TATE COHOMOLOGY IN AXIOMATIC STABLE HOMOTOPY THEORY.
J. P. C. GREENLEES
Contents
1. Introduction *
* 1
2. Axiomatic Tate cohomology in a stable homotopy category. *
* 2
3. Finite localizations. *
* 4
4. The category of G-spectra *
* 6
5. The derived category of a commutative ring. *
* 7
6. The category of modules over a highly structured ring *
* 10
7. The derived category of kG *
* 11
8. Chromatic categories. *
* 12
9. Splittings of the Tate construction. *
* 13
10. Calculation by comparison. *
* 14
11. Calculations by associative algebra. *
* 16
12. Calculations by commutative algebra. *
* 19
13. Gorenstein localizations. *
* 22
References *
*23
1. Introduction
The purpose of the present note is to show how the axiomatic approach to Tate*
* cohomol-
ogy of [18, Appendix B] can be implemented in the axiomatic stable homotopy the*
*ory of
Hovey-Palmieri-Strickland [32]. Much of the work consists of collecting known r*
*esults in a
single language and a single framework. The very effortlessness of the process *
*is an effective
advertisement for the language, and a call for further investigation of other i*
*nstances. The
main point is to recognize and compare incarnations of the same phenomenon in d*
*ifferent
contexts: the splitting and duality phenomena described in Sections 9 and 13 ar*
*e particularly
notable. More practically, Theorem 11.1 is new, and Theorem 12.1 extends result*
*s of [19].
A stable homotopy category [32, 1.1.4] is a triangulated category C with arbi*
*trary coprod-
ucts, and so that all cohomology theories are representable. It is also requir*
*ed to have a
compatible symmetric monoidal structure with unit S and a set G of strongly dua*
*lizable ob-
jects generating all of C using triangles, coproducts and retracts. If in addit*
*ion the objects
of G are small, the stable homotopy category is said to be algebraic.
We shall illustrate our constructions in several contexts specified in greate*
*r detail later.
The following list gives the context followed by an associated stable homotopy *
*category.
Each of these admits a number of variations.
1
2 J. P. C. GREENLEES
o Equivariant topology: the homotopy category of G-spectra (Section 4).
o Commutative algebra: the derived category of a commutative ring R (Section*
* 5).
o Brave new commutative algebra: the homotopy category of modules over a hi*
*ghly
structured commutative ring spectrum R (Section 6).
o Representation theory: the derived category of the group ring kG of a fini*
*te group G
(Section 7).
o The bordism approach to stable homotopy theory: various chromatic categori*
*es (Section
8).
The paper is in three parts.
Part I: General formalities (Sections 2 and 3). In Section 2 we summarize nec*
*es-
sary definitions and give the Tate construction in a stable homotopy category a*
*ssociated to a
smashing localization. We establish the fundamental formal properties that make*
* it reason-
able to call this a Tate construction. In Section 3 we recall from [39] that fi*
*nite localizations
are smashing and hence give rise to Tate theories: the minor novelty is to emph*
*asize the
view that this is an Adams projective resolution in the sense of [1].
Part II: Examples (Sections 4 to 8). We describe the above contexts in more d*
*etail,
and consider the construction in each one, identifying it in more familiar term*
*s.
Part III: Special properties (Sections 9 to 13): The final sections give some*
* more
subtle results about the construction which require additional hypotheses. In S*
*ection 9, we
discuss dichotomy results stating that the Tate construction is either periodic*
* or split. We
then turn to methods of calculation. The first is the familiar calculation usin*
*g associative
algebra, generalizing the use of group cohomology in descent spectral sequences*
* (one uses
homological algebra over the endomorphism ring of the basic building block). We*
* describe
this in Section 11, and give a new example in the case of equivariant topology *
*with a compact
Lie group of equivariance. This method applies fairly generally, provided the s*
*table homotopy
category arises from an underlying Quillen model category. Less familiar is the*
* calculation
in terms of commutative algebra. This arises when the (commutative) endomorphis*
*m ring
of the unit object has a certain duality property (it is `homotopically Gorenst*
*ein'). This is
quite exceptional, but it applies in a surprisingly large number of familiar ex*
*amples: in the
cohomology of groups [19, 9, 8], in equivariant cohomology theories [17, 25], a*
*nd in chromatic
stable homotopy theory (Gross-Hopkins duality). Its occurrence in commutative a*
*lgebra is
investigated in [21], and shown to be very special.
2. Axiomatic Tate cohomology in a stable homotopy category.
In this section we describe the Tate construction. Since it depends on a suit*
*able Bousfield
localization, we briefly recall the terminology in a suitable form (see [32, Se*
*ction 3] for more
detail). We consider a functor L : C -! C on the stable homotopy category C. Th*
*e acyclics
of L are the objects X so that LX ' *. The functor L is a Bousfield localizatio*
*n if it is exact,
equipped with a natural tranformation X -! LX, idempotent, and its class of acy*
*clics is
an ideal.
A Bousfield localization L is determined by its class D of acyclics as follow*
*s: Y is L-local
if and only if [D; Y ]* = 0 for all D in D, and a map X -! Y is the Bousfield l*
*ocalization
if and only if Y is local and the fibre lies in D. The usual notation for the*
* localization
triangle is CX -! X -! LX. Furthermore, any such class of acyclics is a localiz*
*ing ideal
(i.e. it is closed under completing triangles, sums and smashing with an arbitr*
*ary object).
AXIOMATIC TATE COHOMOLOGY 3
A localization is said to be smashing if the natural map X ^ LS -! LX is an equ*
*ivalence
for all X. It is equivalent to require either that L commutes with arbitrary su*
*ms, or that
the class LC of L-local objects is a localizing ideal [32, 3.3.2].
We shall define a Tate construction associated to any smashing localization.
Notation 2.1. (General context)
o C: a stable homotopy category
o G: a set of generators for C
o D: the localizing ideal of acyclics for a smashing Bousfield localization *
*(.)[D-1].
o C[D-1]: the localizing ideal of [D-1]-local objects.
The notation LD is often used for (.)[D-1]; the present notation better refle*
*cts the character
of a smashing localization, and corresponds to that in [24]. The idea is that w*
*e should think
of X[D-1] as a localization away from D. More precisely the archetype is locali*
*zation away
from a closed subset in algebraic geometry. The notation comes from the case w*
*hen the
closed subset is defined by the vanishing of a single function f. In this very*
* special case,
the localization is realized by inverting the multiplicatively closed set {1; f*
*; f2; : :}:in the
sense of commutative algebra. We therefore use the corresponding `sections with*
* support'
notation for the fibre of this localization:
D (X) -! X -! X[D-1]:
We can use this to define an associated completion.
Lemma 2.2. The natural transformation X -! F (D (S); X) is Bousfield completi*
*on whose
class of acyclics is the class of [D-1]-local objects.
Proof.First we must show that if E is [D-1]-local, then [E; F (D (S); X)]* = 0.*
* By [32,
3.1.8], S[D-1] is a ring object in C and E = E[D-1] is a S[D-1]-module. Hence E*
* ^ D (S) is
a retract of E^S[D-1]^D (S); since [D-1] is idempotent and smashing, S[D-1]^D (*
*S) ' *.
Secondly we must show that the fibre, F (S[D-1]; X) is [D-1]-local. However i*
*f_D_lies in
D then D ^ S[D-1] ' D[D-1] ' *. *
*|__|
We write X^D:= F (D (S); X) for this Bousfield completion, and also introduce*
* the fol-
lowing notation for its fibre:
D (X) -! X -! X^D:
We now define the D-Tate construction by
tD (X) = X^D[D-1]:
This gives the diagram
D (X) ---! X - --! X[D-1]
? ? ?
? ? ?
y y y
D (X^D) ---! X^D - --! X^D[D-1] = tD (X)
Lemma 2.3. The map D (X) -! D (X^D) is an equivalence.
Proof.We need only remark that D (D (X)) ' *; however by definition D (X) lies *
*in the__
class of [D-1]-local objects. *
* |__|
4 J. P. C. GREENLEES
Corollary 2.4. (Hasse Principle) The diagram
X ---! X[D-1]
? ?
? ?
y y
X^D ---! tD (X)
*
* __
is a homotopy pullback square. *
* |__|
Corollary 2.5. (Warwick Duality [18]) There is an equivalence
tD (X) = X^D[D-1] ' D (D (X)):
Proof.This is a composite of three equivalences.
X^D[D-1] - D (X^D[D-1]) -! D (D (X^D)) -! D (D (X))
The first is an equivalence since (.)[D-1]^D' * (the class E of acyclics for (.*
*)^Aconsists of
[D-1]-local objects) so that X^D[D-1]^D' *. The second is an equivalence since *
*D (X^D) ' *
(defining property of D ((.)) together with idempotence of (.)^D). The third is*
* an_equivalence
since D (D ((.))) ' * by 2.3 so that D (D (D (X))) ' *. *
* |__|
This shows that the cohomology as well as the homology only depends on the lo*
*cal-
ization away from D. More precisely, the definition tD (X) = F (D (S); X)[D-1]*
* shows
that T ^ tD (X) only depends on the localization T [D-1]. The second avatar tD*
* (X) '
F (S[D-1]; D (X)) gives
[T; tD (X)]* = [T ^ S[D-1]; D (X)]* = [T [D-1]; D (X)]*;
which again only depends on T [D-1].
Remark 2.6. The definition of the Tate construction we have given is at a nat*
*ural level of
generality. One might be tempted to consider LD LEX for arbitrary D and E. Howe*
*ver, if one
wants Warwick Duality, one requires (i) LELD X ' *, so that E LD C and (ii) CD*
* CEX ' *,
so that CEX is LD -local, and E LD C. Thus we require E = LD C, and this must*
* be a
localizing ideal. Thus LD must be smashing, and determines E.
3. Finite localizations.
In this section we describe one very fruitful source of smashing localization*
*s. This is explicit
in Section 3.3 of [32], and especially Theorem 3.3.5. It generalizes the finite*
* localization of
Mahowald-Sadofsky and Miller [36, 39]. We recall the construction for future re*
*ference, and
emphasize the connection with Adams projective resolutions.
Recall that a full subcategory is thick if it is closed under completing tria*
*ngles and taking
retracts. The piece of data we need is a G-ideal A of small objects (ie a thick*
* subcategory
of small objects, closed under smashing with elements of G). If C is not algebr*
*aic, we must
suppose in addition that A is essentially small, consists of strongly dualizabl*
*e objects and is
closed under Spanier Whitehead duality; if C is algebraic these conditions are *
*automatic. In
practice we will specify A by giving a set T of small generators: A = G-ideal(T*
*). We then
need to form the localizing ideal D = locid(A) generated by A: this is the smal*
*lest thick
subcategory containing A which is closed under arbitrary sums and smashing with*
* arbitrary
elements of C.
AXIOMATIC TATE COHOMOLOGY 5
Context 3.1. (for a finite localization)
o C: a stable homotopy category
o G: a set of generators for C
o T: a set of small objects of C
o A = G-ideal(T)
o D = locid(T) = locid(A)
In these circumstances, we write write A or T in place of D in the notation, *
*so that
tA(X) = tT(X) = tD (X) and so forth.
Miller has shown that there is a smashing localization functor (.)[A-1] whose*
* acyclics are
precisely D, and whose small acyclics are precisely A; this is known as a finit*
*e localization and
the notation LfAis used in [32]. The construction is described in 3.3 below. Th*
*e associated
functor (.)^Awhose acyclics are the objects X[A-1] is denoted by LA in [32].
There is a convenient lemma for showing a set of elements in a localizing sub*
*category is
a generating set. It would be more traditional to view it as a convergence the*
*orem for a
projective resolution in the sense of Adams [1].
Proposition 3.2. If T D is a set of objects then D = locid(T) provided one of *
*the two
following conditions holds.
(i) T is a set of small objects and detects triviality in D, in the sense that *
*if X is in D then
[T; X]* = 0 for all T in T implies X ' *.
(ii) The objects of G are small, and for any X 2 D; S0 2 G and any x 2 [S0; X]**
* there is a
map t : nT -! X with x 2 im(t* : [S0; nT ]* -! [S0; X]*) and T in T.
Proof.We need to prove that if X is an arbitrary object of D, we may form X fro*
*m copies
of elements of T using sums and completion of triangles. We give the proof assu*
*ming that
Condition (ii) holds; the proof when Condition (i) holds is similar except that*
* [S0; .]* for
S02 G is replaced by [T; .]* for T 2 T.
By hypothesis we may form a projective resolution in the sense of Adams:
X _______X0- i0--!X1 --i1-!X2 --i2-!: : :
x x x
? ? ?
?t0 ?t1 ?t2
T0 T1 T2
Thus each Ti is a sum of suspensions of elements of T, each ti is surjective in*
* [S0; .]* for all
S02 G and Xi+1is formed as the cofibre of ti: Ti- ! Xi. Note that X1 = telnXn h*
*as trivial
[S0; .]* for all S0 2 G since is is zero in [S0; .]* by construction; thus X1 *
*' * since G gives
a set of generators. Defining Xi as the fibre of X -! Xi we find that Xi is co*
*nstructed
from sums of suspensions of elements of T by a finite number of cofibre sequenc*
*es. Passing_
to direct limits, we obtain a cofibre sequence X1 -! X -! X1 , so that X1 ' X*
*. |__|
Remark 3.3. Note that the argument essentially gives the construction of a fi*
*nite localiza-
tion. Take a set T of small generators of the G-ideal A and the localizing idea*
*l D, and ensure
it is closed under duality. We now form a projective resolution as in the proof*
* of 3.2, but
without assuming that X lies in D. Ensure ti is surjective in [T; .]* for each*
* i. Then the
triangle X1 -! X -! X1 has X1 in D by construction, and [T; X1 ]* = 0 for all*
* T in T.
6 J. P. C. GREENLEES
This completes the discussion of formalities. In the rest of the paper we wan*
*t to discuss
a number of examples from this point of view, and show how comparisons between *
*the
examples give rise to means of calculation.
4.The category of G-spectra
In this section we consider the category C = G-spectra of G-spectra for a com*
*pact Lie
group G, and localizations associated to a family F of subgroups. We recover th*
*e construc-
tions of [23]; indeed these constructions motivated investigation of its other *
*manifestations.
Thus we suppose F is closed under conjugation and passage to subgroups, and w*
*e let
T = {G=H+ | H 2 F}. Thus A is the class of retracts of finite F-spectra, and D*
* is the
class of all F-spectra. We recall that in the homotopy category of G-spectra, *
*the class of
F-spectra can be described in three ways, as is implicit in [34].
Lemma 4.1. The following three classes of G-spectra are equal, and called F-s*
*pectra.
(i) G-spectra formed from spheres G=H+ ^ Sn with H 2 F
(ii) G-spectra X so that the natural map EF+ ^ X -'! X is an equivalence and
(iii) G-spectra X so that the geometric fixed point spectra H X are non-equivar*
*iantly con-
tractible for H 2 F.
Proof: The equality of Classes (i) and (ii) is straightforward.
Since H commutes with smash products [34, II.9.12], and it agrees with H-fixe*
*d point
spaces on suspension spectra [34, II.9.9], it follows that EF+ ^ X lies in the*
* third class, so
Class (ii) is contained in Class (iii). Suppose then that X is in Class (iii); *
*we must show it is
also in Class (ii). By hypothesis, the map EF+ ^ X -! X has the property that *
*applying
H gives a non-equivariant equivalence for all H. It remains to observe that geo*
*metric fixed
points detect weak equivalences. This is well known, but I do not know a good r*
*eference: it
follows from the fact that Lewis-May fixed points tautologically detect weak eq*
*uivalences, by
an induction on isotropy groups. The basis is the relation between geometric an*
*d Lewis-May
fixed points [34, II.9.8]: for any H-spectrum X, H X ' (E"P ^ X)H where P is th*
*e family_
of proper subgroups of H. *
* |__|
From the equality of Classes (i) and (ii) EF+ ^ X lies in D, and from the fac*
*t that "EF is
F-contractible we see that X -! "EF^ X is localization away from D. Hence FS = *
*EF+
and S[F-1] = "EF. Now the equality of Classes (i) and (ii) can be recognized as*
* the statement
that localization away from the class of F-spectra is smashing. It follows that*
* in this case
Diagram 2 is the diagram
EF+ ^ X ---! X ---! "EF^ X
? ? ?
'?y ?y ?y
F (EF+ ; EF+ ^ X) ---! F (EF+ ; X)---! tF(X);
which is Diagram C of [23].
The skeletal filtration gives rise to spectral sequences for calculating the *
*homotopy groups
of these spectra based on group cohomology [23], and we discuss this in more ab*
*stract terms
in Section 11. More interesting is that for well behaved cohomology theories (s*
*uch as those
which are Noetherian, complex orientable and highly structured), one may prove *
*a local
AXIOMATIC TATE COHOMOLOGY 7
cohomology theorem in which case the homotopy groups may be calculated by commu*
*tative
algebra [18]. We discuss this in more abstract terms in Section 12. The formal *
*framework
for such spectral sequences is described in Section 10.
5. The derived category of a commutative ring.
In this section we consider the category C = D(R) for a commutative ring R, a*
*nd local-
izations associated to an ideal I of R. In particular, we obtain a new approach*
* to the results
of [18] and an improved perspective on the role of finiteness conditions.
We wish to consider the class of acyclics for a localization, and there are s*
*everal candidates
for this. The most natural is the class
D = {M | supp(H*(M)) V (I)};
but we should also consider
D0= {M | every element of H*(M) is I-power torsion}:
It is straightforward to check they are both candidates.
*
* __
Lemma 5.1. The classes D and D0are localizing ideals. *
* |__|
It is also easy to see that D0 D.
Lemma 5.2. If I is finitely generated then D0= D, but this is not true in gen*
*eral.
Proof.Suppose M is a module with support in V (I), and xp2_M_haspannihilator_J.*
* Since
R=J has support V (J), we see that V (J) V (I) so that J I I. If I is f*
*initely
generated, some power of I lies in J.
Topgive_an example where equality fails we needponly_display an ideal J so th*
*at no power
of J lies in J, since then we may take I = J and M = R=J. For instance if *
*R is
polynomial on a countably infinite numberpof_generators, x1; x2; x3; : :o:ver a*
* field and
J = (x1; x22; x33; : :):we find that J is the maximal ideal (x1; x2; x3; : :)*
*:no_power_of which
lies in J. *
* |__|
It is useful to have a specific generator for D as a localizing ideal. Perha*
*ps the most
natural candidate for a generator of D is R=I, but this can only generate D0. F*
*or the rest of
the section we assume that I is finitely generated, say I = (x1; : :;:xn), and *
*thus D = D0.
We show that R=I does give a generator, but there are other candidates which ar*
*e usually
convenient.
Warning 5.3. If R=I does not have a finite resolution by finitely generated pr*
*ojectives, it
need not be small.
We may define the unstable Koszul complex for the sequence xd1; xd2; : :;:xdn*
*by
xd1 xdn
UKod(x) = (R -! R) . . .(R -! R):
We also write UKo(x) := UKo1(x). The unstable Koszul complexes have the advanta*
*ge of
being small, and explicitly constructed from free modules.
We may also define the stable Koszul complex
Ko(I) = (R -! R[1=x1]) . . .(R -! R[1=xn])
8 J. P. C. GREENLEES
and define Co(I) by the existence of a fibre sequence Ko(I) -! R -! Co(I). It i*
*s not hard
to check [17] that both Ko(I) and Co(I) are independent of the generators of th*
*e ideal, up
to quasi-isomorphism. Since Ko(I) and Co(I) are only complexes of flat modules *
*and not
projective modules, it is necessary to replace them by complexes P Ko(I) and P *
*Co(I) of
projectives when calculating maps out of them in the derived category.
We start by showing what can be constructed from UKo(x).
Lemma 5.4. (i) Provided d1; d2; : :;:dn 1, the unstable Koszul complex UKo(x*
*d11; xd22; : :x:dnn)
lies in the thick subcategory generated by UKo(x).
(ii) The stable Koszul complex Ko(I) lies in the localizing subcategory generat*
*ed by the un-
stable Koszul complex UKo(x).
Proof.(i) First we deal with the case n = 1. We proceed by induction on d using*
* the square
d-1
R -x--! R
? ?
1?y x?y
d
R -x--! R
to construct a cofibre sequence UKod-1(x) -! UKod(x) -! UKo(x). The general cas*
*e follows
since the argument remains valid after tensoring with any free object.
d
(ii) The map R -! R[1=x] is the direct limit of the maps R -x! R, and hence K*
*o(x) is
equivalent to the homotopy direct limit of the terms UKo(xd). Tensoring these t*
*ogether and
using the fact that holim commutes with tensor products, we find
! d
Ko(I) ' holim UKod(x):
! d
*
* __
*
*|__|
We also need a related result in the other direction.
Lemma 5.5. The unstable Koszul complex UKo(x) lies in the thick subcategory g*
*enerated
by the stable Koszul complex Ko(I).
Proof.Consider the self-map of the cofibre sequence Ko(x) - ! R - ! R[1=x] give*
*n by
multiplication by x. Since x is an equivalence of R[1=x], the octahedral axiom *
*shows there
is a fibre sequence UKo(x) -! Ko(x) -x! Ko(x). We may tensor this argument with*
* any
object X, so that we find a fibre sequence
Ko(x1; : :;:xn-1) UKo(xn) -! Ko(I) -xn!Ko(I):
*
* __
Repeating this, we see that UKo(x) lies in the thick subcategory generated by K*
*o(I). |__|
Proposition 5.6. If I = (x1; x2; : :;:xn) is finitely generated, the class D is*
* generated as a
localizing ideal by R=I, by Ko(I) and by UKo(x).
Proof.We start by showing that D is generated by UKo(x). Since UKo(x) is small,*
* we may
apply Proposition 3.2 (i). It suffices to check that UKo(x) detects triviality*
* of objects D
of D. Suppose then that H*(X) is I-power torsion and t 2 H*(X). It suffices b*
*y 5.4 to
AXIOMATIC TATE COHOMOLOGY 9
show that the corresponding map t : R -! X extends over R -! UKo(xd11; xd22; : *
*:;:xdnn)
for some d1; d2; : :;:dn 1.
Suppose by induction on m that t has been extended to t0: UKo(xd11; xd22; : :*
*;:xdmm) -! X.
This is clear for m = 0, so the induction starts, and we suppose 0 < m < n. Now*
* note that
t0is I-power torsion, since [T; X] is I-power torsion for any finite complex T *
*of free modules.
Choose dm+1 so that xdm+1m+1t0= 0. Construct a cofibre sequence by tensoring
xdm+1m+1 o dm+1
R - ! R -! UK (xm+1 )
with UKo(xd11; xd22; : :;:xdmn). Exactness of [.; X] shows that t0extends along
UKo(xd11; xd22; . .;.xdmm) -! UKo(xd11; xd22; . .;.xdmm; xdm+1m+*
*1);
completing the inductive step. This completes the proof that UKo(x) generates D.
By 5.5 it follows that Ko(I) also generates D, and the fact that R=I generate*
*s D follows
if we can show UKo(x) lies in the localizing ideal generated by R=I.
Lemma 5.7. The localizing ideal containing R=I contains any complex X so that*
* H*(X) is
bounded in both directions and I-power torsion.
Proof.First, we prove by induction on k that a module M (regarded as an object *
*of the
derived category in degree 0 with zero differential) lies in locid(R=I) provide*
*d IkM = 0. If
k = 0 this means M = 0, so we suppose k 1. First, the short exact sequence Ik=*
*Ik+1 -!
R=Ik+1 -! R=Ik gives a triangle, with the first and third term already known to*
* be in
the ideal, so R=Ik+1 lies in the ideal. Now suppose Ik+1M = 0. There is a surje*
*ctive map
T0 -! M0 = M of modules where T0 is a sum of copies of R=Ik+1, and the kernel K*
*0 also
satisfies Ik+1K0 = 0. We may thus iterate the construction and apply 3.2 (ii) t*
*o deduce M
lies in the localizing ideal generated by R=Ik+1.
The modules M are Eilenberg-MacLane objects, and we show that if X is bounded*
*, it has
a finite Postnikov tower. After suspension we may suppose Hi(X) = 0 for i < 0. *
*Since X is
equivalent to the subcomplex X0zero in negative degrees, with X00the 0-cycles, *
*and agreeing
with X in positive degrees, we may suppose X is zero in negative degrees. There*
* is then a
canonical map X = X0 -! M0 which is an isomorphism in degree 0 where M0 = H0(X).
The fibre X1 then has Hi(X1) = 0 for i < 1, and Hi(X1) ~=Hi(X) for i 1, and we*
* may
iterate the construction. Defining Xk by the triangle Xk -! X -! Xk we see that*
* X0 ' 0,
and by the octahedral axiom there is a cofibre seqence
kMk -! Xk+1 -! Xk:
Since Mk lies in the localizing ideal generated by R=I, so does Xk for all k. B*
*y the bound-_
edness hypothesis, XN ' 0 for N sufficiently large, and so XN ' X. *
* |__|
*
* __
Since UKo(x) satisfies the conditions of the lemma, this completes the proof *
*of 5.6. |__|
It is not hard to construct the relevant localizations and completions.
Lemma 5.8. If I is finitely generated,
(i) M[D-1] = M Co(I)
(ii) D (M) = M Ko(I))
(iii) M^D= Hom (P Ko(I); M)
10 J. P. C. GREENLEES
Proof.(i) To see that M Co(I) is local, we need only check it admits no morphi*
*sm from
UKo(x) except zero. However Co(I) admits a finite filtration with subquotients *
*R[1=x] for
x 2 I so it suffices to show [UKo(x); R[1=x]]* = 0. This follows since x is ni*
*lpotent on
UKo(x) and an isomorphism on R[1=x]. To see that M -! M Co(I) is a D-equivalen*
*ce
we need only verify that the M Ko(I) can be built from UKo(x). However M can b*
*e built
from R, and we saw in 5.4 that Ko(I) can be built from UKo(x).
Part (ii) follows from the defining fibre sequence of Co (I), and Part (iii) *
*follows_from
2.2. *
* |__|
We write
H*I(M) := H*(Ko(I) M) = H*(D (M)) :
this is the local cohomology of M, and if R is Noetherian it calculates the rig*
*ht derived
functors of
I(M) = {x 2 M | Inx = 0 forn >> 0}
for modules M [29]. We write
HI*(M) := H*(Hom (P Ko(I); M)) = H*(M^D) :
this is the local homology of M [22]. If, in addition, R is Noetherian or good *
*in the sense
of [22], then this local homology gives the left derived functors of completion*
*. In particu-
lar, if M is of finite type, M^D= M^I. Furthermore, the Tate cohomology coinci*
*des with
that of [18]. As pointed out in [18], Warwick duality is a generalization of th*
*e isomorphism
Z^p[1=p] = lim(Z=p1 ; p).
Remark 5.9. If I is finitely generated, we have described both a construction*
* and a method
of calculation for useful localizations. It would be interesting to have analog*
*ues when I is
not finitely generated.
6. The category of modules over a highly structured ring
In this section we suppose that R is a commutative S-algebra in the sense of *
*[14], and
we allow the equivariant case. Such objects are essentially equivalent to E1 ri*
*ng spectra, so
there is a good supply: in particular, any commutative ring R gives rise to the*
* Eilenberg-
MacLane S-algebra HR. We then let C denote the homotopy category of highly stru*
*ctured
module spectra over R and consider localizations and completions associated to *
*a finitely
generated ideal I of the coefficient ring R*
Much of the discussion of the previous section applies in the present case, a*
*nd was pre-
sented in [24], so we shall be brief. Thus we may form the stable and unstable*
* Koszul
modules by using cofibre sequences and smash products. Thus for example, UKo(x*
*) is
the fibre of R -x! R; we avoid the common notation -1R=x for fear of confusio*
*n.
Now UKo(x) = UKo(x1) ^R UKo(x2) ^R : :^:RUKo(xn); similarly Ko(x) is the fibre *
*of
R -! R[1=x], and Ko(I) = Ko(x1) ^R Ko(x2) ^R : :^:RKo(xn) where I = (x1; x2; : *
*:;:xn).
We take A to be the class of retracts of finite R-modules M so that M* is I-pow*
*er torsion
This is generated by T = {UKo(x)}, and generates the localizing ideal of all M *
*so that each
element of M* is I-power torsion (i.e. M* is in the class D(R*; I) in the sense*
* of Section 5).
We write I(M) := D (M), M[I-1] := M[D-1] and tI(M) := tD (M).
AXIOMATIC TATE COHOMOLOGY 11
The statement and proof of Lemma 5.8 apply without change. Because the constr*
*uction
comes with an evident filtration we may obtain spectral sequences by taking hom*
*otopy, and
the E1 term is a chain complex representing the corresponding constructions of *
*Section 5.
This gives spectral sequences
H*I(R*; M*) =) M[I-1]*
^H*I(R*; M*) =) tI(M)*
H*I(R*; M*) =) I(M)*
HI*(R*; M*) =) (M^I)*
for calculating their homotopy.
7. The derived category of kG
For a finite group G and a field k we consider the derived category C = D(kG)*
*, and take
A to be the category of finite complexes of projectives. This is generated by T*
* = {kG}, and
the generation is so systematic algebraically that it leads to the usual method*
* for calculating
Tate cohomology using projective resolutions and their duals. The relationship *
*of the derived
category D(kG) to the category of G-spectra is analogous to the relationship of*
* D(R*) to
the category of highly structured modules over R.
It is proved in [32, 9.6] that the localization M -! M[A-1] is obtained by te*
*nsoring with a
Tate resolution. Since any Tate resolution admits a finite filtration with subq*
*uotients R[1=x]
as in [19], it follows that every object of C with bounded cohomology is alread*
*y complete.
Thus we find that if M has bounded cohomology, tA(M) = M[A-1] = M tA(k) and so*
* the
Tate construction defined by localization agrees with Tate homology in the clas*
*sical sense.
There are at least three other examples to consider here, but some work is ne*
*eded to give
them substance. Recall that an indecomposable module M has vertex H if it is a *
*summand
in a module induced from H but not from any proper subgroup of H.
Variation 7.1. Consider a family F of subgroups, and the category AF of finite *
*complexes
of modules with vertex in F. The case F = {1} is that given above. The G-idea*
*l AF
is generated by TF = {k[G=H] | H 2 F}. It is then appropriate to use Amitsur-D*
*ress F-
cohomology [13]. Perhaps there is again a local cohomology theorem in the sense*
* of Section 12
below, using the ideal of positive degree elements, but the appropriate theory *
*of varieties has
not been developed. It would also be interesting to know the relationship to or*
*dinary group
cohomology and the ideal IF of cohomology elements restricting to zero in the c*
*ohomology
of H for all H 2 F.
Variation 7.2. We choose a block fi of kG and take Afito be the category of fin*
*ite complexes
of projectives in fi.
Variation 7.3. We may consider the stable module category C = StMod(kG), which *
*is
proved in [32, 9.6.4] to be a localization of D(kG). It would then be interesti*
*ng to investigate
complexity quotients in the sense of Carlson-Donovan-Wheeler [10, 11, 5, 6] fro*
*m the present
point of view.
12 J. P. C. GREENLEES
8. Chromatic categories.
Another important class of examples arises in the approach to stable homotopy*
* theory
through bordism. For background and further information see [40]. Thus we wor*
*k in the
stable homotopy category of spectra in the sense of algebraic topology, and we *
*choose a
prime p > 0. For 0 n 1 we shall need the spectrum E(n) representing Johnson-W*
*ilson
cohomology theory and the Morava K-theory spectrum K(n). For 0 < n < 1 these ha*
*ve
coefficient rings E(n)* = Z(p)[v1; v2; : :;:vn-1; vn; v-1n], and K(n)* = Z=p[vn*
*; v-1n]. The cases
n = 0; 1 are somewhat exceptional: by convention, for n = 0 we have E(0) = K(0)*
* = HQ
and for n = 1 we have E(1) = BP and K(1) = HZ=p. Recall that a spectrum is of t*
*ype
n if K(i)*(X) = 0 for i < n and K(n)*(X) 6= 0.
Bousfield localization Ln with respect to E(n) is the localization whose acyc*
*lics are the
spectra X with E(n) ^ X ' *. A well known theorem of Hopkins-Ravenel states tha*
*t Ln is
smashing. The usual notation is CnX -! X -! LnX: The completion X^D= F (CnS; X)
is more mysterious, but when n = 0 it is profinite completion F (S-1Q=Z; X).
_____________________________________________________________________||||||||
|_n_|E|(n)_|K(n)______|F_(n)_____L|n-1_____________|LK(n)___________|_
| 0 H|Q| |HQ | S0 |* |rationalization |
|| |||||| || (p)0k || | | | |
|| 1 K|(|p)|K=p||| |S =p|0k |invertlp| -L|K(1)1| ||
| 2 E|l|l El|l=(p; v1)S|=(p ; v1)L|1= "(.)[(p; v1) L]"K|(2) |
| ...||...|... |... |... |... |
|| |||| || || || | | | |
|_1_|B|P___|HFp______|*_________|p-localization_____p-|adic_completion_|
Following [33], let us consider a slightly simpler example. Let C be the E(n)*
*-local category,
and A the thick subcategory generated by LnF (n) for a finite type n spectrum F*
* (n). In
this case X[A-1] = Ln-1X [33, 6.10] and X^A= LK(n)X [33, 7.10]. The fibre of X*
* - !
Ln-1X isusually known as the nth monochromatic piece when X is E(n)-local, so w*
*e have
A(X) = MnX. The fibre of K(n) completion is sometimes known as CK(n), but we si*
*mply
write A(X) = K(n)(X).
Corollary 8.1. (Warwick Duality) If X is E(n)-local then
__
Ln-1LK(n)X ' K(n)(MnX): |__|
We note that if n = 0 this states M0X is rational, and if n = 1 it states tha*
*t the cofibre
of M1X -! LK(1)M1X is the rationalization of LK(1)X.
If we take C to be the entire category of p-local spectra there are two relat*
*ed examples.
Indeed, we may still consider the smashing localization Ln-1 = LE(n-1), but it *
*does not
seem so easy to describe the associated completion. In particular it is not eq*
*ual to LK(n)
(indeed, although Ln-1S0 is K(n)-acyclic, there are many spectra, such as F (n *
*+ 1), which
are K(n)-acyclic but not E(n-1)-local). We may also consider spectra T el(n) = *
*F (n)[1=vn],
and the smashing localization Lfn-1which is Bousfield localization with respect*
* to T el(0) _
T el(1)_: :_:T el(n-1); this is finite localization with respect to F (n) [39] *
*and it is therefore
smashing, and we may again consider the associated completion, which is again d*
*ifferent from
LK(n)for similar reasons. There is a natural transformation Lfn-! Ln; which is *
*believed
not to be an isomorphism for n 2 [35].
AXIOMATIC TATE COHOMOLOGY 13
9. Splittings of the Tate construction.
We describe two different classes splittings of the Tate construction. Each r*
*equires special
properties of the localization.
First, continuing with the notation of the previous section, note that tA(X) *
*= Ln-1LK(n)X
is the subject of Hopkins's chromatic splitting conjecture [31, 4.2]. When X =*
* S0 (albeit
not in the E(n)-local category C) this is conjectured to split into 2n pieces. *
*More precisely
there is a cofibre sequence
Ln-1S0p-! Ln-1LK(n)S0 -! F (Ln-1S0; LnS0p);
which is conjectured to split, and furthermore, F (Ln-1S0; LnS0p) is also conje*
*ctured to split
as a wedge of 2n - 1 suitable localizations of spheres. To obtain the cofibre s*
*equence, apply
F (.; X)[D-1] to the cofibre sequence -1S[D-1] -! D (S) -! S to obtain
X[D-1] -! tD (X) -! F (S[D-1]; X)
since F (S[D-1]; X) is already [D-1]-local.
Secondly, there is a dichotomy between the periodic and split behaviour of th*
*e Tate con-
struction, typified by the cohomology of finite groups. Although Tate cohomolog*
*y is often
associated with periodic behaviour, it is the split case that is generic. On t*
*he one hand,
when G has periodic cohomology there is a `periodicity element' x in H*(G) and *
*the Tate
cohomology ^H*(G) = H*(G)[1=x] is periodic under multiplication by x. By contra*
*st, when
group cohomology H*(G) has a regular sequence of length 2, Benson-Carlson [4] a*
*nd Benson-
Greenlees [7] have shown that the mod p Tate cohomology ^H*(G) of a finite grou*
*p splits
^H*(G) = H*(G) 1H*(G)
(where the suspension is homological) both as a module over H*(G) and as a modu*
*le over
the Steenrod algebra. Even this context does not provide a true dichotomy, sinc*
*e there are
groups with depth 1 which are not periodic, but this mixed behaviour is excepti*
*onal.
The analogous statement concerns the standard cofibre sequence
X^D-! tD (X) -! D (X)
when X = S. The dichotomy principle would suggest that in most cases, either t*
*D (S) is
obtained from S^Dby inverting some multiplicatively closed subset of ss*(S^D), *
*or else the
cofibre sequence splits, and that the split case is generic. The hypotheses fo*
*r a splitting
must include the requirement that the norm map -1D (X) -! X^Dis zero in homotop*
*y,
and probably also that ss*(X^D) is of depth at least 2. However the proofs from*
* the case of
group cohomology do not extend in any simple way since they use the fact that h*
*omology
and cohomology are identified in the Tate cohomology by their occurrence in pos*
*itive and
negative degrees.
A second case where the dichotomy holds is in commutative algebra [18]. When *
*the ring
is Noetherian and of Krull dimension 1, the rationality theorem [18, 7.1] holds*
*: H^*I(R) =
S-1(R^I) where S is the set of regular elements of R. This is the periodic case*
*. It is immediate
that if the ring is of I-depth two or more the Tate cohomology splits since the*
* local homology
is in degree 0 whilst the local cohomology is only non-zero at or above the dep*
*th.
14 J. P. C. GREENLEES
10. Calculation by comparison.
We discuss two quite different methods of calculation. To introduce the disc*
*ussion, we
explain the two methods as they apply to calculating the homology H*(G; M) of a*
* finite
group with coefficients in a chain complex M of kG-modules. The first is quite *
*familiar, and
states there is a spectral sequence
H*(G; H*(M)) =) H*(G; M):
The second method is the local cohomology theorem, stating that there is a spec*
*tral sequence
H*I(H*(G; M)) =) H*(G; M)
where I is the ideal of positive codegree elements of H*(G) [19], and H*I(.) de*
*notes local
cohomology in the sense of Grothendieck [29] (the definition was recalled in Se*
*ction 5).
The generalization we have in mind concerns finite localizations in the case *
*that A is
generated by a single object A. We require that A is a commutative comonoid in *
*the sense
that it has a commutative and associative coproduct A -! A^A and a counit A -! *
*S. We
require in addition that A is strongly dualizable and self-dual up to an invert*
*ible element, in
the sense that DA ' A ^ S-o for some object S-o admitting a smash inverse S-o ^*
* So ' S.
Example 10.1. (i) The motivating example has C = D(kG) for a finite group G an*
*d A =
kG. Note that we have an augmentation kG -! k, and a diagonal map kG -! kG kG.
Furthermore kG is self-dual.
(ii) Alternatively, for a compact Lie group G, we may take C to be a category o*
*f G-spectra
(or of module G-spectra over a ring G-spectrum R) and A = G+ (or R ^ G+). Again*
* we
have an augmentation G+ -! S0, and a diagonal map G+ -! G+ ^ G+. We also have t*
*he
duality statement DG+ ' -dG+ where d = dim(G). This helps explain the notation *
*S-o,
which is chosen since, in the geometric context, Atiyah duality shows o corresp*
*onds to the
tangent bundle.
(iii) Rather differently, we may take C to be the category of p-local spectra, *
*(or of p-local
R-module spectra over a ring spectrum R) and A = -dF (n) (or A = R ^ -dF (n)) w*
*here
F (n) = S0=(pi0; vi11; vi22; : :;:vin-1n-1) for suitable i0; i1; : :;:in-1 and *
*d = dim(F (n)). Collapse
onto the top cell gives an augmentation -dF (n) -! S0. In favourable cases we h*
*ave the
duality statement DF (n) ' -dF (n), and F (n) may be taken to be a commutative *
*ring
spectrum [12], and the dual to the product gives a coproduct map -dF (n) -! -dF*
* (n) ^
-dF (n).
We need to consider the graded commutative ring k* = [S; S]*, where S is the *
*unit in C, and
two k*-algebras. Firstly, since A is a commutative comonoid, l* = [A; S]* is a *
*commutative
k*-algebra, and [A; Z]* is a module over l* for any Z. Secondly, we consider th*
*e k*-algebra
E* = End(A)*, which need not be commutative.
Context 10.2. (for calculation)
o A a commutative comonoid object
o A generated by A
o DA ' S-o ^ A
o k* = [S; S]*
o l* = [A; S]*
AXIOMATIC TATE COHOMOLOGY 15
o I = ker(k* -! l*)
o E* = [A; A]*
In our examples these are as follows.
Example 10.3. (i) When A = kG we have k* = l* = k and End (kG)* = kG.
(ii) When A = R ^ G+ we have k* = RG*, l* = R* and End (R ^ G+)* = R*(G+).
(iii) When A = -dF (n) we have k* = R*, l* = R*(F (n)) and End (R ^ -dF (n))* =
R*(F (F (n); F (n))).
Given these data, there are two functors we can apply:
[A; .]* : C -! End(A)*-mod
(corresponding to non-equivariant homotopy in Example (ii)), and
[S; .]* : C -! k*-mod
(corresponding to equivariant homotopy in Example (ii)).
It is then natural to seek spectral sequences reversing these two functors.
In the first case we may hope they take the form
10.4.
H*(End (A)*; [A; So ^ X]*) =) (A(X))*
H*(End (A)*; [A; X]*) =) (X^A)*
and
H^*(End (A)*; [A; X]*) =) tA(X)*:
A construction in some cases is given in Section 11, and the twisting So in the*
* first spectral
sequence will be explained.
In the second case we let
I = ker(k* = [S; S]* -! [A; S]*)
be the augmentation ideal, and apply local cohomology, local homology and local*
* Tate
cohomology as appropriate and hope the spectral sequences take the form
10.5.
H*I(X*) =) (A(X))*
HI*(X*) =) (X^A)*
and
H^I*(X*) =) tA(X)*:
A construction in some cases is given in Section 12. When the first spectral se*
*quence exists
we say that the local cohomology theorem holds. Provided this happens for good*
* enough
reasons, the other two spectral sequences exist as a consequence.
The content should be clearer when we give some examples. It is not surprisin*
*g that to
prove the existence of either set of spectral sequences we have to assume the e*
*xistence of
additional structure beyond that present in the stable homotopy category.
16 J. P. C. GREENLEES
11.Calculations by associative algebra.
The point of this section is to generalize the Atiyah-Hirzebruch Tate spectra*
*l sequence of
[16]:
^H*(G; E*(X)) =) t(E)*G(X)
for finite groups G, or in other words to prove the expectations suggested in 1*
*0.4 hold
under suitable circumstances. The construction does not work entirely in a stab*
*le homotopy
category, but rather relies on the existence of a suitable Quillen model catego*
*ry from which
the stable homotopy category is formed by inverting weak equivalences.
We thus suppose given a stable homotopy category, and consider the G-ideal A *
*generated
by a single object A. The aim is to find ways to calculate A(X)*, (X^A)* and t*
*A(X)* in
terms of the [A; A]*-module [A; X]*. In view of the notational conflict we remi*
*nd the reader
that in the context of G-spectra, where A = G+ the group [S; .]* is equivariant*
* homotopy
and [A; .]* is non-equivariant homotopy. The present discussion covers a numbe*
*r of new
examples: the generalization is cruder than that of [23], but more general. The*
* discussion
of convergence in [23, Appendix B] applies without change.
To avoid the appearance of empty generalization, we state an unequivocal theo*
*rem in the
equivariant homotopy context of Section 4 (with A generated by R ^ G+).
Theorem 11.1. Suppose G is a compact Lie group of dimension d, R is an equiva*
*riant
S-algebra, and M an R-module. Provided we have the K"unneth theorem
(KT1)
M*(G+ ^ T ^ Y ) = R*(G+) R* M*(T ^ Y )
and the universal coefficient theorem
(UCT)
[G+ ^ T; M ^ Y ]G*= [T; M ^ Y ]* = Hom R*(R*(T ); M*(Y ))
when T = G^s+for s 0, there are spectral sequences
H*(R*(G+); M*(Sd ^ Y )) =) MG*(EG+ ^ Y );
H*(R*(G+); M*(Y )) =) M*G(EG+ ^ Y );
and
H^*(R*(G+); M*(Y )) =) t(M)G*(Y );
where the homology and cohomology on the left is that of the Frobenius algebra *
*R*(G+).
We return to this particular case at the end of the section. The rest of the *
*discussion is
conducted in general terms.
We want to view the construction of AS as a "resolution" for X = S using sums*
* of
objects of A. More precisely, we use the method of 3.2 (i) without assuming X i*
*s in D. The
dual resolution is thus
j0 1 j1 2 j2
* _______(S)0---! (S) ---! (S) ---! : : :
? ? ?
? ? ?
y yq0 yq1
* T0 T1
where each Tiis a sum of suspensions of objects of A. This is associated with t*
*he sequence
T0 -ffi1-1T1 -ffi2-2T2 -ffi3-3T3 - :
AXIOMATIC TATE COHOMOLOGY 17
We want to apply simplicial methods, so we suppose there is an underlying mod*
*el category,
from which the stable homotopy category is formed by passage to homotopy. Furth*
*ermore,
we require a compatible symmetric monoidal structure and that A is a strict com*
*onoid
object.
Example 11.2. The example relevant to the theorem is the homotopy category of *
*modules
over the equivariant S-algebra R for a compact Lie group G, and A = R ^ G+. Th*
*is is
the homotopy category of the model category of equivariant R-modules [14]. Howe*
*ver, in
this case it is more elementary to make the construction described below at the*
* space level,
apply the suspension spectrum functor and take the extended R-module: this stra*
*tegy will
give parts of the theorem under weaker hypotheses.
We form the homogeneous bar construction [38] as a simplicial object, and tak*
*e its geo-
metric realization
AS = S1 = B(A; A; S):
This ensures Ti= iA^(i+1).
By smashing with X we obtain a resolution for arbitrary X. Thus we may define
tA(X) = F (B(A; A; S); X) ^ "B(A; A; S);
where "B(A; A; S) is the mapping cone of B(A; A; S) -! B(S; S; S) = S.
To relate the resolution to an algebraic one, we apply a homology theory to o*
*btain
(ffi1)*-1 (ffi2)*-2 (ffi3)*-3
[A; T0]* - [A; T1]* - [A; T2]* - [A; T3]* - :
In the equivariant context we have [A; -iTi]* = R*(G^i+1+). To ensure it is a r*
*esolution, we
assume there is a K"unneth theorem
(KT1)
[A; A ^ Z] = [A; A]* [A;S]*[A; Z]*
for relevant Z (namely Z = A^i). In the equivariant context this is a K"unneth*
* theorem
for the (non-equivariant) homology theory R*(.). This ensures that the simplici*
*al contrac-
tion in geometry is converted to one in algebra and the bar filtration spectral*
* sequence
for calculating [A; B(A; A; S)]* has its E1 term given by the algebraic bar con*
*struction
B([A; A]*; [A; A]*; [A; X]*): To calculate A(X)*, we need the second K"unneth t*
*heorem
(KT2)
[S; Z]* = [A; S]* [A;A]*[A; So ^ Z]*
for relevant Z (namely Z = A^(i+1)^ X). In the equivariant context, this states*
* that the
change of groups isomorphisms [S; G+ ^T ]G*= [S; Sd^T ]* = [G+; T ]G*are reflec*
*ted in algebra.
Lemma 11.3. The K"unneth theorem (KT2) for Z = A ^ T follows from the K"unnet*
*h theo-
rem (KT1) for Z = So ^ T .
Proof: Assuming (KT1) for Z = So ^ T , we calculate
[A; S]* [A;A]*[A; A ^ So ^ T=]*[A; S]* [A;A]*[A; A]* [A;S]*[A; So ^ T ]*
= [A; S]* [A;S]*[A; So ^ T ]*
= [A; So ^ T ]*
= [S; So ^ DA ^ T ]*
= [S; A ^ T ]*
*
* __
as required. *
* |__|
18 J. P. C. GREENLEES
This is enough to give a spectral sequence with
E1 = [A; S]* [A;A]*B([A; A]*; [A; A]*; [A; So ^ X]*);
it therefore takes the form
E2*;*= H*(End (A)*; [A; So ^ X]*) =) A(X)*:
It is easy to see this spectral sequence is conditionally convergent in the sen*
*se of Boardman
[2]. The homology in the E2-term is defined to be the homology of the bar const*
*ruction, but
in favourable cases it can be calculated in various other ways. For example in *
*the case of
G-spectra this spectral sequence takes the form
H*(R*(G+); (Sd ^ X)*) =) XG*(EG+):
Note that we have two possible definitions of the R*(G+) module structure on X**
* a diagram
chase verifies they agree.
Lemma 11.4. The action of R*(G+) on X* = [G+; X]G*= [R ^ G+; X]R;G*implied by*
* the
K"unneth theorem and the action of G on X agrees with the action of [R ^ G+; R *
*^ G+]R;G*_
by composition. *
* |__|
For cohomology we want to have universal coefficient theorem
(UCT)
[A ^ Z; X]* = Hom [A;A]*([A; A ^ Z]*; [A; X]*) = Hom [A;S]*([A; Z]*; [A*
*; X]*);
where the second equality is (KT1) and a change of rings isomorphism. This is e*
*nough to
get a spectral sequence with
E1 = Hom [A;A]*(B([A; A]*; [A; A]*; [A; S]*); [A; X]*);
it therefore takes the form
E*;*2= H*(End (A)*; [A; X]*) =) [A(S); X]* = (X^A)*:
Convergence is again conditional in the sense of Boardman. In the equivariant *
*case this
spectral sequence becomes
H*(R*(G+); X*) =) X*G(EG+):
When it comes to Tate cohomology we need to ask about splicing, both in topol*
*ogy and
algebra. In topology we have
- DA2 - DA - A - A2 - : : :
where the splicing is via
DA Dt0-DS = S -t0 A:
To obtain a spectral sequence we may either apply [A; . ^ X]* and use the first*
* avatar
tA(X) = F (B(A; A; S); X) ^ "B(A; A; S), or apply [A ^ .; X]* and use the secon*
*d avatar
tA(X) = F (B"(A; A; S); X ^ B(A; A; S)). The first will make the relation to h*
*omology
clearer and the second will make the relation to cohomology clearer, but since *
*the resolution
is self-dual, the two are essentially equivalent, and we only discuss the first*
*. Convergence is
again covered by the relevant argument (10.8) from [23].
AXIOMATIC TATE COHOMOLOGY 19
In view of the equality [A ^ A; S]* = [A; DA]*, we conclude that the E2-term *
*agrees with
the homological one in positive filtration degrees, and with the cohomological *
*one (shifted
by one degree) in filtration degrees -2. More precisely, if E* = End (A)* = [A*
*; A]*, and
"E*= [A ^ A; S]* = [A; DA]*, we have the algebraic resolution
- "E2*- "E*- E* - E2* - : :::
Using this particular resolution to define the E2-term we have a spectral seque*
*nce
H^*([A; A]*; [A; X]*) =) tA(X)*:
This is again conditionally convergent in the sense of Boardman. In the equivar*
*iant case this
spectral sequence becomes
H^*(R*(G+); X*) =) tR (X)G*:
For a more satisfactory account of the algebra, we assume E* is projective as*
* an l*-module.
Next, we express this in terms of a single type of resolution. Thus, by (KT1),
E"*= [A; DA]* = [A; A ^ S-o]* = [A; A]* [A;S]*[A; S-o]* = E* l**
where * = [A; S-o]*. On the other hand, by (UCT),
"E*= [A ^ A; S]* = Hom ([A; A]*; [A; S]*) = Hom (E*; l*);
so we conclude
Hom (E*; l*) = E* *:
Next, we assume that the first K"unneth theorem (KT1) applies also to So ^ S-o,*
* so that
* is invertible and hence projective projective. Then we can specify a projecti*
*ve complete
resolution by taking a resolution of l*, dualizing and splicing. This is essen*
*tially the Tate
cohomology of a Frobenius algebra, but with the twisting module inserted.
Proof of 11.1 We work in the category of R-modules and take X = M ^ Y in the fi*
*rst and_
third case, and X = F (Y; M) in the second. *
* |__|
12. Calculations by commutative algebra.
In this section we discuss the more subtle question of when the local cohomol*
*ogy theo-
rem holds for A so that there is a calculation by commutative algebra in the se*
*nse of 10.5.
This requires better behaviour of the cohomology theory concerned, and consider*
*ably more
substance to the proof. We discuss two somewhat different methods for proving *
*a local
cohomology theorem. In a sense, the second method is a partial unravelling of *
*the first:
cellular constructions are replaced by multiple complexes. Both methods apply *
*to the lo-
cal cohomology theorem for finite groups, but beyond this they have different d*
*omains of
relevance.
We discuss the more sophisticated example first [17, 24], because the formal *
*machinery
highlights the structure of the proof whilst hiding the technical difficulties.
Indeed if R is a highly structured commutative ring G-spectrum we have seen i*
*n Section
6 that, by construction, for any finitely generated ideal I in RG*we have spect*
*ral sequences
H^*I(RG*; MG*) =) tI(M)G*
H*I(RG*; MG*) =) I(M)G*
20 J. P. C. GREENLEES
HI*(RG*; MG*) =) (M^I)G*:
What we really want is to obtain similar spectral sequences for calculating tA(*
*M)G*; A(M)G*
and (M^A)G*for the class A generated by G+ using the ideal I = ker(R*G-! R*). W*
*e assume
here that R*Gis Noetherian, so that I is finitely generated, but see [25] for a*
*n example where
this is not true. To obtain the desired spectral sequences we need to check tha*
*t each of the
constructions with A is equivalent to the corresponding construction on R-modul*
*es for the
ideal I. In fact, we need only check that
I(R) ' I(R ^ EG+ ) ' {G+}R = R ^ EG+ :
The second equivalence is a formal consequence of the fact that I restricts to *
*zero non-
equivariantly. The first equivalence contains the real work: it is equivalent t*
*o the statement
that I(R ^ "EG) ' *, where E"G is the unreduced suspension of EG. If G acts fr*
*eely
on a product of spheres (for example if it is a p-group) this follows from the *
*existence of
Euler classes (obviously elements of I) and the construction of "EG in terms of*
* representation
spheres [17]. To extend this to other groups some sort of transfer argument is *
*necessary (see
[20, 27] for examples).
This construction will give means of calculation whenever we have two suitabl*
*y related
smashing localizations. For example we may consider the localization (.)[D-1] w*
*ith acyclics
D and the localization (.)[I-1] for an ideal I in the coefficient ring S*. The *
*requirements are
then
o I(S) ^ S[D-1] ' * and
o S[I-1] ^ D (S) ' *
Together, these give the equivalence
I(S) ' D (S);
and hence the corresponding equivalences of other localization and colocalizati*
*on functors.
If we suppose D is generated by the single augmented object A as before, and de*
*fine I =
ker([S; S]* -! [A; S]*), then the second requirement is again a formal conseque*
*nce of the
fact that elements of I restrict to zero. One expects the first requirement to*
* use special
properties of the context, as it did in the equivariant case.
We now turn to the second method for proving a local cohomology theorem, and *
*work with
the group cohomology of a finite group in the derived category D(kG) as in Sect*
*ion 7. We are
considering the relationship with the derived category of the graded ring R = H*
**(G; k) and
the ideal I of positive dimensional elements as in [19]. We may view these resu*
*lts as relating
various completions and Tate cohomologies in the two categories by spectral seq*
*uences. We
take this opportunity to extend the results of [19] to unbounded complexes. Sin*
*ce H*(G; M)
is already I-complete if M is bounded below, the second spectral sequence is on*
*ly of interest
in the unbounded case.
Theorem 12.1. Suppose G is a finite group, and M is a complex of kG-modules, *
*and let I
denote the ideal of positive codegree elements of the graded ring H*(G). There *
*are spectral
sequences
H*I(H*(G; M)) =) H*(G; M);
HI*(H*(G); H*(G; M)) =) H*(G; M^{kG})
and
^HI*(H*(G); H*(G; M)) =) H*(G; t{kG}(M)):
AXIOMATIC TATE COHOMOLOGY 21
We explain the changes that need to be made to the arguments of [19] to cover*
* the
unbounded case. The idea is to use the algebraic spheres of Benson-Carlson [3] *
*to construct
algebraic analogues of tori B on which G acts freely. Thus if k is of character*
*istic p > 0 and
G is of p-rank r, then B is a complex graded over Zr concentrated in a box with*
* the lowest
corner at the origin. From B we may construct a multigraded projective resoluti*
*on T of k
by stacking boxes in the region with all coordinates 0. More generally, if oe *
* {1; 2; : :;:r}
we may form T [oe] by stacking boxes to fill the region defined by requiring ni*
* 0 if i 6= oe.
Thus T [;] = T , and T {1; 2; : :;:r} fills all of Zr. We then form a dual Kosz*
*ul complex Lo of
multigraded chain complexes:
0 1
M M M
Lo = @ T [oe] -! T [oe] -! . .-.! T [oe]A:
|oe|=r |oe|=r-1 |oe|=0
The idea of the proof is to consider the double complex
Hom (Lo; M)G:
If one takes homology in the Koszul direction first one obtains Hom (T !; M)G, *
*where T !is
the complex concentrated in negative multidegrees; provided M is bounded below *
*this is
isomorphic to the rth suspension of T G M, and this has homology H*(G; M) by de*
*finition.
If M is not bounded below, the first complex has infinite products where the se*
*cond has
infinite sums.
Now
Hom (T [oe]; M) = Hom (lim -k|oe|T; M)
k
and, provided M is bounded below, this is equal to lim Hom (-k|oe|T; M) because*
* the limit
! k
is achieved in each total degree. Thus, if one takes homology in the kG-resolut*
*ion direction
first, one obtains the stable Koszul complex of H*(G; M). To avoid the require*
*ment of
boundedness we simply use the double complex
lim Hom (L[ s]; M)
! s
from the start, where L[ s] is the quotient of L by the subcomplex of boxes whi*
*ch are at
least s boxes below zero in some coordinate.
As is familiar from the case of commutative algebra, to construct the second *
*spectral
sequence we should consider the double complex
holim L[ s] G M
s
If we take homology in the Koszul direction first we obtain
holimT ![ s] G M = holimHom ((T ![ s])*; M)G
s s
= Hom (holim (T ![ s])*; M)G
! s
' Hom (rT; M)G
On the other hand, if we take homology in the kG-resolution degree we obtain a *
*homo-
topy inverse limit of complexes, each term of which is a suspension of H*(G; M)*
*, and
so that the differentials are products of the chosen generators of I. By defin*
*ition this is
holim UKs(x) H*(G)H*(G; M), and by definition, its homology is the local homolo*
*gy in
s
the statement.
22 J. P. C. GREENLEES
For the Tate spectral sequence we combine these methods to form the double co*
*mplex
holim Hom (L[ t]; holim T ![ s] G M):
! t s
13. Gorenstein localizations.
In this final section we point out that a local cohomology theorem in the sen*
*se of Section
12 implies a strong duality theorem in certain cases. The idea is that the loca*
*l cohomology
theorem gives a covariant equivalence of two objects that are quite generally c*
*ontravari-
antly equivalent using a universal coefficient theorem. The composite contrava*
*riant self-
equivalence is the duality.
To motivate the name, we recall that under mild hypotheses, a commutative com*
*plete
local k-algebra (R; I; k) of dimension d is Gorenstein if H*I(R) = HdI(R) (i.e.*
* R is Cohen-
Macaulay) and in addition
R = Hom R(HdI(R); R_) = Hom R=I(HdI(R); R=I)
where M_ = Hom R=I(M; R=I). We want to consider a homotopy level version of the*
* Goren-
stein condition on the unit object S in a stable homotopy_category C. _To_make *
*sense of
this we need (i) a second_stable homotopy category C with unit object S, (ii) a*
* `restric-
tion' functor r : C -! C, thought of as_a forgetful map, and required to be lax*
* monoidal,
and (iii) an `inflation' functor i : C -! C, splitting the forgetful_map,_and a*
*lso required
to be lax monoidal._ This gives sense to the statement that S is an S-algebra. *
* Now take
I = ker(S* -! S*), and say that S is homotopically I-Gorenstein if it is comple*
*te and there
is an equivalence __
S ' F (I(S); S_) = F__S(I(S); S)
__ __
where X_ = F__S(X; S), and where the S-function object is an additional_piece o*
*f structure.
To see that the homotopical Gorenstein statement has force, suppose S* is a f*
*ield. We
then remark that if S is homotopically Gorenstein and S* is Cohen-Macaulay then*
* S* is
Gorenstein. Indeed, if S* is Cohen-Macaulay of dimension d, then ss*(I(S)) = Hd*
*I(S*) from
the spectral sequence of Section 6, and in the presence of a universal coeffici*
*ent theorem we
find a spectral sequence __
Ext *;*_S*(HdI(S*); S*) ) S*:
__
If in addition S* is a field, this states that S* is the dual of HdI(S*) and so*
* S* is Gorenstein.
See [21] for further investigation.
The principal example of the present formal setup is when C_is the category o*
*f equivariant
R-modules for a highly structured split ring spectrum R and C is the category o*
*f non-
equivariant R-modules. The relevant functors have been constructed by Elmendor*
*f and
May [15, 37].
In this case the augmentation is right adjoint to product with A = G+, and th*
*ere is
additional structure since the completion X^A= F (EG+; X) and the torsion A(X) *
*= EG+ ^
X both have homotopy described in nonequivariant terms. It is pointed out in th*
*e appendix
to [21] that when there is a local cohomology theorem, R^A= F (EG+;_R) is homot*
*opically
Gorenstein. Recalling that S = R in the equivariant category and S = R in the*
* non-
equivariant category, we may summarize the proof as follows
__ ^ __ __ ^
F__S(I(S^A); S) ' F__S(A(SA); S) ' F__S(A(S); S) ' FS(A(S); S) = SA:
AXIOMATIC TATE COHOMOLOGY 23
The first equivalence is the local cohomology theorem, the second is 2.3 and th*
*e third is the
split condition.
We remark that one expects a twisting in the application of the universal coe*
*fficient
theorem when G is not a finite group. For example with a compact Lie group G, *
*the
twisting is given by the adjoint bundle in the Adams isomorphism. Similarly the*
* twisting is
given by the dualizing module for a virtual Poincare duality group as in [8]. T*
*he twisting
is built from the invertible object So in the sense that it is essentially So o*
*n each copy of
A used to build A(S). Thus, when G is a compact Lie group of dimension d, the a*
*djoint
bundle is a trivial d-dimensional bundle over any cell Sn ^ G+.
The existence and implications of the homotopy Gorentstein duality statement *
*has been
investigated for the cohomology of groups [19, 9, 8, 21], and for coefficients *
*of equivariant
cohomology theories in [17, 25, 26, 27]. We remark here that there is a precis*
*e formal
similarity with Gross-Hopkins duality [28, 30, 41], which states that the Brown*
*-Comenetz
dual IMnX of the monochromatic section MnX is a twisted suspension of LK(n)DX f*
*or
suitable finite spectra X, where MnX is the fibre of LnX -! Ln-1X. Hopkins and *
*Ravenel
have proved there are spectral sequences for calculating the homotopy of MnX an*
*d LK(n)X
whose E2-terms are the cohomology of the profinite group = SnoGal(Fpn=Fp) with*
* suitable
coefficients, where Sn is the Morava stabilizer group. Furthermore, is a p-adi*
*c Lie group;
if it is p-torsion free it is a Poincare duality group, and in general its coho*
*mology has a local
cohomology theorem as in [8] (the proof in the discrete case carries over to th*
*e profinite case
in the category of Symonds-Weigel [42]). The local cohomology theorem at the E2*
* level is
the precise counterpart of the Gross-Hopkins duality between the spectra.
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AXIOMATIC TATE COHOMOLOGY 25
School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK.
E-mail address: j.greenlees@sheffield.ac.uk