"; "R) = "F. *
* |__|
The other useful fact [18] about is that it satisfies Serre's Condition S2 (*
*its localization at "
has depth 2 if ht(") 2 and depth 1 if ht(") = 1). We will have occasion to in*
*clude a proof
of this in 5.2 below.
An immediate corollary of local duality allows us to discuss Hilbert series.
Corollary 2.4. If M is a finitely generated R-module then for each i the local *
*cohomology module
Him(M) is finite in each degree.
*
* __
Proof: It suffices to observe DHim(M) = Extr-i"R(M; "R) is a finitely generate*
*d "R-module. |__|
Although local cohomology modules are all supported at m, we can use duality *
*to give a useful
way of localizing local cohomology modules. Grothendieck's dual localization fu*
*nctor
L" : R-mod - ! R"-mod
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
* 5
for a prime " is defined by
L"M = D((DM)"):
Here the innermost D is duality for R and the outer one is duality for R". Not*
*e that if " is
graded and the localization and duality are both interpreted in the graded sens*
*e, L" takes graded
modules to graded modules, and Matlis duality works as usual [7, 3.6.16].
Lemma 2.5. If the prime " has dimension d and M is a finitely generated R-mo*
*dule then
L"Him(M) = Hi-dm(M")
Proof: Note that "R"is a regular local ring of dimension r - d, and then apply*
* local duality:
(DHim(M))" = aExtr-i"R(M; "R)""= aExtr-i"R"(M"; "R") = DH(r-d)-(i-d)m(*
*M")
*
* __
*
* |__|
Reasonable behaviour in a module is reflected in the small size of its lower *
*local cohomology.
Lemma 2.6. If M is an R-module of dimension n with no associated primes of di*
*mension < n
then dim DHdm(M) d - 1 for d < n.
Proof: Suppose d < n and " is a prime of dimension d. By hypothesis, M" is of *
*depth 1._
Therefore DHdm(M)" = DH0m(M") = 0 and " is not in the support of DHdm(M). *
* |__|
3. The LCT approximation map.
We now begin the investigation of LCT v rings as defined in 1.2. Thus we sup*
*pose R is a
graded connected local ring and there is a spectral sequence
Es;t2= Hs;tm(R) =) vDR:
Note immediately that we have an edge homomorphism
rHrm(R) -! vDR
with dual
ff : R -! v-r:
We call ff the LCT-approximation, and it is of central importance.
Lemma 3.1.
ker(ff) = {x 2 R | dim(Rx) r - 1}
Proof: The spectral sequence shows that the kernel has a finite filtration by *
*subquotients of
DH0m(R); DH1m(R); : :;:DHr-1m(R). It therefore consists of elements generating *
*a submodule of
dimension r - 1 by 2.2. Conversely, since every associated prime of is r-dime*
*nsional by 2.3,_
every element generating a submodule of dimension r - 1 lies in the kernel. *
* |__|
It follows that ff is injective if R is unmixed (i.e. if all associated prime*
*s of R have dimension
r).
We may make a weaker statement about cok(ff).
6 J.P.C.GREENLEES AND G.LYUBEZNIK
Lemma 3.2. cok(ff) is of dimension r - 2.
Proof: The spectral sequence shows that the cokernel has a finite filtration b*
*y subquotients_of
DH0m(R); DH1m(R); : :;:DHr-2m(R). It therefore has dimension r - 2 by 2.2. *
* |__|
There is a very convenient way to package a refinement of these observations,*
* and we turn to
it in Section 7. However in the meanwhile we give a criterion for ff to be an i*
*somorphism.
4. When is R quasi-Gorenstein?
We say that R is quasi-Gorenstein if the canonical module = DHrm(R) is isomo*
*rphic to a
suspension of R. Thus an r-dimensional ring is Gorenstein if and only if it is *
*quasi-Gorenstein
and of depth r.
Lemma 4.1. Suppose R is equidimensional and unmixed. The following three cond*
*itions on a
prime " in R of dimension d are equivalent.
1. ht(") 2 and depth(R") = 1.
2. There is a regular a 2 R such that " is an embedded prime of (a).
3. " is a minimal prime of Supp(DHd+1m(R))
Proof: (1) () (2): If an element a 2 " is a non-zero divisor it is regular o*
*n R". If R"
is of depth 1, the regular sequence a cannot be extended, and hence R=(a) is of*
* depth 0. The
argument is reversible. The conditions on height and embeddedness correspond.
(1) () (3): The requirement depth(R") = 1 is equivalent to H1"(R) 6= 0. Bu*
*t H1"(R) =
D(DHd+1m(R)") and hence DHd+1m(R)" 6= 0, or equivalently, " is in the support o*
*f DHd+1m(R).
It is minimal in this support since the fact that R is equidimensional and unmi*
*xed implies_that
DHd+1m(R) is supported in dimension (d + 1) - 1 by 2.6. *
* |__|
Remark 4.2. An R-module M is S2 in the sense of Serre if depth(M") min{2; dim*
*(R")} for
every prime " of R. It follows that if R is unmixed, then R is S2 if and only i*
*f no prime satisfies
Condition (1) of Lemma 4.1.
Corollary 4.3. If R is equidimensional and unmixed, then the set of primes " sa*
*tisfying the
equivalent conditions of Lemma 4.1 is finite.
Proof: For each i, the set of minimal primes of Supp(DHim(R)) is finite becaus*
*e DHim(R)_is
finitely generated. *
* |__|
Theorem 4.4. Suppose R is equidimensional and unmixed, and that it satisfies a*
* local cohomol-
ogy theorem with shift v. The LCT-approximation map ff : R -! v-r is an isomorp*
*hism if
and only if R is S2.
Proof: We have remarked that R is S2 if and only if it satisfies the equivalen*
*t conditions of the
lemma.
It is known that is S2 [18] (or see 5.2 below), hence if ff is an isomorphis*
*m, R is S2.
For the converse we show that if ff is not an isomorphism then R is not S2. *
*Since R is
equidimensional and unmixed, the canonical map is injective, and the cokernel h*
*as dimension
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
* 7
r - 2 by 3.2. Now if ff is not an isomorphism =R 6= 0. Let a 2 R be a regular *
*element such
that a R. Thus
aR a R;
and hence a=aR R=aR. But R=aR has equi-dimension dim(R) - 1, while a=aR ~==R h*
*as
dimension r - 2 by 3.2. If " is a minimal prime of a=aR, then " is associated *
*to R=aR, and __
" is an embedded associated prime of (a), so that by 4.1 R is not S2. *
* |__|
5.Interaction with Grothendieck's spectral sequence.
We begin by recalling Grothendieck's spectral sequence from [18]. We shall se*
*e that its form
is quite different to that of the local cohomology theorem. Firstly, it is cont*
*ravariant in R, both
at the E2 and in the target. Secondly, each entry Ep;q2is itself an R module. I*
*ts construction is
quite formal and applies to any ring.
Proposition 5.1. There is a spectral sequence
Ep;q2= Hpm(DHr-qm(R)) =) DR;
where the target is concentrated in total degree r. The differentials du : Ep;q*
*u-! Ep+u;q-u+1uare
maps of R-modules.
Proof: Given a finitely generated module M we have mHom "R(M; N) = Hom R"(M; *
*mN).
Furthermore, m preserves injectives and Hom "R(M; .) takes injectives to m-acyc*
*lic modules [18].
Hence we obtain two composite functor spectral sequences converging to the same*
* cohomology
Ep;q2= Hpm(Extq"R(M; N)) =) Rp+q(mHom "R(M; .))(N) (= Extp"R(M; HqmN) = *
*Ep;q2:
In particular if N is Cohen-Macaulay, the second spectral sequence collapses. W*
*e are interested
in the special case M = R and N = "R.
Next observe that Hrm(R") = DaR", where a is the sum of the degrees of the po*
*lynomial
generators of "R. This is injective, and there is an isomorphism Hom "R(R; DR")*
* ~=DR, so we find
the spectral sequence takes the form
Ep;q2= Hpm(DHr-qm(R)) =) DR;
*
* __
where the target is concentrated in total degree r. *
* |__|
It is useful preparation to recall the following application.
Lemma 5.2. [18] The canonical module satisfies Serre's Condition S2.
Proof: Consider the above spectral sequence for the localized ring R" where " *
*is a prime of
height h. The entries E0;02and E1;02necessarily survive to E1 . If h 1 this sh*
*ows E0;02= 0, and
*
* __
if h 2 we also have E1;02= 0. Thus is S2. *
* |__|
Case 0: R is Cohen-Macaulay.
Lemma 5.3. If R is Cohen-Macaulay, then is of depth r and Hrm() = DR. If in *
*addition R
is an LCT v-ring then DR ~=r-vD and hence
Hrm() ~=r-vD:
8 J.P.C.GREENLEES AND G.LYUBEZNIK
Proof: The first statement is immediate from the fact that Grothendieck's spec*
*tral sequence_
collapses. The next isomorphism is the LCT-approximation. *
* |__|
Case 1: R is almost Cohen-Macaulay.
It is the analysis of this case which gives us our main theorem. One innovati*
*on of our approach
is that we deduce module theoretic consequences, rather than simply conditions *
*on Hilbert series.
To state the condition we recall that a module M of dimension d is said to be G*
*orenstein if it is
finitely generated R-module of depth d and Hdm(M) is a suspension of DM.
We also use the abbreviation = DHr-1mR.
Theorem 5.4. If R is almost Cohen-Macaulay and satisfies a local cohomology th*
*eorem with
shift v then
(i) is of depth r and Hrm() ~=r-vD.
(ii) is of depth r - 1 and Hr-1m() ~=r-1-vD.
Thus is Gorenstein of dimension r and is Gorenstein of dimension r - 1.
Proof: First consider Grothendieck's spectral sequence: it is concentrated on*
* the two rows
q = 0 and q = 1. Since it converges to DR concentrated in total degree r, we co*
*nclude that for
p r - 1 the d2 differential gives an isomorphism
Hp-2m() ~=Hpm();
and there is an exact sequence
0 -! Hr-2m -d2!Hrm -! -aDR -! Hr-1m -! 0:
On the other hand, if R has a local cohomology theorem we have the exact sequ*
*ence
0 -! v-r+1 -! R -! v-r -! 0:
Because R has depth r - 1, and has dimension r - 1, applying local cohomology*
* gives the
isomorphisms
Hp-1m ~=Hpm
for p r - 2, the exact sequence
0 -! Hr-2m(v-r) -! Hr-1m(v-r+1) -! Hr-1m(R) -! Hr-1m(-r) -! 0;
and the isomorphism
Hrm(R) ~=v-rHrm():
Combining the first two displayed isomorphisms we obtain
Hpm() ~=-1Hp-3m() and Hp-1m() ~=-1Hp-4m()
for p r - 1. Hence is of depth r and is of depth r - 1 as required. The four*
* term exact_
sequence now becomes the isomorphism of Part (ii). *
* |__|
Examples of Aoyama [1] show that the depth d of for an almost Cohen-Macaulay*
* ring may
take any value 2 d r, so the theorem is a significant restriction.
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
* 9
6.Hilbert functions.
If M is a graded k-module of finite dimension in each degree, we may consider*
* its Hilbert
function (or Poincare series). We view this as an element of the Grothendieck g*
*roup of Z -graded
k-modules, and accordingly write [M] for it. We view this as a doubly infinite *
*power series in t
so that [sM] = ts[M]. If M is a finitely generated R-module then [M] may be vie*
*wed as the
expansion of a rational function [M]rf(t) about t = 1, but we retain separate n*
*otation for the
Hilbert function and the associated rational function.
Once again we consider a polynomial subring R" of R. Notice that if R" has g*
*enerators in
degrees a1; a2; : :a:r, the Poincare series [R"]rf(t) = 1=(1 - ta1)(1 - ta2) . *
*.(.1 - tar). This gives the
equality [R"]rf(1=t) = (-1)rta[R"]rf(t) of rational functions, where a = a1+ a2*
*+ . .+.ar. Note also
that DHrm(R") = aR".
Case 0: R is Cohen-Macaulay. Any ring with local cohomology theorem which is *
*Cohen-
Macaulay is automatically Gorenstein. Stanley has shown [24] that any graded Go*
*renstein ring
satisfies a functional equation. In the presence of a local cohomology theorem,*
* we may give a
more direct proof, which also prepares the way for our result in the almost Coh*
*en-Macaulay case.
Proposition 6.1. If R is a Cohen-Macaulay ring of dimension r with a local coho*
*mology theorem
then its Hilbert rational function satisfies the functional equation
[R]rf(1=t) = t-v(-t)r[R]rf(t):
Proof: In the Cohen-Macaulay case R ~="R F0 for some finite graded k-module F0.*
* This gives
the two equations
1. [R] = [R"][F0]
2. [DHrm(R)] = [DHrm(R")][F0_] = ta[R"][F0_];
where F0_is the k-dual of F0. If in addition R has a local cohomology theorem w*
*ith suspension
v then v-rDHrm(R) = R, and this allows us to combine the above statements to gi*
*ve
[R"][F0] = tv-r+a[R"][F0_]:
In short, [F0] is self-dual up to suspension in the sense that
[F0] = tv-r+a[F0_]:
In terms of rational functions this states
[R]rf(t) = [R"]rf(t)[F0]rf(t) = tv-r+a[R"]rf(t)[F0_]rf(t) = (-1)rtv-*
*r[R]rf(1=t)
*
* __
as required. *
* |__|
Case 1: R is almost Cohen-Macaulay. Benson and Carlson [3] have shown that the *
*coho-
mology ring of a finite group satisfies a pair of functional equations, and it *
*has been conjectured
by Benson and Greenlees [4] that this also holds for cohomology rings of classi*
*fying spaces of
compact Lie groups (although a sign in [4] is incorrect for odd dimensional rin*
*gs). We prove
an arbitrary almost Cohen-Macaulay ring satisfying a local cohomology theorem s*
*atisfies such
functional equations.
10 J.P.C.GREENLEES AND G.LYUBEZNIK
Theorem 6.2. Suppose R is of depth r - 1 and dimension r, and that it satisfie*
*s a local co-
homology theorem with shift v. Then the Hilbert function of R satisfies the fo*
*llowing pair of
functional equations:
[R]rf(1=t) - t-v(-t)r[R]rf(t) = (-1)r-1(1 + t)[DHr-1m(R)]rf(t)
and
[DHr-1m(R)]rf(1=t) = t-v(-t)r-1[DHr-1m(R)]rf(t);
Proof: In the almost Cohen-Macaulay case we have a short exact sequence
0 -! "R F1 -! "R F0 -! R -! 0
for suitable finite graded k-modules F1 and F0. Taking local cohomology and du*
*alizing, this
gives the exact sequence
0 - DHr-1m(R) - aR" F1_- aR" F0_- DHrm(R) - 0:
In terms of Grothendieck groups these give two equations
1. [R] = [R"]([F0] - [F1])
2. [DHrm(R)] = ta[R"]([F0_] - [F1_]) + [DHr-1m(R)]
If R has a local cohomology theorem with suspension v, then we obtain a short e*
*xact sequence
0 -! v-r+1DHr-1m(R) -! R -! v-rDHrm(R) -! 0:
In terms of Hilbert functions this gives
[R] = tv-r[DHrm(R)] + tv-r+1[DHr-1m(R)]:
This allows us to combine the Equations 1 and 2 to give
i j
[R"]([F0] - [F1])=tv-r ta[R"]([F0_] - [F1_]) + [DHr-1m(R)]+ tv-r+1[DHr-1*
*m(R)]
= tv+a-r[R"]([F0_] - [F1_]) + tv-r(1 + t)[DHr-1m(R)]
In terms of rational functions this gives the first of the conjectured equations
[R]rf(t) - (-1)rtv-r[R]rf(1=t) = tv-r(1 + t)[DHr-1m(R)]rf(t):
We have seen in 5.4 that = DHr-1m(R) enjoys the Gorenstein duality property.
The module is of dimension r - 1 and we have ~=r-1-vDHr-1m(). It is theref*
*ore of
dimension r - 1 if it is nonzero. As before is a finitely generated module ov*
*er the (r - 1)-
dimensional ring R=ann(), and hence by Noether normalization it is a finitely g*
*enerated module
over an (r - 1)-dimensional polynomial ring with generators in degrees b1; b2; *
*: :;:br-1. Since it
is Cohen-Macaulay, it is a free module, so we conclude as before that
[]rf(t) = (-1)r-1tv-(r-1)[]rf(1=t):
*
* __
*
* |__|
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
*11
7.Localization of the spectral sequence.
So far we have concentrated on what can be said about the whole ring, at the *
*expense of
adding hypotheses. In this section, we note that for any ring with a local coho*
*mology theorem,
its localizations at primes of height 0 and 1 are very well behaved.
Lemma 7.1. If R has a local cohomology theorem with suspension v and " is a *
*prime of
dimension d then R" admits a local cohomology theorem with shift v - d.
Proof: Apply the exact functor L" to the spectral sequence. We obtain a new spe*
*ctral sequence
with
L"Es;*2= L"Hsm(R) = Hs-dm(R") =) vL"DR:
Now we calculate L"DR = D((DDR)") = D(R"): Finally note that it remains only to*
* regrade
__s-d;* *
* __
the spectral sequence with E u = L"Es;*u. *
* |__|
One advantage is that if if we choose a prime of dimension r or r - 1 we obta*
*in a collapsed
spectral sequence. We first examine the 0 and 1 dimensional cases to which we a*
*re reduced by
localization, and then return to deduce conclusions for arbitrary rings.
Proposition 7.2. If R is zero dimensional with a local cohomology theorem of sh*
*ift v then
R = H0m(R) ~=vDR;
*
* __
and R is a Gorenstein ring with dualizing shift v. *
* |__|
Proposition 7.3. If R is one dimensional with a local cohomology theorem of shi*
*ft v then H0m(R)
is Gorenstein with dualizing shift v:
DH0m(R) = -vH0m(R)
The dualizing module is also Gorenstein: it has the property that H0m() = 0 as*
* usual and
H1m() ~=1-vD:
Proof: If R is a one dimensional ring the spectral sequence amounts to a short*
* exact sequence
0 -! H1m(R) -! vDR -! H0m(R) -! 0
or dualizing, to a sequence
0 - -1DH1m(R) - -vR - DH0m(R) - 0:
and DH0m(R) has only m associated to it. Now consider the six term exact seque*
*nce of local
cohomology modules: it splits into two isomorphisms. Indeed, by 2.3, = DH1m(R*
*) only has
one dimensional associated primes, so that H0mDH1mR = 0 and hence
-vH0m(R) ~=H0m(DH0m(R)) = DH0m(R);
where the equality arises since DH0m(R) has only m associated to it. This stat*
*es that H0m(R) is
self-dual with dualizing dimension v. Since DH0m(R) is 0-dimensional, H1m(DH0m(*
*R)) = 0 and so
-1H1m() = -1H1m(DH1m(R)) ~=-vH1m(R) = -vD:
*
* __
*
* |__|
12 J.P.C.GREENLEES AND G.LYUBEZNIK
This leads to the following general conclusion about the good behaviour of a *
*general ring R
with local cohomology theorem in codimension 0 and 1.
Corollary 7.4. If R is an r dimensional ring with local cohomology theorem of s*
*hift v then
(1) R is generically Gorenstein (ie its localization at any minimal prime is Go*
*renstein).
(2) R is almost Gorenstein in codimension 1 in the sense that R has the followi*
*ng behaviour
localized at a prime " of height 1. If " is of dimension d, let 0= DHd+1m(R) an*
*d 0= DHd-1m(R)
(so that 0= and 0= if d = r - 1). There is a a short exact sequence
0 -! v-d0"-! R" -! v-d+10"-! 0
where both 0"= DHd+1m(R)" and both 0"= DHd-1m(R)" are Gorenstein. More precisely
v-d0"~=D0";
and 0has the property that DHdm(0)" = 0 and
v-d-10"= DHd+1m(0)":
Proof: Part (1) is clear, since the localized spectral sequence gives
L"Hdm(R) ~=v-dL"DR:
For Part (2) we give more details. The one dimensional case gives the short e*
*xact sequence
0 -! L"Hd+1m(R) -! v-dDR" -! L"Hdm(R) -! 0
or equivalently, a short exact sequence
0 - -10"- d-vR" - 0"- 0:
Considering the six term local cohomology exact sequence we obtain two isomorph*
*isms: since
DHd+1m(R)" ~=DH1m(R") only has associated primes of dimension 1 it has zero H0m*
*, and since
DHdm(R)" = DH0m(R") is zero dimensional it has zero H1m. Thus we find
d-vL"Hdm(R) = d-vH0m(R") ~=H0m(DHdm(R)") = DHdm(R)";
or dualizing
v-d0"= v-dDHdm(R)" ~=D(DHdm(R)") = D0":
Similarly,
d-vL"Hd+1m(R) = d-vH1m(R") ~=-1H1m(DHd+1m(R)") = -1L"Hd+1m(DHd+1m(R));
or dualizing
v-d0"= v-dDHd+1m(R)" ~=DHd+1m(DHd+1m(R))" = DHd+1m(0)":
*
* __
*
* |__|
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
*13
8. Minimal associated primes of dual local cohomology.
In this section we formalize the geometric content of our results, and in the*
* special case of mod
p cohomology of groups, we use a theorem of Quillen to relate this in turn to t*
*he group theory.
We have been concerned with the defect of local rings, defined by def(R) = di*
*m(R)-depth(R),
and we now consider the defect stratification of X = Spec(R). Thus, we let
Xi= {" 2 Spec(R) | def(R") i}
Evidently this gives a chain of inclusions
X = X0 X1 X2 . . .Xffi Xffi+1= ;:
We write X0i= Xi\ Xi+1for the ith pure stratum. Thus X00= X0 \ X1 is the Cohen-*
*Macaulay
locus (which we have shown is also the Gorenstein locus) and X01= X1 \ X2 is th*
*e almost
Cohen-Macaulay locus (which we have shown is the almost Gorenstein locus).
Now suppose " is a prime of height h and dimension d, so that r h+d. Because*
* DHim(R)" =
DHi-dm(R") we see that if " 2 Supp(DHim(R)) then " 2 Xh+d-i, and conversely, if*
* " 2 X0h+d-i
then " 2 Supp(DHim(R)). This shows the defect only ever decreases under localiz*
*ation and hence
the chain terminates with ffi = def(R).
For the remainder of the discussion we assume that R is equidimensional, so t*
*hat h + d = r
for all primes. Thus
Xi= Supp(DHr-im(R)) [ Supp(DHr-i-1m(R)) [ : :[:Supp(DHr-ffim(R)):
Corollary 8.1. If R is equidimensional and aj = ann(DHjm(R)) = ann(Hjm(R)) then
Xi= V (ar-i) [ V (ar-i-1) [ : :[:V (ar-ffi):
*
* __
In particular Xi is a closed set. *
* |__|
From 2.2 the minimal primes over aj have dimension j, and hence Xiis of dime*
*nsion r-i.
We now focus on the case when R is the mod p cohomology of a classifying spac*
*e of a compact
Lie group or virtual duality group. Quillen [21, 22] has given a description o*
*f the variety of
R, and in particular its minimal primes. We briefly recall Quillen's stratifica*
*tion of the variety.
For an elementary abelian group A and an algebraically closed fieldSK of charac*
*teristic p, let
XA(K) = A K. This defines a variety XA, and we take X+A= XA \ BA XB. Restric*
*tion
defines a map
a
XA -! XG
A
where the coproduct is over elementary abelian subgroups of G and XG is the var*
*iety of H*(BG).
Quillen shows that
a
XG = X+G;A;
(A)
where X+G;Ais the image of XA, and the coproduct_is now over conjugacy classes *
*of elementary
___
abelian subgroups. Furthermore X+G;A= X+A=WG(A) where W G(A) = NG(A)=CG(A).
The following lemma is useful in understanding Quillen's filtration.
14 J.P.C.GREENLEES AND G.LYUBEZNIK
Lemma 8.2. For any prime ", let m(") denote the set of minimal primes contain*
*ed in ". The
set m(") depends only on the Quillen pure stratum X+G;Acontaining ".
Proof: Closed Quillen strata XG;A and XG;B intersect in unions of strata XG;C *
*for subgroups_
C of conjugates of A and B. *
* |__|
The significant feature for us is that the mod p Steenrod algebra Ap acts on *
*R. Quillen has
shown [22, 12.1] that any prime invariant under the Steenrod operations P ifor *
*i 0 is equal
to the prime "(E) obtained by pullback from the minimal prime of H*(BE) for an *
*elementary
abelian subgroup under the restriction map R = H*(BG) -! H*(BE). This shows tha*
*t any
P *-invariant prime determines a conjugacy class of elementary abelian subgroup*
*s, and therefore
has a group theoretic counterpart.
Proposition 8.3. The minimal primes over ai are invariant under the Steenrod al*
*gebra, and
hence define a union of Quillen closed strata, namely the one defined by the set
Ei= {E | "(E) is minimal overai}
of elementary abelian subgroups of G closed under conjugacy.
Proof: First note that Him(R) is an Ap-module: to see this, observe that Ap act*
*s via differential
operators, and apply the result of [19] that Him(R) is a D-module. It follows t*
*hat its annihilator
ai is Ap-invariant, and the first clause is immediate from [23, 11.2.3]. The re*
*st follows from_the
description of the Quillen stratification. *
* |__|
This shows that the defect stratification is subordinate to the Quillen strat*
*ification, and in
particular the minimal primes of Xi are Quillen strata. This generalizes Duflo*
*t's theorem [8]
that the associated primes of R are Quillen strata. Indeed, an associated prime*
* of R of height h
is a minimal prime of Hd-hm(R), by local duality.
Appendix A. Topological background
Although equivariant topology plays no role at all in this paper except to pr*
*ovide examples
which show the theory is not vacuous, it still seems worth providing a little b*
*ackground. For
simplicity, suppose G is a finite group. There are certainly interesting adapt*
*ions to the cases
of compact Lie groups, discrete virtual duality groups, and also to p-adic anal*
*ytic groups, and
we comment below on how they affect the discussion. There are various interesti*
*ng examples of
cohomology theories F *(.). Three to bear in mind are (1) ordinary cohomology w*
*ith coefficients
in a field k, (2) K-theory and (3) bordism. These may be applied to the classif*
*ying space BG,
and we consider the ring R = F *(BG), and the R-module F*(BG). It is natural t*
*o expect a
universal coefficient theorem
Exts;tF*(F*(BG); F*) =) F *(BG):
Note in particular that both the E2 term and the target are contravariant in BG*
*. In case (1) the
universal coefficient theorem collapses to state D(F*(BG)) = F *(BG), and since*
* the cohomology
is finite in each degree, this gives F*(BG) = DR. It is natural to expect more*
* generally that
the universal coefficient theorem arises from an equivalence RD(F*(BG)) ' R, in*
* some derived
category, where D(.) = Hom F*(.; F*): this will be given a precise meaning belo*
*w.
RINGS WITH A LOCAL COHOMOLOGY THEOREM *
*15
So far these are formal properties which are to be expected in any theory. Ho*
*wever in many
examples (such as the three listed above [12, 11, 16]) a much more interesting *
*relationship arises
out of good behaviour of an associated equivariant theory: there is a spectral *
*sequence
Hs;tI(F *(BG)) =) F*(BG);
where I = ker(F *(BG) -! F *). Note that this too connects F *(BG) with F*(BG) *
*but its E2
term is contravariant in G whilst it converges to F*(BG), which is covariant. A*
* derived category
form of this would be an equivalence
RI(R) ' F*(BG):
Combining the local cohomology theorem with the universal coefficient theorem w*
*e obtain a form
of duality for the ring R = F *(BG):
RD O RI(R) ' R
in some derived category of R-modules.
In the discussion so far we were not obliged to discuss equivariant cohomolog*
*y theories at all.
However, the natural statements are in terms of the equivariant cohomology FG*(*
*EG x X) and
homology FG*(EG x X). For cohomology the change of groups isomorphism FG*(EG x*
* X) ~=
F *(EG xG X) is straightforward, and generalizes directly to the various classe*
*s of infinite group.
However, in homology the isomorphism F*G(EG x X) ~=F*(EG xG X) involves a trans*
*fer and
therefore a suspension or twisting. For example if G is a compact Lie group
F*G(EG x X; EG x *) ~=F*(EG xG ((X; *) x (D(ad(G)); S(ad(G)))
where (D(ad(G)); S(ad(G))) is the pair consisting of the unit disc and unit bal*
*l of the adjoint
representation. The effect of this is to give a statement like
RD O RI(R) ' -vR
where -v is some invertible functor on the derived category. It would be intere*
*sting to investi-
gate the implications of such an algebraic statement along the lines of the pre*
*sent paper.
We finish by giving a context in which the heuristic derived category stateme*
*nts are true. For
this we need to work with highly structured ring and module spectra, and more p*
*recisely, in the
category of S-algebras of Elmendorf-Kriz-Mandell-May [9]. We thus suppose that *
*the represent-
ing spectrum F is an S-algebra, and use the highly structured inflation of Elme*
*ndorf-May [10]
to view it as an equivariant S-algebra. To emphasize the algebraic content, we *
*write RHom for
function spectra. The articles [14, 15] provide an introduction to some of the *
*constructions used
here.
Proposition A.1. Suppose F is a complex orientable S-algebra, and G is a compa*
*ct Lie group.
If F *(BG) is Noetherian and I = ker(F *(BG) -! F *) then there is an equivalen*
*ce
RD O RI O RHom (EG+; F ) ' RHom (EG+; F )
of equivariant RHom (EG+; F )-modules, where RD(M) = RHom F (M; F ).
Remark A.2. The equivariant homotopy of the right hand side is R = F *(BG+). *
*There are
various spectral sequences for calculating the equivariant homotopy of the left*
* hand side, their
initial terms are related to Ext*;*F*(H*;*m(R); F*).
16 J.P.C.GREENLEES AND G.LYUBEZNIK
Proof: By [9, IV.4] the universal coefficient theorem is realized by an equiva*
*lence
RHom F (F ^ BG+; F ) ' RHom (BG+; F )
of non-equivariant F -modules. We need to work with G-spectra so we note that t*
*he universal
coefficient equivalence is obtained by passage to Lewis-May fixed points from t*
*he equivalence
RHom F (F ^ EG+; F ) ' RHom (EG+; F )
of equivariant RHom (EG+; F )-modules.
Now the local cohomology theorem (see [15] for a proof in this context) is an*
* equivalence
F ^ EG+ ' RI O RHom (EG+; F );
*
* __
and the result follows by combining the two. *
* |__|
It may be worth comparing this statement with a more elementary one, which ca*
*n be viewed
as a generalization of Anderson duality. If R is any S-algebra, with R* Noether*
*ian and I is an
ideal of R* we may form the R-module RIR. There is a spectral sequence
H*;*I(R*) ) ss*(RIR)
for calculating its homotopy. Given an R-module E we may consider the duality *
*functor
RD(M) = RHom R (M; E), and the composite RD O RI(R). A particularly familiar ca*
*se occurs
when (R*; I) is local and E is the injective envelope of the residue field. If*
* R* is Gorenstein,
RD O RI(R) has homotopy -vR* for some v, and hence RD O RI(R) ' vR. More genera*
*lly
we remark that if M* is an R*-module of projective dimension 1 there is an R m*
*odule M with
ss*M = M*, and M is unique up to equivalence. Thus if R* is Cohen-Macaulay and *
*DHrI(R*)
is of projective dimension 1, we can again identify the R-module RD O RIR. Usua*
*lly this will
not be a suspension of R, but rather a wedge of several. Thus A.1 states that R*
*Hom (EG+; F )
behaves very much like a Gorenstein ring.
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Department of Pure Mathematics, Hicks Building, Sheffield, S3 7RH, UK.
E-mail address: j.greenlees@sheffield.ac.uk
Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
E-mail address: gennady@math.umn.edu