TATE COHOMOLOGY IN AXIOMATIC STABLE HOMOTOPY THEORY.

J. P. C. GREENLEES

Contents

1. Axiomatic Tate cohomology in a stable homotopy category. 1 2. Finite localizations. 3 3. The category of G-spectra 4 4. The derived category of a commutative ring. 5 5. The category of modules over a highly structured ring 8 6. The derived category of kG 8 7. The chromatic category. 9 8. Splittings of the Tate construction. 10 9. Calculation by comparison. 11 10. Calculations by associative algebra. 12 11. Calculations by commutative algebra. 15 References 17

1. Axiomatic Tate cohomology in a stable homotopy category.

The purpose of the present note is to show how the axiomatic approach to Tate cohomology can be implemented in the axiomatic stable homotopy theory of Hovey-Palmieri-Strickland [22]. There are few new results, and much of the work consists simply of collecting known results in a single language and a single framework. On the other hand the very effortlessness of the process is an effective advertisement for the language, and a call for further investigation of other instances. Nonetheless Theorem 10.1 is new, and Theorem 11.1 extends results of [14].

A stable homotopy category [22, 1.1.4] is a triangulated category 𝒞 with arbitrary coproducts, and so that all cohomology theories are representable. It is also required to have a compatible symmetric monoidal structure and a set 𝒢 of strongly dualizable objects generating all of 𝒞 using triangles, coproducts and retracts. If in addition the objects of 𝒢 are small, the stable homotopy category is said to be algebraic.

Recall that a Bousfield localization L is determined by its class 𝒟 of acyclics as follows: Y is L-local if and only if [D, Y]_{∗} = 0 for all D in 𝒟, and that a map X ⟶ Y is the Bousfield localization 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 localizing ideal (i.e. it is closed under completing triangles, sums and smashing with an arbitrary object). A localization is said to be smashing if the natural map X ∧ LS ⟶ LX is an equivalence for all X. It is equivalent to require either that L commutes with arbitrary sums, or that the class L𝒞 of L-local objects is a localizing ideal.

We shall define a Tate construction associated to any smashing localization.

Notation 1.1. (General context)

• 𝒞: a stable homotopy category

• 𝒢: a set of generators for 𝒞

• 𝒟: the localizing ideal of acyclics for a smashing Bousfield localization (·)[𝒟^{−1}].

• 𝒞[𝒟^{−1}] the localizing ideal of [𝒟^{−1}]-local objects.

The notation L_{𝒟} is often used for (·)[𝒟^{−1}]; the present notation better reflects the character of a smashing localization. It reflects the fact that we should think of X[𝒟^{−1}] as a localization away from 𝒟, and corresponds to that in [18]. We therefore use the following notation for the fibre of this localization:

Γ_{𝒟} (X) ⟶ X ⟶ X[𝒟^{−1}].

We can use this to define an associated completion.

Lemma 1.2. The natural transformation X ⟶ F (Γ_{𝒟} (S), X) is Bousfield completion 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 (Γ_{𝒟} (S), X)]_{∗} = 0. However E = E[𝒟^{−1}] is a S[𝒟^{−1}]-module spectrum and hence E ∧Γ_{𝒟} (S) is a retract of E ∧S[𝒟^{−1}]∧Γ_{𝒟} (S); since [𝒟^{−1}] is idempotent and smashing, S[𝒟^{−1}] ∧ Γ_{𝒟} (S) ≃ ∗.

Secondly we must show that the fibre, F (S[𝒟^{−1}], X) is [𝒟^{−1}]-local. However if D lies in 𝒟 then D ∧ S[𝒟^{−1}] ≃ D[𝒟^{−1}] ≃ ∗.

We write X_{𝒟}^{∧} := F (Γ_{𝒟} (S), X) for this Bousfield completion, and also introduce the following notation for its fibre:

Δ_{𝒟} (X) ⟶ X ⟶ X_{𝒟}^{∧}.

We now define the 𝒟-Tate construction by

t_{𝒟} (X) = X_{𝒟}^{∧}[𝒟^{−1}].

This gives the diagram

Γ_{𝒟} (X) ⟶ X ⟶ X[𝒟^{−1}]
↓  ↓  ↓
Γ_{𝒟} (X_{𝒟}^{∧}) ⟶ X_{𝒟}^{∧} ⟶ X_{𝒟}^{∧}[𝒟^{−1}]

Lemma 1.3. The map Γ_{𝒟} (X) ⟶ Γ_{𝒟} (X_{𝒟}^{∧}) is an equivalence.

Proof. We need only remark that Γ_{𝒟} (Δ_{𝒟} (X)) ≃ ∗; however by definition Δ_{𝒟} (X) lies in the class of [𝒟^{−1}]-local objects.

Corollary 1.4. (Hasse Principle) There is a homotopy pullback square

X ⟶ X[𝒟^{−1}]
↓  ↓
X_{𝒟}^{∧} ⟶ t_{𝒟} (X)

Corollary 1.5. (Warwick Duality) There is an equivalence

t_{𝒟} (X) = X_{𝒟}^{∧}[𝒟^{−1}] ≃ Δ_{𝒟} (ΣΓ_{𝒟} (X)).

Proof. This is a composite of three equivalences.

X_{𝒟}^{∧}[𝒟^{−1}] ⟵ Δ_{𝒟} (X_{𝒟}^{∧}[𝒟^{−1}]) ⟶ Δ_{𝒟} (ΣΓ_{𝒟} (X_{𝒟}^{∧})) ⟶ Δ_{𝒟} (ΣΓ_{𝒟} (X))

The first is an equivalence since (·)[𝒟^{−1}]_{𝒟}^{∧} ≃ ∗ (the class ℰ of acyclics for (·)_{𝒜}^{∧} consists of [𝒟^{−1}]-local objects) so that X_{𝒟}^{∧}[𝒟^{−1}]_{𝒟}^{∧} ≃ ∗. The second is an equivalence since Δ_{𝒟} (X_{𝒟}^{∧}) ≃ ∗ (defining property of Δ_{𝒟} ((·)) together with idempotence of (·)_{𝒟}^{∧}). The third is an equivalence since Γ_{𝒟} (Δ_{𝒟} ((·))) ≃ ∗ by 1.3 so that Δ_{𝒟} (ΣΓ_{𝒟} (Δ_{𝒟} (X))) ≃ ∗.

The usefulness arises since the definition shows that

t_{𝒟} (X) = F (Γ_{𝒟} (S), X)[𝒟^{−1}] ≃ F (S[𝒟^{−1}], ΣΓ_{𝒟} (X)).

The second form gives [T, t_{𝒟} (X)]_{∗} = [T ∧ S[𝒟^{−1}], ΣΓ_{𝒟} (X)]_{∗}.

Remark 1.6. One might be tempted to define a Tate construction for 𝒟 and ℰ as L_{𝒟} L_{ℰ}X. However, if one wants Warwick Duality, one requires (i) L_{ℰ}L_{𝒟} X ≃ ∗, so that ℰ ⊇ L_{𝒟} 𝒞 and (ii) C_{𝒟} C_{ℰ}X ≃ ∗, so that C_{ℰ}X is L_{𝒟} -local, and ℰ ⊆ L_{𝒟} 𝒞. Thus we require ℰ = L_{𝒟} 𝒞, and this must be a localizing ideal. Thus L_{𝒟} must be smashing, and determines ℰ.

This completes the discussion of formalities. In the rest of the paper we want to discuss a number of examples from this point of view, and show how comparisons between the examples give rise to means of calculation.

2. Finite localizations.

This example is explicit in Section 3.3 of [22], and especially Theorem 3.3.5. It generalizes the finite localization of Mahowald–Sadofsky and Miller [24, 25].

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 algebraic, we must suppose in addition that 𝒜 is essentially small, consists of strongly dualizable 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 generated by 𝒜: this is the smallest thick subcategory closed under arbitrary sums and smashing with arbitrary elements of 𝒞.

Context 2.1. (for a finite localization)

• 𝒞: a stable homotopy category

• 𝒢: a set of generators for 𝒞

• 𝒯: a set of small objects of 𝒞

• 𝒜 = 𝒢-ideal(𝒯)

• 𝒟 = locid(𝒯) = locid(𝒜)

In these circumstances, we write write 𝒜 in place of 𝒟 in the notation, so that t_{𝒜} (X) = t_{𝒟} (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 finite localization and the notation L_{𝒜}^{f} is used in [22]. The construction follows easily from the discussion below. The associated functor (·)_{𝒜}^{∧} whose acyclics are the objects X[𝒜^{−1}] is denoted by L_{𝒜} in [22].

There is a convenient lemma for showing a set of elements in a localizing subcategory is a generating set. It would be more traditional to view it as a convergence theorem for a projective resolution in the sense of Adams [1].

Proposition 2.2. If 𝒯 ⊆ 𝒟 is a set of objects then 𝒟 = loc(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 ∈ 𝒟, S ∈ 𝒢 and any x ∈ [S, X]_{∗} there is a map t : Σ^{n}S ⟶ X with x ∈ im(t_{∗} : [S, Σ^{n}T]_{∗} ⟶ [S, X]_{∗})

Proof. We need to prove that if X is an arbitrary object of 𝒟, we may form X from copies of elements of 𝒯 using sums and cofibre sequences. We give the proof assuming that Condition (ii) holds; the proof when Condition (i) holds is similar except that [S, ·]_{∗} for S ∈ 𝒢 is replaced by [T, ·]_{∗} for T ∈ 𝒯.

By hypothesis we may form a projective resolution in the sense of Adams:

X  X_{0} ⟶^{i_{0}} X_{1} ⟶^{i_{1}} X_{2} ⟶^{i_{2}} …
↑^{t_{0}}  ↑^{t_{1}}  ↑^{t_{2}}
T_{0}  T_{1}  T_{2}

Thus each T_{i} is a sum of suspensions of elements of 𝒯, each t_{i} is surjective in homotopy and X_{i+1} is formed as the mapping cone of t_{i} : T_{i} ⟶ X_{i}. Note that X_{∞} = tel_{n}X_{n} has trivial [S, ·]_{∗} for all S ∈ 𝒢 since i_{s} is zero in [S, ·]_{∗} by construction; thus X_{∞} ≃ ∗. Defining X^{i} as the mapping fibre of X ⟶ X_{i} we find that X^{i} is constructed from sums of suspensions of T by a finite number of cofibre sequences. Passing to direct limits, we obtain a cofibre sequence X^{∞} ⟶ X ⟶ X_{∞}, so that X^{∞} ≃ X.

Remark 2.3. Note that the argument essentially gives the construction of a finite localization. Take a set 𝒯 of small generators of the 𝒢-ideal 𝒜 and the localizing ideal 𝒟, and ensure it is closed under duality. Ensure t_{i} is surjective in [T, ·]_{∗} for each i. Then the triangle X^{∞} ⟶ X ⟶ X_{∞} has X^{∞} in 𝒟 by construction, and [T, X^{∞}]_{∗} = 0 for all T in 𝒯.

3. The category of G-spectra

In this section we consider the category 𝒞 = G-spectra of G-spectra for a compact Lie group G, and localizations associated to a family ℱ of subgroups. We recover the constructions of [17]; indeed these constructions motivated investigation of its other manifestations.

Thus we suppose ℱ is closed under conjugation and passage to subgroups, and we let 𝒯 = {G/H_{+} | H ∈ ℱ}. Thus 𝒜 is the class of retracts of finite ℱ-spectra, and 𝒟 is the class of all ℱ-spectra. We recall that the class of ℱ-spectra, can be described in three ways (i) the class of spectra equivalent to spectra formed from spheres G/H_{+} ∧ S^{n} with H ∈ ℱ, (ii) as the class of G-spectra X so that the natural map Eℱ_{+} ∧ X ⟶^{≃} X is an equivalence and (iii) as the category of G-spectra X so that the geometric fixed point spectra Φ^{H}X are non-equivariantly contractible for H ∈ ℱ.

We leave the reader to check that in this case Diagram 1 is the diagram

Eℱ_{+} ∧ X  ⟶  X  ⟶ Ẽℱ ∧ X
≃↓  ↓  ↓
F (Eℱ_{+}, Eℱ_{+} ∧ X) ⟶ F (Eℱ_{+}, X) ⟶ t_{ℱ}(X)

ie Diagram C of [17].

It is obvious that the skeletal filtration gives rise to spectral sequences for calculating the homotopy groups of these spectra based on group cohomology [17], and we discuss this in more abstract terms in Section 10. More interesting is that for well behaved cohomology theories (such as those which are Noetherian, complex orientable and highly structured), one may prove a local cohomology theorem in which case the homotopy groups may be calculated by commutative algebra [13]. We discuss this in more abstract terms in Section 11. The formal framework for such spectral sequences is discussed in Section 9

4. The derived category of a commutative ring.

In this we consider the category 𝒞 = D(R) for a commutative ring R, and localizations associated to an ideal I of R. In particular we obtain a new approach to the results of [13] associated to an ideal of R. In particular we obtain an improved perspective on the role of finiteness conditions.

We wish to consider the class of acyclics for a localization, and there are several candidates for this. The most natural is the class

𝒟 = {M | supp(H_{∗}(M)) ⊆ V (I)},

but we should also consider

𝒟′ = {M | every element of H_{∗}(M) is I-power torsion}.

It is easy to see that 𝒟′ ⊆ 𝒟.

Lemma 4.1. If I is finitely generated then 𝒟′ = 𝒟, but this is not true in general.

Proof. Suppose M is a module with support in V (I), and x ∈ M has annihilator J. Since R/J has support V (J), we see that V (J) ⊆ V (I) so that \sqrt{J} ⊇ \sqrt{I} ⊇ I. If I is finitely generated, some power of I lies in J.

To give an example where equality fails we need only display an ideal J so that no power of \sqrt{J} lies in J, since then we may take I = \sqrt{J} and M = R/J. For instance if R is polynomial on a countably infinite number of generators, x_{1}, x_{2}, x_{3}, … over a field and J = (x_{1}, x_{2}^{2}, x_{3}^{3}, …) we find that \sqrt{J} is the maximal ideal (x_{1}, x_{2}, x_{3}, …) no power of which lies in J.

Lemma 4.2. The classes 𝒟 and 𝒟′ are localizing ideals.

Proof. The class 𝒟 is thick, by exactness of localization, and localizing since localization commutes with sums. To see it is also an ideal we use the fact that localization commutes with tensor products, and that the left derived tensor product is an operation on the derived category.

We leave the reader to verify that 𝒟′ is also a localizing ideal.

It is useful to have a specific generator for 𝒟 as a localizing ideal. Perhaps the most natural candidate for a generator of 𝒟 is A/I, but this only generates 𝒟′. When I is finitely generated 𝒟 = 𝒟′, so A/I does give a generator, but there are other candidates which are often more convenient.

Lemma 4.3. The class 𝒟′ is generated as a localizing ideal by A/I.

Proof. We prove by induction on k that all objects X with I^{k}H_{∗}(X) = 0 lie in locid(A/I). When k = 1 we have A/I ∈ locid(A/I) by hypothesis. For k ≥ 2 we also have A/I^{k} ∈ locid(A/I) since the short exact sequence I^{k−1}/I^{k} ⟶ A/I^{k−1} ⟶ A/I^{k} gives a triangle, with the first and second term already known to be in the ideal. Now we apply the argument of Proposition 2.2, using 𝒯 = {A/I^{k}}. Indeed, if x ∈ H_{∗}(X) is I^{k}-torsion, the map x : A ⟶ X factors through A/I^{k} in the derived category.

Furthermore, any submodule of a sum of suspensions of A/I^{k} is I^{k} torsion, so the process may be repeated.

By induction this shows that A/I^{k} lies in the localizing ideal for all k, and now we apply the argument of Proposition 2.2 with 𝒯 = {A/I^{k} | k ≥ 1}.

Warning 4.4. Note that if A/I does not have a resolution by finitely generated projectives, it need not be small.

If I is finitely generated, say I = (x_{1}, …, x_{n}) there are also other candidates for generators of 𝒟. Indeed, we may define the unstable Koszul complex for the sequence x_{1}^{d}, x_{2}^{d}, …, x_{n}^{d} by

UK_{d}^{•}(x) = (R ⟶^{x_{1}^{d}} R) ⊗ ⋯ ⊗ (R ⟶^{x_{n}^{d}} R).

However, note that UK^{•}(x) is small. We may also define the stable Koszul complex

K^{•}(I) = (R ⟶ R[1/x_{1}]) ⊗ ⋯ ⊗ (R ⟶ R[1/x_{n}])

and define Č^{•}(I) by the cofibre sequence Č^{•}(I) ⟶ R ⟶ K^{•}(I). It is not hard to check that both K^{•}(I) and Č^{•}(I) are independent of the generators of the ideal, up to quasiisomorphism.

We want to show that Koszul complexes provide generators of the localizing ideal 𝒟.

Proposition 4.5. If I = (x_{1}, x_{2}, …, x_{n}) is finitely generated, the class 𝒟 is generated as a localizing ideal by A/I, by K^{•}(I) and by UK^{•}(x).

Proof. We have seen that 𝒟 is generated by A/I. Since UK^{•}(x) is small, it is easy to apply Proposition 2.2 to see that it also generates 𝒟. To see that K^{•}(I) is also a generator, we apply the following elementary lemma.

Lemma 4.6. (i) The unstable Koszul complex UK^{•}(x) lies in the thick subcategory generated by the stable Koszul complex K^{•}(I). (ii)The stable Koszul complex K^{•}(I) lies in the localizing subcategory generated by the stable Koszul complex UK^{•}(x).

Proof. (i) Consider the self-map of the cofibre sequence K^{•}(x) ⟶ R ⟶ R[1/x] given by multiplication by x. Since x is an equivalence of R[1/x], the octahedral axiom shows there is a fibre sequence UK^{•}(x) ⟶ K^{•}(x) ⟶^{x} K^{•}(x). We may tensor this argument with any object X, so that we find a fibre sequence

K^{•}(x_{1}, …, x_{n−1}) ⊗ UK^{•}(x) ⟶ K^{•}(I) ⟶^{x_{n}} K^{•}(I)

Repeating this, we see UK^{•}(x) lies in the thick subcategory generated by K^{•}(I).

(ii) First we show that UK_{k}^{•}(x) can be constructed from UK^{•}(x). We proceed by induction, using the square

\begin{matrix}R & ⟶^{x^{k−1}} & R \\ ^{1}↓ & & ^{x}↓ \\ R & ⟶^{x^{k}} & R \end{matrix}

to construct a cofibre sequence UK_{k−1}^{•}(x) ⟶ UK_{k}^{•}(x) ⟶ UK^{•}(x). This remains true after tensoring with any object, so we may increase the power of each x_{i} in turn.

Now we see that the map R ⟶ R[1/x] is the direct limit of the maps R ⟶^{x^{k}} R, and hence K^{•}(x) is equivalent to the homotopy direct limit of the terms UK^{•}(x^{k}). Tensoring these together and using the fact that holim_{→}_{n} commutes with tensor products, we find

K^{•}(I) = holim_{→}_{k} UK_{k}^{•}(x).

Now that we understand the localizing ideal 𝒟 quite well when I is finitely generated, it is not hard to construct the relevant localizations and completions.

Lemma 4.7. If I is finitely generated,

(i) M[𝒜^{−1}] = M ⊗ Č^{•}(I)

(i) M_{𝒜}^{∧} = Hom(K^{•}(I), M)

Proof. (i) To see that M ⊗ Č^{•}(I) is local, we need only check it admits no morphism from A/I except zero. However Č^{•}(I) admits a finite filtration with subquotients A[1/x] for x ∈ I so it suffices to show [UK^{•}(x), A[1/x]]_{∗} = 0. This is clear since x is nilpotent on UK^{•}(x) and an isomorphism on A[1/x]. To see that M ⟶ M ⊗ Č^{•}(I) is a 𝒜-equivalence we need only verify that the M ⊗ K^{•}(I) can be built from UK^{•}(x). However M can be built from A, and we saw in 4.6 that K^{•}(I) can be built from UK^{•}(x).

(ii) Since K^{•}(I) ⊗ Č^{•}(I) ≃ ∗ it is clear that Hom(K^{•}(I), M) admits no maps from 𝒞[𝒜^{−1}] except 0. One then needs to check that the fibre, Hom(Č^{•}(I), M) lies in 𝒞[𝒜^{−1}]; for this it is convenient to note that 𝒞[𝒜^{−1}] is the class of objects X with X ⊗ UK^{•}(x) ≃ 0. Now the result follows since UK^{•}(I) is self-dual up to suspension, and Č^{•}(I) ⊗ UK^{•}(x) ≃ 0.

It follows that if in addition R is Noetherian or good in the sense of [16], then the homology of M_{𝒜}^{∧} is calculated by the left derived functors of completion. In particular, if M is of finite type, M_{𝒜}^{∧} = M_{I}^{∧}. Furthermore the Tate cohomology coincides with that of [13]. As pointed out in [13], Warwick duality is a generalization of the isomorphism ℤ_{p}^{∧}[1/p] = lim_{←}(ℤ/p^{∞}, p).

If I is not finitely generated, the construction it is interesting to ask for an analogue. Of course we need to decide whether to seek a localization with 𝒟 or with 𝒟′ as the class of acyclics. If A/I is small, the construction by finite localization gives a localization with 𝒟′ as acyclics, and it is interesting to ask for algebraic calculations. However, if A/I does not admit a resolution by finitely generated projectives, it is not small, and we do not have a construction.

5. 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 [10], and we allow the equivariant case. Such objects are essentially equivalent to E_{∞} ring 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 structured module spectra over R and consider localizations and completions associated to a finitely generate ideal I of the coefficient ring R_{∗}

Much of the discussion of the previous section applies in the present case, and was presented in [18]. Thus we may form the stable and unstable Koszul modules by using cofibre sequences and smash products. Thus for example, UK^{•}(x) is the fibre of R ⟶^{x} R; we avoid the common notation Σ^{−1}R/x for fear of confusion with the previous section. Now UK^{•}(x) = UK^{•}(x_{1}) ∧_{R} UK^{•}(x_{2}) ∧_{R} … ∧_{R} UK^{•}(x_{n}); similarly K^{•}(x) is the fibre of R ⟶ R[1/x], and K^{•}(I) is formed using smash products. We take 𝒜 to be the class of retracts of finite R-modules M so that M_{∗} has support in V (I). This is generated by 𝒯 = {UK^{•}(x)}, and generates the localizing ideal of all M so that M_{∗} has support in V (I).

The statement and proof of Lemma 4.7 apply without change. Because the construction comes with an evident filtration we may obtain spectral sequences by taking homotopy, and the E_{1} term is a chain complex representing the analogous constructions in the previous section. We thus obtain spectral sequences

Ȟ_{I}^{∗}(R_{∗}; M_{∗}) ⟹ M[I^{−1}]_{∗}
Ĥ_{I}^{∗}(R_{∗}; M_{∗}) ⟹ t_{I}(M)_{∗}
H_{I}^{∗}(R_{∗}; M_{∗}) ⟹ Γ_{I}(M)_{∗}
H_{∗}^{I}(R_{∗}; M_{∗}) ⟹ (M_{I}^{∧})_{∗}

and so forth.

6. The derived category of kG

The relationship of this example to the category of G-spectra is analogous to the relationship between the category of highly structured modules over R and the derived category D(R_{∗}). For a finite group G and a field k we take 𝒞 = D(kG), and 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.

It is proved in [22, 9.6] that the localization M ⟶ M[𝒜^{−1}] is obtained by tensoring with a Tate resolution. Since any Tate resolution admits a finite filtration with subquotients R[1/x] as in [14], it follows that every object of 𝒞 with bounded cohomology is already complete. Thus we find that if M has bounded cohomology, t_{𝒜} (M) = M[𝒜^{−1}] = M ⊗ t and so the Tate construction defined by localization agrees with Tate homology in the classical sense.

There are at least three other obvious examples to consider here, but some work is needed to give them substance.

Variation 6.1. Consider a family ℱ of subgroups, and the category 𝒜_{ℱ} of finite complexes of modules with source in ℱ. The case ℱ = {1} is that given above. The 𝒢-ideal 𝒜_{ℱ} is generated by 𝒯_{ℱ} = {kG/H | H ∈ ℱ}. It is then appropriate to use Amitsur-Dress ℱcohomology [9]. Presumably one again obtain a local cohomology theorem 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 ordinary group cohomology and the ideal I_{ℱ} of cohomology elements restricting to zero in the cohomology of H for all H ∈ ℱ.

Variation 6.2. We choose a block β of kG and take 𝒜_{β} to be the category of finite complexes of projectives in β.

Variation 6.3. We may consider the stable module category 𝒞 = StMod(kG), which is proved in [22, 9.6.4] to be a localization of 𝒟(kG). It would then be interesting to investigate complexity quotients in the sense of Carlson-Donovan-Wheeler [7, 8, 4, 5] from the present point of view.

7. The chromatic category.

Another important example arises in the approach to stable homotopy theory through bordism. For background and further information see [26]. Thus we work in the stable homotopy category of spectra in the sense of algebraic topology, and we choose a prime p > 0. For 0 ≤ n ≤ ∞ we shall need the spectrum E(n) representing Johnson-Wilson cohomology theory and the Morava K-theory spectrum K(n). For 0 < n < ∞ these have coefficient rings E(n)_{∗} = ℤ_{(p)}[v_{1}, v_{2}, …, v_{n−1}, v_{n}, v_{n}^{−1}], and K(n)_{∗} = ℤ/p[v_{n}, v_{n}^{−1}]. The cases n = 0, ∞ are somewhat exceptional: by convention, for n = 0 we have E(0) = K(0) = Hℚ and for n = ∞ we have E(∞) = BP and K(∞) = Hℤ/p. Recall that a spectrum is of type n if K(i)_{∗}(X) = 0 for i < n and K(n)_{∗}(X) ≠ 0.

Bousfield localization L_{n} with respect to E(n) is the localization whose acyclics are the spectra X with E(n) ∧ X ≃ ∗. A well known theorem of Hopkins-Ravenel states that L_{n} is smashing. The usual notation is C_{n}X ⟶ X ⟶ L_{n}X. The completion X_{𝒟}^{∧} = F (C_{n}S, X) is more mysterious. When n = 0 the completion is profinite completion F (Sℚ/ℤ, X).

n E(n) K(n)  F (n)  L_{n−1}  L_{K(n)}
0 Hℚ Hℚ  S_{(p)}^{0}  *  rationalization
1 K_{(p)} K/p  S^{0}/p^{k}  invert p  L_{K(1)}
2 Ell Ell/(p, v_{1}) S^{0}/(p^{k}, v_{1}^{l}) L_{1} = “(·)[(p, v_{1})^{−1}]” L_{K(2)}
⋮ ⋮  ⋮  ⋮  ⋮  ⋮
∞ BP H𝔽_{p}  *  p-localization  p-adic completion

Following [22], let us consider a slightly simpler example. Let 𝒞 be the E(n)-local category, and 𝒜 the thick subcategory generated by L_{n}F (n) for a finite type n spectrum F (n). It follows that X[𝒜^{−1}] = L_{n−1}X and X_{𝒜}^{∧} = L_{K(n)}X. On the E(n)-local category, the fibre of X ⟶ L_{n−1}X is usually known as the nth monochromatic piece, so we have Γ_{𝒜} (X) = M_{n}X. The fibre of K(n) completion is sometimes known as C_{K(n)}, but we simply write Δ_{𝒜} (X) = Δ_{K(n)}(X).

Corollary 7.1. (Warwick Duality) If X is E(n)-local then

L_{n−1}L_{K(n)}X ≃ ΣΔ_{K(n)}(M_{n}X).

We note that if n = 0 this states M_{0}X is rational, and if n = 1 it states that the cofibre of M_{1}X ⟶ L_{K(1)}M_{1}X is the rationalization of L_{K(1)}X.

If we take 𝒞 to be the entire category of p-local spectra there are two related examples. Indeed, we may still consider the smashing localization L_{n−1} = L_{E(n−1)}, but it does not seem so easy to describe the associated completion. In particular it is not equal to L_{K(n)} (indeed, although L_{n−1}S^{0} 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/v_{n}], and the smashing localization L_{n−1}^{f} which 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) [25] and it is therefore smashing, and we may again consider the associated completion, which is again different from L_{K(n)} for similar reasons. There is a natural transformation L_{n}^{f} ⟶ L_{n}, which is not generally a natural isomorphism for n ≥ 2 [23].

8. Splittings of the Tate construction.

It seems worth mentioning two rather different splittings, even though we have no general results about them. First, continuing with the notation of the previous section, note that t_{𝒜} (X) = L_{n−1}L_{K(n)}X is the subject of Hopkins’s chromatic splitting conjecture [21, 4.2]. When X = S^{0} (albeit not in the E(n)-local category 𝒞) this is conjectured to split into 2^{n} pieces; the cofibre sequence

L_{n−1}S_{p}^{0} ⟶ L_{n−1}L_{K(n)}S^{0} ⟶ ΣF (L_{n−1}S^{0}, L_{n}S_{p}^{0})

splits, and F (L_{n−1}S^{0}, L_{n}S_{p}^{0}) splits as a wedge of 2^{n} − 1 suitable localizations of spheres. To obtain the cofibre sequence, suppose given localizing ideals 𝒟 and ℰ, with L_{𝒟} S ∈ ℰ this is enough to ensure a cofibre sequence F (L_{𝒟} S, L_{𝒟∩ℰ} X) ⟶ L_{𝒟∩ℰ} X ⟶ L_{ℰ}X [21, 4.1] to which we may apply L_{𝒟} .For the case relevant to the Tate construction 𝒟 has a smashing localization and ℰ = 𝒞[𝒟^{−1}], so that 𝒟 ∩ ℰ = {∗}. We then have a more direct construction: apply F (·, X)[𝒟^{−1}] to the cofibre sequence Σ^{−1}S[𝒟^{−1}] ⟶ Γ_{𝒟} (S) ⟶ S to obtain

X[𝒟^{−1}] ⟶ t_{𝒟} (X) ⟶ ΣF (S[𝒟^{−1}], X)

since F (S[𝒟^{−1}], X) is already [𝒟^{−1}]-local.

Secondly, there is the Benson-Carlson [3] and Benson-Greenlees [6] splitting of the Tate cohomology ring Ĥ^{∗}(G). These state that when group cohomology H^{∗}(G) has a regular sequence of length 2, the Tate cohomology of a finite group admits a splitting

Ĥ^{∗}(G) = H^{∗}(G) ⊕ Σ^{1}H_{∗}(G)

(where the suspension is homological) both as a module over H^{∗}(G) and as a module over the Steenrod algebra. This is to be contrasted with the case when G has periodic cohomology when Ĥ^{∗}(G) = H^{∗}(G)[1/x] where x is the periodicity element. The analogous statement would be that the standard cofibre sequence

X_{𝒟}^{∧} ⟶ t_{𝒟} (X) ⟶ ΣΓ_{𝒟} (X)

splits in some sense when X = S. The regularity hypotheses must include the requirement that the norm map Σ^{−1}Γ_{𝒟} (X) ⟶ X_{𝒟}^{∧} is zero, and probably also that X_{𝒟}^{∧} is of depth at least 2. This holds rather trivially for the case of Tate cohomology in commutative algebra [13], and is to be contrasted with the case when the ring is Noetherian and of Krull dimension 1, where the rationality theorem [13, 7.1] holds: Ĥ^{∗}(R) = S^{−1}(R_{I}^{∧}) where S is the set of regular elements of R.

9. Calculation by comparison.

We intend to discuss two methods of calculation. Perhaps we may introduce the discussion by explaining 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 spectral sequence

H_{I}^{∗}(H^{∗}(G; M)) ⟹ H_{∗}(G; M)

of [14], where I is the ideal of positive codegree elements of H^{∗}(G).

The generalization we have in mind concerns finite localizations in the case that 𝒜 = {A} consists of a single commutative cogroup object A. We also require that A is strongly dualizable and self-dual up to an invertible element, in the sense that DA ≃ A ∧ S^{−τ} for some object S^{−τ} admitting a smash inverse S^{−τ} ∧ S^{τ} ≃ S.

Example 9.1. (i) The motivating example has 𝒞 = 𝒟(kG) and A = kG. Note that we have an augmentation kG ⟶ k, and a diagonal map kG ⟶ kG ⊗ kG. Furthermore kG is self-dual. (ii) Alternatively we may take 𝒞 to be a category of 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_{+} ⟶ S^{0}, and a diagonal map G_{+} ⟶ G_{+} ∧ G_{+}. We also have the duality statement DG_{+} ≃ Σ^{−d}G_{+} where d = dim(G). This helps explain the notation S^{−τ}, which is chosen since, in the geometric context, Atiyah duality shows τ corresponds 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 = Σ^{−d}F (n) (or A = R ∧ Σ^{−d}F (n)) where F (n) = S^{0}/(p^{i_{0}}, v_{1}^{i_{1}}, v_{2}^{i_{2}}, …, v_{n−1}^{i_{n−1}}) for suitable i_{0}, i_{1}, …, i_{n−1} and d = dim(F (n)). Collapse onto the top cell gives an augmentation Σ^{−d}F (n) ⟶ S^{0}. In favourable cases we have the duality statement DF (n) ≃ Σ^{−d}F (n), and F (n) may be taken to be a commutative ring spectrum, and the dual to the product gives a diagonal map Σ^{−d}F (n) ⟶ Σ^{−d}F (n)∧Σ^{−d}F (n).

We need to consider the graded commutative ring k_{∗} = [S, S]_{∗}, where S is the unit in 𝒞, and two augmented k_{∗}-algebras. Firstly there is the commutative augmented k_{∗}-algebra L_{∗} = [A, S]_{∗}, and secondly there is the augmented k_{∗}-algebra End(A)_{∗}, which need not be commutative.

In our examples these are as follows.

Example 9.2. (i) When A = kG we have k_{∗} = L_{∗} = k and End(kG)_{∗} = kG. (ii) When A = R ∧ G_{+} we have k_{∗} = R_{∗}^{G}, L_{∗} = R_{∗} and End(R ∧ G_{+})_{∗} = R_{∗}(G_{+}). (iii) When A = Σ^{−d}F (n) we have k_{∗} = R_{∗}, L_{∗} = R_{∗}(F (n)) and End(R ∧ Σ^{−d}F (n))_{∗} = R_{∗}(F (F (n), F (n))).

Given this 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.

We may hope the spectral sequences reversing the first functor of the following form:

H_{∗}(End(A)_{∗}; [A, X]_{∗}) ⟹ (Γ_{𝒜} (X))_{∗}
H^{∗}(End(A)_{∗}; [A, X]_{∗}) ⟹ (X_{𝒜}^{∧})_{∗}

and

Ĥ^{∗}(End(A)_{∗}; [A, X]_{∗}) ⟹ t_{𝒜} (X)_{∗}.

A construction in some cases is given in Section 10.

We may hope those reversing the second are of the following form. 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:

H_{I}^{∗}(X_{∗}) ⟹ (Γ_{𝒜} (X))_{∗}
H_{∗}^{I}(X_{∗}) ⟹ (X_{𝒜}^{∧})_{∗}

and

Ĥ_{∗}^{I}(X_{∗}) ⟹ t_{𝒜} (X)_{∗}.

A construction in some cases is given in Section 11.

The content should be clearer when we give some examples. It is not surprising that to prove the existence of either set of spectral sequences we have to assume the existence of additional structure beyond that present in the stable homotopy category.

10. Calculations by associative algebra.

The point of this section is to generalize the Atiyah-Hirzebruch Tate spectral sequence of [11]:

Ĥ^{∗}(G; E^{∗}(X)) ⟹ t(E)_{G}^{∗}(X)

for finite groups G, or in other words to prove the expectations suggested in the previous section under suitable circumstances. It does not work entirely in a stable homotopy category, but rather relies on the existence of an unstable model category from which the stable homotopy category is formed.

The aim is to find ways to calculate Γ_{A}(X)_{∗}, (X_{A}^{∧})_{∗} and t_{A}(X)_{∗} in terms of the augmented [A, S]_{∗}-algebra [A, A]_{∗}, and the module [A, X]_{∗}. In view of the notational conflict we remind the reader that in the context of G-spectra, homotopy [S, ·]_{∗} is equivariant homotopy and [A, ·]_{∗} is non-equivariant homotopy. The present discussion covers a number of new examples. The generalization is cruder than that of [17], but more general. The discussion of convergence in [17, Appendix B] applies without change.

To avoid the appearance of empty generalization, we state an unequivocal theorem.

Theorem 10.1. If G is a compact Lie group, R is a ring spectrum, and M an R-module up to homotopy, and if we have universal coefficient theorems

M_{∗}(G_{+} ∧ T ∧ Z) = R_{∗}(G_{+}) ⊗_{R_{∗}} M_{∗}(T ∧ Z)

and

[G_{+} ∧ T, M ∧ Z]_{∗}^{G} = [T, M ∧ Z]_{∗} = Hom_{R_{∗}}(R_{∗}(T), M_{∗}(Z))

where T = G_{+}^{∧s} then there are spectral sequences

H_{∗}(R_{∗}(G_{+}); M_{∗}(Z)) ⟹ M_{∗}^{G}(EG_{+} ∧ Z);

H_{∗}(R_{∗}(G_{+}); M^{∗}(Z)) ⟹ M_{G}^{∗}(EG_{+} ∧ Z);

Ĥ_{∗}(R_{∗}(G_{+}), M_{∗}(Z)) ⟹ t(M)_{∗}^{G}(Z);

where the cohomology is that of a Frobenius algebra R_{∗}(G_{+}).

We return to the proof of the theorem at the end of the section. The rest of the discussion is conducted in general terms.

We want to construct a resolution for X = S as in 2.2 to X = S, or rather the dual resolution

∗  (S)^{0} ⟶^{j^{0}} (S)^{1} ⟶^{j^{1}} (S)^{2} ⟶^{j^{2}} …
↓  ↓^{q_{0}}  ↓^{q_{1}}
∗  T_{0}  T_{1}

This is associated with the sequence

T_{0} ⟵^{δ_{1}} Σ^{−1}T_{1} ⟵^{δ_{2}} Σ^{−2}T_{2} ⟵^{δ_{3}} Σ^{−3}T_{3} ⟵

We want to apply simplicial methods, so we suppose there is an underlying model category, from which the stable homotopy category is formed by stabilization and passage to homotopy, and we require that A is a strict cogroup object. For the example of the theorem, we take the category of G-spectra for a compact Lie group G, and A = G. We may then form the homogeneous bar construction as a simplicial object, and take its geometric realization

Γ_{A}S = S^{∞} = B(A, A, S).

This ensures T_{i} = Σ^{i}A^{∧(i+1)}. One may imagine cases where this can still be done if A is only a cogroup up to all higher homotopy, but we do not pursue this. By smashing with X we obtain a resolution for arbitrary X, Thus we may define

t_{A}(X) = F (B(A, A, S), X) ∧ B̃(A, A, S),

where B̃(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 obtain

[A, T_{0}]_{∗} ⟵^{(δ_{1})_{∗}} [A, Σ^{−1}T_{1}]_{∗} ⟵^{(δ_{2})_{∗}} [A, Σ^{−2}T_{2}]_{∗} ⟵^{(δ_{3})_{∗}} [A, Σ^{−3}T_{3}]_{∗} ⟵ .

In the equivariant context we have [A, Σ^{−i}T_{i}]_{∗} = R_{∗}(G_{+}^{∧i+1}). To ensure it is a resolution, we assume there is a Künneth theorem

[A, A ∧ Z] = [A, A]_{∗} ⊗_{[A,S]_{∗}} [A, Z]_{∗}

for relevant Z. In the equivariant context this is a Künneth theorem for non-equivariant R_{∗}(·). This ensures that the simplicial contraction in geometry is converted to one in algebra and the bar filtration spectral sequence for calculating [A, B(A, A, S)]_{∗} has its E^{1} term given by the algebraic bar construction B([A, A]_{∗}, [A, A]_{∗}, [A, X]_{∗}). To calculate Γ_{A}(X)_{∗}, we need the second Künneth theorem

[S, Z]_{∗} = [A, S]_{∗} ⊗_{[A,A]_{∗}} [A, Z]_{∗}

for relevant Z. In the equivariant context, this states that the change of groups isomorphism [G_{+}, Z]_{∗}^{G} = [S, Z]_{∗} is reflected in algebra. This is enough to give a spectral sequence with E^{1} term [A, S]_{∗} ⊗_{[A,A]_{∗}} B([A, A]_{∗}, [A, A]_{∗}, [A, X]_{∗}) and hence it takes the form

E_{∗,∗}^{2} = H_{∗}(End(A)_{∗}; [A, X]_{∗}) ⟹ Γ_{A}(X)_{∗}.

It is easy to see this spectral sequence is conditionally convergent in the sense of Boardman. The homology in the E_{2}-term is defined to be the homology of the bar construction, 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_{+}); X_{∗}) ⟹ X_{∗}^{G}(EG_{+}).

Note that we have two possible definitions of the R_{∗}(G_{+}) module structure on X_{∗} a diagram chase verifies they agree.

Lemma 10.2. The action of R_{∗}(G_{+}) on X_{∗} = [G_{+}, X]_{∗}^{G} = [R ∧ G_{+}, X]_{∗}^{R,G} implied by the Künneth 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

[A ∧ Z, X]_{∗} = Hom_{[A,A]_{∗}}([A, A ∧ Z]_{∗}, [A, X]_{∗}) = Hom_{[A,S]_{∗}}([A, Z]_{∗}, [A, X]_{∗}).

This is enough to get a spectral sequence with E_{1} term Hom_{[A,A]_{∗}}(B([A, A]_{∗}, [A, A]_{∗}, [A, S]_{∗}), [A, X]_{∗}) and hence it takes the form

E_{2}^{∗,∗} = H^{∗}(End(A)_{∗}; [A, X]_{∗}) ⟹ [Γ_{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 topology and algebra. In topology we have

⟵ DA^{2} ⟵ DA ⟵ A ⟵ A^{2} ⟵ …

where the splicing is via

DA ⟵^{Dt_{0}} DS = S ⟵^{t_{0}} A.

To obtain a spectral sequence we may either apply [A, · ∧ X]_{∗} and use the first avatar t_{A}(X) = F (B(A, A, S), X) ∧ B̃(A, A, S), or apply [A ∧ ·, X]_{∗} and use the second avatar t_{A}(X) = F (B̃(A, A, S), ΣX ∧ B(A, A, S)). The first will make the relation to homology 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 [17].

In view of the equality [A ∧ A, S]_{∗} = [A, DA]_{∗}, we conclude that the E^{2}-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 ℰ̃_{∗} = [A ∧ A, S]^{∗} = [A, DA]_{∗}, we have the algebraic resolution

⟵ ℰ̃_{∗}^{⊗2} ⟵ ℰ̃_{∗} ⟵ ℰ_{∗} ⟵ ℰ_{∗}^{⊗2} ⟵ ….

Using this particular resolution to define the E_{2}-term we have a spectral sequence

Ĥ_{∗}([A, A]_{∗}, [A, X]_{∗}) ⟹ t_{A}(X)_{∗}.

This is again conditionally convergent in the sense of Boardman. In the equivariant case this spectral sequence becomes

Ĥ_{∗}(R_{∗}(G_{+}), X_{∗}) ⟹ t^{R}(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, since

ℰ̃_{∗} = [A, DA]_{∗} = [A, A ∧ S^{−τ}]_{∗} = [A, A]_{∗} ⊗ [A, S^{−τ}]_{∗} = ℰ_{∗} ⊗ I

where I = [A, S^{−τ}]_{∗}. On the other hand

ℰ̃_{∗} = [A ∧ A, S]_{∗} = Hom([A, A]_{∗}, [A, S]_{∗}) = Hom(ℰ_{∗}, L_{∗}),

so we conclude

Hom(ℰ_{∗}, L_{∗}) = ℰ_{∗} ⊗ I.

Since I is invertible, it is projective. Then we can specify a projective complete resolution by taking a resolution of L_{∗}, dualizing and splicing. This is essentially the Tate cohomology of a Frobenius algebra, but with the twisting module inserted. Proof of 10.1 We work in the category of R-modules and take X = M ∧ Z in the first and third case, and X = F (Z, M) in the second.

11. Calculations by commutative algebra.

In this section we discuss the more subtle process of calculation by commutative algebra. This requires better behaviour of the cohomology theory concerned, and considerably more substance to the proof. On the other hand we discuss two somewhat different methods of proof: they both apply to the local cohomology theorem for finite groups, but beyond this they have different domains of relevance.

We discuss the more sophisticated example first, because the formal machinery available makes it easy to present it so as to highlight the formal structure of the proof whilst hiding the technical difficulties.

Indeed if R is a highly structured ring G-spectrum we have seen that, by construction, for any finitely generated ideal I in R_{∗}^{G} we have spectral sequences

Ĥ_{I}^{∗}(R_{∗}^{G}; M_{∗}^{G}) ⟹ t_{I}(M)_{∗}^{G}
H_{I}^{∗}(R_{∗}^{G}; M_{∗}^{G}) ⟹ Γ_{I}(M)_{∗}^{G}
H_{∗}^{I}(R_{∗}^{G}; M_{∗}^{G}) ⟹ (M_{I}^{∧})_{∗}^{G}.

What we want is to obtain spectral sequences with these E_{2} terms for the particular ideal I = ker(R_{G}^{∗} ⟶ R^{∗}), but converging to the corresponding constructions for the class A = {G_{+}}. (We assume here that R_{G}^{∗} is Noetherian, so that I is finitely generated, but see [19] for an example where this is not true.) In other words we need only check that

Γ_{I}(R) ≃ Γ_{I}(R ∧ EG_{+}) ≃ Γ_{{G_{+}}}R = R ∧ EG_{+}.

The second equivalence is a formal consequence of the fact that I restricts to zero nonequivariantly. The first equivalence contains the real work: it is equivalent to the statement that Γ_{I}(R∧ẼG) ≃ ∗. If G acts freely 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 ẼG in terms of representation spheres [12]. To extend this to other groups some sort of transfer argument is necessary (see [15, 20] for examples).

Evidently this construction will give means of calculation whenever we have two suitably related smashing localizations. For example we may continue to suppose one is defined using I-torsion objects, defined by means of an augmentation to commutative algebra. The requirements are then

• Γ_{I}(S) ∧ S[𝒟^{−1}] ≃ ∗ and

• S[I^{−1}] ∧ Γ_{𝒟} (S) ≃ ∗ Together, these give the equivalence

Γ_{I}(S) ≃ Γ_{𝒟} (S),

and hence the corresponding equivalences of other localization and colocalization functors. If we suppose 𝒟 is generated by the single object A as before, and define I = ker([S, S]_{∗} ⟶ [A, S]_{∗}), then the second is again a formal consequence of the fact that elements of I restrict to zero.

For the second method, let us return to the group cohomology of a finite group, and work in the derived category D(kG). 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 [14]. We may view these results as relating various completions and Tate cohomologies in the two categories by spectral sequences. We take this opportunity to extend the results of [14] to unbounded complexes. Of course H_{∗}(G; M) is already I-complete if M is bounded below, so the second spectral sequence is only of interest in the unbounded case.

Theorem 11.1. Suppose G is a finite group G, and a complex M of kG-modules. Let I denote the ideal of positive codegree elements of the graded ring H^{∗}(G). There are spectral sequences as follows where I is the ideal of positive dimensional elements.

H_{I}^{∗}(H^{∗}(G; M)) ⟹ H_{∗}(G; M).
H_{∗}^{I}(H^{∗}(G); H_{∗}(G; M)) ⟹ H^{∗}(G; M_{𝒜}^{∧})
Ĥ_{∗}^{I}(H^{∗}(G); H_{∗}(G; M)) ⟹ H^{∗}(G; t_{𝒜} (M))

We explain the changes that need to be made to the arguments of [14] to cover the unbounded case. The idea is to use the algebraic spheres of Benson-Carlson [2] to construct algebraic analogues of tori B on which G acts freely. Thus if k is of characteristic p > 0 and G is of p-rank r, then B is a complex graded over ℤ^{r} concentrated in a box with the lowest corner at the origin. From B we may construct a multigraded projective resolution R of k by stacking boxes in the orthant with all coordinates ≥ 0. More generally, if σ ⊆ {1, 2, …, r} we may form R[σ] by stacking boxes to fill the region defined by requiring n_{i} ≥ 0 if i ≠ σ. Thus R[∅] = R, and R{1, 2, …, r} fills all of ℤ^{r}. We then form a dual Koszul complex L_{•} of multigraded chain complexes:

L_{•} = (⨁_{|σ|=r} R[σ] ⟶ ⨁_{|σ|=r−1} R[σ] ⟶ ⋯ ⟶ ⨁_{|σ|=0} R[σ]) .

The idea of the proof is to consider the double complex

Hom(L_{•}, M)^{G}.

If one takes homology in the Koszul direction first one obtains Hom(R^{!}, M)^{G}, where R^{!} is the complex concentrated in negative multidegrees; provided M is bounded below this is isomorphic to the rth suspension of R⊗_{G} M, and this has homology H_{∗}(G; M) by definition. If M is not bounded below, the first complex has infinite products where the second has infinite sums.

Now

Hom(R[σ], M) = Hom(lim_{←} Σ^{−k|σ|}R, M)

and, provided M is bounded below, this is equal to lim_{→}_{k} Hom(Σ^{−k|σ|}R, M) because the limit is achieved in each total degree. Thus, if one takes homology in the kG-resolution direction first, one obtains the stable Koszul complex of H^{∗}(G; M). To avoid the requirement of boundedness we simply use the double complex

lim_{→}_{s} Hom(L[≥ s], M)

from the start, where L[≥ s] is the quotient of L by the subcomplex of boxes which 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_{←}_{s} L[≥ s] ⊗_{G} M

If we take homology in the Koszul direction first we obtain

holim_{←}_{s} R^{!}[≥ s] ⊗_{G} M = holim_{←}_{s} Hom((R^{!}[≥ s])^{∗}, M)^{G}
= Hom(holim_{→}_{s}(R^{!}[≥ s])^{∗}, M)^{G}
≃ Hom(Σ^{r}R, M)^{G}

On the other hand, if we take homology in the kG-resolution degree we obtain a homotopy 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 definition this is holim_{←}_{s} UK_{s}(x) ⊗_{H^{∗}(G)} H_{∗}(G; M), and by definition, its homology is the local homology in the statement.

For the Tate spectral sequence we combine these methods to form the double complex

holim_{→}_{t} Hom(L[≤ t], holim_{←}_{s} R^{!}[≥ s] ⊗_{G} M).

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School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH, UK.

E-mail address: j.greenlees@sheffield.ac.uk