TATE COHOMOLOGY OF THEORIES WITH ONE DIMENSIONAL COEFFICIENT RING.

J.P.C.GREENLEES AND HAL SADOFSKY

Abstract. For a finite group G we calculate the E-Tate cohomology t(E)_{G}^{∗} and the E-homology E_{∗}(BG_{+}) as functors of the augmented commutative ring E^{∗}(BG_{+}) when E^{∗}(·) is a complex oriented, v_{n} periodic cohomology theory with one dimensional graded coefficient ring E^{∗}.

1. Introduction.

Let G be a finite group and p be a prime. Our aim is to calculate the coefficient ring of the G-equivariant Tate theory for a complex oriented v_{n}-periodic theory E. In [7] we showed that if E is mod p Morava K-theory (whose coefficient ring K(n)^{∗} = 𝔽_{p}[v_{n}, v_{n}^{−1}] is a graded field) then the associated Tate theory is trivial and the representing spectrum is equivariantly contractible

t(K(n)) ≃ ∗.

The basis of this is Ravenel’s result [11] that K(n)^{∗}(BG_{+}) is a finitely generated module over the graded field K(n)^{∗}, and the simple form of the answer relies on the fact that K(n)^{∗} has Krull dimension 0 as a graded ring. In the present paper we consider the case when the coefficient ring E^{∗} is of Krull dimension 1, thinking particularly of the case with coefficients

ℤ_{(p)}[v_{1}, v_{2}, …, v_{n}, v_{n}^{−1}]/I_{n,i}

or a suitable completion, where I_{n,i} = (p, v_{1}, …, v̂_{i}, …, v_{n−1}) is generated by all but one of the elements v_{0} = p, v_{1}, …, v_{n−1}. For instance if n = 1 and i = 0 this gives the principal Adams summand of p-local K-theory. For all these theories t(E)_{G}^{∗} is nonzero and we give an exact calculation in (6.1).

We begin by explaining the importance of Krull dimension, and the place of the Tate theory in calculating E_{∗}(BG_{+}) from E^{∗}(BG_{+}). This also directs attention to the important features we shall need of E to be able to give definite answers.

2. Philosophy.

Let E be a complex oriented v_{n}-periodic theory and consider the ring R^{∗} = E^{∗}(BG_{+}). Using the maps induced by ∗ ⟶ BG ⟶ ∗ we see that R^{∗} is an augmented E^{∗}-algebra, and we shall have to consider the augmentation ideal J^{∗} = ker(R^{∗} ⟶ E^{∗}). Because E^{∗} is periodic we may work with the degree 0 part of all rings and apply standard commutative algebra. To begin with, note that J^{0} generates an ideal in R^{∗} with the same radical as J^{∗}. Indeed if x ∈ J^{∗} then x^{2(p^{n}−1)} lies in a degree containing a power of the unit v_{n}, and is thus a unit multiple of an element of J^{0}. From now on we use the abbreviations R = R^{0} = E^{0}(BG_{+}) and J = J^{0}.

The first author thanks the Nuffield Foundation for making possible his visit to Johns Hopkins where this work was begun, and Johns Hopkins for its hospitality.

The following method only works easily if J is finitely generated (at least up to taking radicals). In the present context we are content to assume rather more.

Finiteness Hypothesis 2.1. The ring E^{0} is Noetherian and the map E^{0} ⟶ R makes R into a finite E^{0}-module.

Under this hypothesis R is also Noetherian, and hence J is finitely generated. Furthermore, E^{0} and R have the same Krull dimension, which we shall denote by d. We may therefore apply the standard constructions of commutative algebra to R. In particular, for any R-module M we may form the local cohomology groups H_{J}^{∗}(M) in the sense of Grothendieck [9]; see [3] for a summary of definitions in a convenient form. The essential property from our point of view is Grothendieck’s vanishing theorem, which states that H_{J}^{i}(M) = 0 if i > d. The importance of this in equivariant topology comes from the local cohomology theorem which states there is a finite spectral sequence for calculating the homology of classifying spaces:

E_{2}^{∗,∗} = H_{J}^{∗,∗}(R^{∗}) ⟹ E_{∗}(BG_{+}). (2-2)

Some constructions give an E_{2} term with J replaced by a subideal J′ ⊆ J, but these ideals all have the same radical and hence [2, p. 297] the same local cohomology groups. For p-groups 2-2 was proved in [2], but we shall give an outline below. For general finite groups it may be proved using the machinery of highly structured ring spectra [1], although we do not need this case for our main result.

This gives us an interest in calculating the local cohomology groups of R^{∗}. These are bigraded since R^{∗} is graded, but because J is in degree 0 the local cohomology can be calculated one degree at a time in the sense that H_{J}^{s,t}(R^{∗}) = H_{J}^{s}(R^{t}). We simplify our notation accordingly.

It is common practice to calculate local cohomology groups using the Čech cohomology groups Ȟ_{J}^{∗}(R^{∗}) together with the exact sequence

0 ⟶ H_{J}^{0}(R^{∗}) ⟶ R^{∗} ⟶ Ȟ_{J}^{0}(R^{∗}) ⟶ H_{J}^{1}(R^{∗}) ⟶ 0 (2-3)

and the isomorphisms

H_{J}^{i}(R^{∗}) ≅ Ȟ_{J}^{i−1}(R^{∗}) for i ≥ 2. (2-4)

The advantages are twofold. Firstly the Čech groups vanish in degree d as well as above it (i.e. Ȟ_{J}^{i}(R^{∗}) = 0 for i ≥ d), and secondly if

Γ_{J}R^{∗} = {x ∈ R^{∗} | J^{N}x = 0 for sufficiently large N}

denotes the J-power torsion ideal it is easy to see that Ȟ_{J}^{∗}(R^{∗}) = Ȟ_{J}^{∗}(R^{∗}/Γ_{J}R^{∗}), so that one may simplify calculations by factoring out the ideal Γ_{J}R^{∗} from the start.

The topological counterpart of Čech cohomology is the Tate theory for E in the sense of [5], indeed there is a spectral sequence

E_{2}^{∗,∗} = Ȟ_{J}^{∗}(R^{∗}) ⟹ t(E)_{G}^{∗} (2-5)

which collapses more often than (2-2) because of the vanishing theorem Ȟ_{J}^{d}(R^{∗}) = 0. Again we shall construct the spectral sequence in case G is a p-group in 4.2 below, and this will suffice for our purposes, but it can be constructed for arbitrary finite groups by the methods of [1]. The topological counterpart of (2-3) is the norm sequence

⋯ ⟶ E_{∗}(BG_{+}) ⟶^{Norm} E^{∗}(BG_{+}) ⟶ t(E)_{G}^{∗} ⟶ · · · . (2-6)

Thus our strategy in calculating E_{∗}(BG_{+}) is to use the local Tate spectral sequence (2-5) to calculate t(E)_{G}^{∗} and then the norm sequence (2-6).

We illustrate this in some cases with low Krull dimension below.

• dim k = 0: This is the case dealt with in [7] by using a topological argument. In the spirit of the present article we give a more algebraic proof using the local Tate cohomology of [3] in Section 8 below. In any case t(E)_{G}^{∗} = 0, and the norm gives an isomorphism E_{∗}(BG_{+}) ≅ E^{∗}(BG_{+}) = R^{∗}.

• dim k = 1: This is the principal case we deal with in the present note. Here the Čech cohomology is concentrated in degree zero and we have

t(E)_{G}^{∗} = Ȟ_{J}^{0}(R^{∗}).

To explain the resulting calculation of E_{∗}(BG_{+}) we need to be clear about grading. We regard all objects as graded in the cohomological fashion. Homology theories are given cohomological grading by taking the nth grading (E_{∗}(X))^{n} = E_{−n}(X) in the standard way. All suspensions refer to this cohomological grading so that (ΣR^{∗})^{n} = R^{n−1}. With these conventions we have an exact sequence

0 ⟶ cok {ΣR^{∗} ⟶ Ȟ_{J}^{0}(ΣR^{∗})} ⟶ E_{∗}(BG_{+}) ⟶ Γ_{J}R^{∗} ⟶ 0.

In particular if R^{∗} is concentrated in even degrees

E_{even}(BG_{+}) = Γ_{J}R^{∗} and E_{odd}(BG_{+}) = cok {ΣR^{∗} ⟶ Ȟ_{J}^{0}(ΣR^{∗})} .

• dim k = 2: If R^{∗} is concentrated in even degrees then

t(E)_{G}^{even} = Ȟ_{J}^{0}(R^{∗}) and t(E)_{G}^{odd} = Ȟ_{J}^{1}(ΣR^{∗}),

and the norm sequence gives

0 ⟶ Ȟ_{J}^{1}(R^{∗}) ⟶ E_{even}(BG_{+}) ⟶ Γ_{J}R^{∗} ⟶ 0

and

0 ⟶ ΣR^{∗}/Γ_{J}R^{∗} ⟶ Ȟ_{J}^{0}(ΣR^{∗}) ⟶ E_{odd}(BG_{+}) ⟶ 0.

• dim k = 3: If R^{∗} is concentrated in even degrees the spectral sequence (2-5) collapses to give

t(E)_{G}^{odd} = Ȟ_{J}^{1}(ΣR^{∗})

and an exact sequence

0 ⟶ Ȟ_{J}^{2}(Σ^{2}R^{∗}) ⟶ t(E)_{G}^{even} ⟶ Ȟ_{J}^{0}(R^{∗}) ⟶ 0.

In this case there is a possible differential d_{3} : H_{J}^{0}(R^{∗}) ⟶ H_{J}^{3}(Σ^{2}R^{∗}) in the local cohomology spectral sequence, and the image of the norm map E_{∗}(BG_{+}) ⟶ E^{∗}(BG_{+}) = R^{∗} is ker(d_{3}), which may therefore be a proper submodule of Γ_{J}R^{∗}.

• dim k = 4: If R^{∗} is concentrated in even degrees there is only one possible differential in the spectral sequence so that we obtain an exact sequence

0 ⟶ Ȟ_{J}^{2}(Σ^{2}R^{∗}) ⟶ t(E)_{G}^{even} ⟶ Ȟ_{J}^{0}(R^{∗}) ⟶^{d_{3}} Ȟ_{J}^{3}(Σ^{2}R^{∗}) ⟶ t(E)_{G}^{odd} ⟶ Ȟ_{J}^{1}(R^{∗}) ⟶ 0.

It would be interesting to understand this first differential.

3. Finiteness

In this section we establish a basic finiteness result for suitably complete theories. The finiteness does not hold for arbitrary v_{n}-periodic theories as we shall see in Section 7, but the finiteness for complete theories is sufficient to let us proceed more generally.

Let E_{n} be the completion of the Johnson-Wilson theory E(n) at (p, v_{1}, …, v_{n−1}) made 2-periodic. If n = 1 we obtain p-adic K-theory, and in general E_{n} has coefficients

E_{n}^{∗} = ℤ_{p}^{∧}[[u_{1}, u_{2}, …, u_{n−1}]][u, u^{−1}],

where u_{i} is of degree 0 and u is of degree −2. These generators are related to the v_{i} by v_{i} = u^{1−p^{i}}u_{i} for 1 ≤ i ≤ n − 1 and v_{n} = u^{1−p^{n}}. The advantages of using this theory will be apparent when we give a very simple example with the uncompleted theory in Section 7 below.

Theorem 3.1. For any finite group G, E_{n}^{∗}(BG_{+}) is a finite module over E_{n}^{∗}.

Notice that E(1)^{∗}(BG_{+}) is a sum of E(1)^{∗} = Z_{(p)}[v_{1}, v_{1}^{−1}] and a number of copies of E(1)_{p}^{∗}ˆ = Z_{p}^{∧}[v_{1}, v_{1}^{−1}] by the Atiyah-Segal completion theorem, so it is not finite over E(1)^{∗} unless G is trivial. This shows some form of completion is necessary for such a finiteness theorem to hold.

We shall prove 3.1 by induction on the number of spectral lines in the coefficients (i.e. the number of polynomial generators v_{j} or u_{j}), starting from Ravenel’s finiteness theorem [11].

Let E be a ring spectrum and v : E → E a map of spectra induced by an element v ∈ E^{∗}. In our applications this element will be a power of v_{i}. Let E/v^{k} denote the cofibre of v^{k} : E ⟶ E, and let

Ê = holim_{⟵}_{k} E/v^{k}.

Finally let K = E/v.

Proposition 3.2. Suppose that K^{∗} is Noetherian and that π_{r}(E/v^{k}) is a compact topological space for all r, k, such that the maps π_{k}(E/v^{k}) ⟶ π_{k}(E/v^{k−1}) are continuous. If X is a locally finite, bounded below spectrum and K^{∗}(X) is finitely generated as a module over K^{∗}, then Ê^{∗}(X) is finitely generated as a module over Ê^{∗}.

Proof. The proof is based on the Bockstein spectral sequence. We form the tower

… → E/v^{3} → E/v^{2} → E/v = K → ∗.

Of course the cofibres at each stage are just copies of K. If we map X into the tower, we get in the usual way a Bockstein spectral sequence

E_{1} = K^{∗}(X) ⊗_{K^{∗}} K^{∗}[v] ⇒ Ê^{∗}(X).

Our compactness assumptions guarantee convergence. First note that Ê/v^{k} ≃ E/v^{k} for all k. It follows that [T, Ê/v^{k}] is compact for any finite spectrum T. Now we calculate

[X, Ê] = lim_{⟵}_{n} [X^{(n)}, Ê] = lim_{⟵}_{n} [X^{(n)}, holim_{⟵}_{k} E/v^{k}]

= lim_{⟵}_{n} lim_{⟵}_{k} [X^{(n)}, E/v^{k}] = lim_{⟵}_{k} [X, E/v^{k}].

The first equality follows since X is locally finite and the homotopy of Ê is compact. The second equality is a definition. The third follows from the assumption about the compactness of π_{k}(E/v^{k}) and the finiteness of X^{(n)} giving compactness of [X^{(n)}, E/v^{k}]. The fourth commutes some limits and uses the compactness of the homotopy of E/v^{k} again.

Returning to the argument, let \overline{E}_{1} ⊆ E_{1} be the subgroup of permanent cycles. Here we make use of the only nontrivial property of the Bockstein spectral sequence. The differentials commute with multiplication by v in the sense that d_{r}(vx) = vd_{r}(x). To see this we note that we may form a second copy of the original tower related by multiplication by v. It follows that \overline{E}_{1} is actually a sub-K^{∗}[v]-module of E_{1}.

By hypothesis E_{1} is finitely generated over K^{∗}[v]; K^{∗} is Noetherian so is K^{∗}[v]. Therefore \overline{E}_{1} is finitely generated over K^{∗}[v]. It follows that the quotient, E_{∞} is also finitely generated over K^{∗}[v]. Thus we have an associated graded module finitely generated over the associated graded ring, and both are complete for their respective filtrations. The usual argument about lifting generators tells us that the original module, Ê^{∗}(X) is finitely generated over the original ring Ê^{∗}.

Now we can apply the proposition to prove the theorem. We use the theories E_{n,i} with coefficients

E_{n,i}^{∗} = E_{n}^{∗}/(p, v_{1}, …, v_{n−1−i}),

interpolating between E_{n,0} = K(n) and E_{n,n} = E_{n}. Now we argue by induction on i, taking E = E_{n,i+1}, K = E_{n,i} and v = v_{n−i−1}. Ravenel’s finiteness theorem supplies the base for the induction, and since E_{n} is complete with respect to (p, v_{1}, …, v_{n−1}) we have E = Ê at each stage.

4. Reduction to Euler classes.

Our method of calculation relies on the local Tate spectral sequence for calculating t(E)_{G}^{∗} from E^{∗}(BG_{+}) with various Euler classes inverted. We take advantage of the fact that the Tate spectrum of a split G-spectrum depends only on the underlying nonequivariant spectrum. This means that whatever the best model for equivariant E-cohomology, it has the same Tate spectrum as the Borel theory X ⟼ E^{∗}(EG_{+} ∧_{G} X) represented by F (EG_{+}, E). We simply write t(E) for the G-spectrum representing this Tate theory, and the homological description is

t(E) = F (EG_{+}, E) ∧ ẼG.

Thus any good model for ẼG gives rise to a means of calculating t(E)_{G}^{∗} from R^{∗} = E^{∗}(BG_{+}) = π_{∗}^{G}(F (EG_{+}, E)).

For instance, if G acts freely on the unit sphere S(V) of a complex representation V we find S(∞V) = ⋃_{k} S(kV) is a model for EG. Since the one point compactification S^{V} of V is the mapping cone of S(V)_{+} ⟶ S^{0}, we find S^{∞V} = ⋃_{k} S^{kV} is a model for ẼG. In general if G acts freely on S(V_{1}) × S(V_{2}) × ⋯ × S(V_{n}) for some n, the suspension spectrum of ẼG admits a filtration so that for s = 0, 1, … n − 1 the sth subquotient is the s-fold desuspension of the wedge of all (s + 1)-fold smash products S^{∞V_{i(0)}} ∧ S^{∞V_{i(1)}} ∧ … ∧ S^{∞V_{i(s)}} where i(0) < i(1) < ⋯ < i(s). To see this, one notes that ΣS(∞V)_{+} = S^{∞V} ∪ CS^{0}, so that EG_{+} = S(∞V_{1})_{+} ∧ S(∞V_{2})_{+} ∧ … ∧ S(∞V_{n})_{+} admits the product filtration with top quotient S^{0} and fibre Σ^{−1}ẼG. Explicitly, in the case where G acts freely on S(V) × S(W) then S(∞V) × S(∞W) is a model for EG and there is a stable cofibre sequence ẼG ⟶ S^{∞V} ∨ S^{∞W} ⟶ S^{∞V ⊕W}.

Therefore once we can calculate π_{∗}^{G}(F (EG_{+}, E) ∧ S^{∞V}), we have a means to calculate t(E)_{G}^{∗} whenever G acts freely on a product of spheres.

Lemma 4.1. For any complex representation V with Euler class χ(V) ∈ E^{|V |}(BG_{+}) we have

π_{∗}^{G}(F (EG_{+}, E) ∧ S^{∞V}) = R^{∗}[1/χ(V)].

Proof. For any G-spectrum X we have π_{∗}^{G}(X ∧ S^{∞V}) = lim_{⟶}_{k} π_{∗}^{G}(X ∧ S^{kV}). If X is a ring spectrum with Thom isomorphisms for V then the image of the unit from π_{0}^{G}X in the diagram

π_{n}^{G}(X ∧ S^{0}) ⟶^{e(V)_{∗}} π_{n}^{G}(X ∧ S^{V})
≅↓  ↓≅
π_{n}^{G}(X)  ⟶^{χ(V)} π_{n−|V |}^{G}(X)

defines the Euler class χ(V) ∈ π_{−|V |}(X). One checks that χ(kV) = χ(V)^{k}, and hence

π_{∗}^{G}(X ∧ S^{∞V}) = lim_{⟶}_{k} {π_{∗}^{G}(X), χ(V)} = π_{∗}^{G}(X)[1/χ(V)]

as required.

It will again be useful to use the periodicity of E^{∗} to shift the Euler classes into degree 0, as is familiar for equivariant K-theory. Replacing V by a multiple if necessary, we may suppose that the real dimension of V is a multiple of the degree of v_{n}. Thus χ(V) is a v_{n}-multiple of an element λ(V) in degree 0, so that inverting χ(V) has the same effect as inverting λ(V)

Now for any elements x_{1}, x_{2}, …, x_{n} in R we may form the Čech complex

⨁_{i(0)} R[1/x_{i(0)}] ⟶ ⨁_{i(0)<i(1)} R[1/(x_{i(0)}x_{i(1)})] ⟶ ⋯ ⟶ ⨁_{i(0)<i(1)<···<i(s)} R[1/(x_{i(0)}x_{i(1)} ⋯ x_{i(s)})].

It is not hard to see that the cohomology only depends on the radical of the ideal I = (x_{1}, …, x_{n}) [2]; this cohomology is the Čech cohomology Ȟ_{I}^{∗}(R). By construction we therefore have a local Tate spectral sequence in case G acts freely on a product of spheres.

Corollary 4.2. If G acts freely on a product of spheres and J′ = (λ(V_{1}), …, λ(V_{n})) is the ideal generated by their degree 0 Euler classes there is a spectral sequence

E_{2}^{s,∗} = Ȟ_{J′}^{s}(R^{∗}) ⟹ t(E)_{G}^{∗}.

If J′ has the same radical as J then the E_{2} term is Ȟ_{J}^{s}(R^{∗}).

In fact, if R^{∗} is Noetherian, the augmentation ideal has the same radical as the Euler class ideal, so that this is consistent with (2-5). Quite explicitly the following is a special case of Theorem 1.4 of [4].

Theorem 4.3. If F^{∗}(·) is a Noetherian cohomology theory with F^{∗} an integral domain, and if G acts freely on S(V_{1})×· · · · · · S(V_{n}) then the radical of the ideal J′ = (λ(V_{1}), …, λ(V_{n})) is the augmentation ideal J(G).

Henceforth we shall use the augmentation ideal rather than J′ without further comment. The reader content to restrict attention to p-groups can work entirely with the more concrete Euler class ideal J′, and need never invoke this comparison theorem. The calculational consequences are identical. However, even for p-groups the augmentation ideal seems more natural, and for general finite groups explicit substitutes for it are clumsier than J′. The theorem is also essential when comparing the ideals for different groups under restriction.

5. Inverting Euler classes

We have now established that the calculation depends on an understanding of R localized away from the Euler classes. The key statement we need is that inverting an Euler class λ(V) ∈ R is essentially the same as inverting an element \overline{λ}(V) of the ground ring E^{0}. The geometric flavour of this is that the map of varieties associated to E^{0} ⟶ R is injective on the hypersurface defined by any Euler class. This geometric point of view is developed further in [8]. As a special bonus the element \overline{λ^{∗}}(V) is independent of V in the one dimensional case.

In this section E^{∗} has a single spectral line v_{i} for some 0 ≤ i ≤ n − 1; for example the coefficients may be ℤ_{(p)}[v_{1}, v_{2}, …, v_{n−1}, v_{n}, v_{n}^{−1}]/(p, v_{1}, … v_{i−1}, v_{i+1}, …, v_{n−1}) or some completion. As in the introduction, we abbreviate the ideal (p, v_{1}, … v_{i−1}, v_{i+1}, …, v_{n−1}) by I_{n,i}.

Let a(V) = ker(R^{∗} ⟶ R^{∗}[1/χ(V)]) and Q(V) = R^{∗}/a(V). It is immediate that R^{∗}[1/χ(V)] = Q(V)[1/χ(V)].

Theorem 5.1. If E = E(n)/I_{n,i} or E_{n}/I_{n,i} then v_{i} is invertible on R^{∗}[1/χ(V)] for any complex representation V; if E = E_{n} then

R^{∗}[1/χ(V)] = Q(V)[1/v_{i}].

The expectation is that v_{i} should be invertible for any theory with a single spectral line, and that if it is also complete then one has a precise identification of R^{∗}[χ(V)]. It is clear that the method of proof below will work in many other cases, but the correct level of generality is not clear.

Proof. The proof is in two steps; firstly we show that v_{i} is invertible on R^{∗}[1/χ(V)], and secondly that in the complete case Q(V)[1/v_{i}] is finite enough that any monomorphism is an isomorphism.

Lemma 5.2. The element v_{i} is invertible on R^{∗}[1/χ(V)].

Proof. Because the Euler for K(n)^{∗} is the image of the Euler class from E^{∗}, the cofibre sequence E ⟶^{v_{i}} E ⟶ K(n) gives a long exact sequence

⋯ ⟶ R^{∗}[1/χ(V)] ⟶^{v_{i}} R^{∗}[1/χ(V)] ⟶ K(n)^{∗}(BG_{+})[1/χ(V)] ⟶ · · · .

But K(n)^{∗}(BG_{+})[1/χ(V)] = 0 by Lemma 2.2 of [7].

Now χ(V) : Q(V) ⟶ Q(V) is injective, and therefore so is χ(V) : Q(V)[1/v_{i}] ⟶ Q(V)[1/v_{i}]. Furthermore, Q(V) is finite over E^{0} so Q(V)[1/v_{i}] is finite over E^{0}[1/v_{i}].

The completeness assumption on E is necessary to obtain the conclusion of the following lemma.

Lemma 5.3. For E = E_{n}/I_{n,i}, E^{∗}[1/v_{i}] is a graded field.

Proof. In fact it is easy to verify that for any graded field k then for an indeterminate y the ring k[[y]][1/y] is again a graded field.

Therefore the monomorphism χ(V) : Q(V)[1/v_{i}] ⟶ Q(V)[1/v_{i}] is also an epimorphism.

6. The proof.

In this section we prove our main theorem. Note in particular that despite the necessity of completeness hypotheses in the previous section, none is necessary here. This is because although R^{∗} need not be v_{i}-complete, the relevant quotient R^{∗}/Γ_{J}R^{∗} is always v_{i}-complete. This raises the general question for an arbitrary v_{n}-periodic cohomology theory of whether R^{∗}/Γ_{J}R^{∗} is I_{n}-complete in the higher dimensional case.

Theorem 6.1. If E = E(n)/I_{n,i} or E_{n}/I_{n,i} then

t(E)_{G}^{∗} = (R^{∗}/Γ_{J}R^{∗})[1/v_{i}]

where Γ_{J}R^{∗} is the ideal of J-power torsion elements of R^{∗}.

Corollary 6.2. If E is a complex oriented v_{n}-periodic theory with one dimensional coefficients satisfying (2.1) and R^{∗} is concentrated in even degrees then

E_{even}(BG_{+}) = Γ_{J}R^{∗} and E_{odd}(BG_{+}) = Σ{(R^{∗}/Γ_{J}R^{∗})/v_{i}^{∞}}.

Again one expects this behaviour for any complex oriented v_{n}-periodic theory with a single spectral line, and the given proof will work in many cases of this type.

Proof. We shall begin by proving the theorem when E is complete, and then explain why the general result follows.

If G is a p-group it acts freely on a product of linear spheres and hence by Corollary 4.2 it is enough to show Ȟ_{J}^{0}(R^{∗}) = (R^{∗}/Γ_{J}R^{∗})[1/v_{i}]. The general case will follow from a transfer argument.

The exact sequence (2-3) gives a short exact sequence

0 ⟶ R^{∗}/Γ_{J}R^{∗} ⟶ Ȟ_{J}^{0}(R^{∗}) ⟶ H_{J}^{1}(R^{∗}) ⟶ 0.

From Theorem 5.1, v_{i} is invertible on Ȟ_{J}^{∗}(R^{∗}), so that we obtain an injective map

(R^{∗}/Γ_{J}R^{∗})[1/v_{i}] ⟶ Ȟ_{J}^{0}(R^{∗}).

The cokernel is the group H_{J}^{1}(R^{∗})[1/v_{i}] = H_{J}^{1}(R^{∗}[1/v_{i}]), which is zero since R^{∗}[1/v_{i}] (or rather its degree 0 subring) is 0-dimensional; this is where we use completeness. This proves the theorem when G is a p-group.

Now for a general group G with Sylow p-subgroup P we first need the fact that t(E)_{G}^{∗} is a retract of t(E)_{P}^{∗}.

Lemma 6.3. If G is a group with Sylow p-subgroup P and E is a p-local G-spectrum then t(E)_{G}^{∗} is a retract of t(E)_{P}^{∗}.

Proof. We work with G-spectra and use the map π : G/P_{+} ⟶ S^{0} and its dual Dπ : S^{0} ⟶ DG/P_{+} ≃ G/P_{+} to construct the diagram in which the horizontals are cofibre sequences

E ∧ EG_{+}  ⟶  F (EG_{+}, E)  ⟶  t(E)
↓  ↓  ↓
G/P_{+} ∧ E ∧ EG_{+} ⟶ G/P_{+} ∧ F (EG_{+}, E) ⟶ G/P_{+} ∧ t(E)
↓  ↓  ↓
E ∧ EG_{+}  ⟶  F (EG_{+}, E)  ⟶  t(E).

We argue that the first and second vertical composites are equivalences, and deduce that the third is by the 5-lemma. Hence the G-spectrum t(E) is a retract of G/P_{+} ∧ t(E); the result follows by taking homotopy groups.

From the identification of [S^{0}, S^{0}]^{G} with the Burnside ring A(G) the vertical composites are multiplication by [G/P] ∈ A(G). Now [G/P] acts as multiplication by the index |G : P | on both G/1_{+} ∧ E and F (G/1_{+}, E); since E is p-local this is an equivalence. Now any free spectrum T can be constructed out of cells G/1_{+} ∧ S^{n} and hence using wedges, cofibre sequences and limits we see that [G/P] is an equivalence of T ∧ E and F (T, E). Taking T = EG_{+} we obtain the required result.

Since the retractions are all maps of E_{∗}-modules it follows that v_{i} acts invertibly on t(E)_{G}^{∗}. From the lemma we obtain a diagram

E_{∗}(BG_{+}) ⟶ R_{G}^{∗} ⟶^{α} t(E)_{G}^{∗}
↓ ↓ ↓
E_{∗}(BP_{+}) ⟶ R_{P}^{∗} ⟶ t(E)_{P}^{∗}

which is part of a long exact ladder, and in which the verticals are injective and the four rightmost maps are ring homomorphisms. From the case of the p-group P we see that inverting v_{i} in the diagram gives a short exact sequence in the second row, and hence also in the first row. It therefore suffices to identify the kernel K of α with Γ_{J(G)}R_{G}^{∗}. Now by [4, 1.2], since R_{G}^{∗} and R_{P}^{∗} are Noetherian, \sqrt{res_{P}^{G}(J(G))} = J(P), and hence the two ideals have the same torsion functors: Γ_{J(G)}M = Γ_{J(P)}M for any R_{P}^{∗} -module M. Now K = R_{G}^{∗} ∩ Γ_{J(P)}R_{P}^{∗} = R_{G}^{∗} ∩ Γ_{J(G)}R_{P}^{∗} = Γ_{J(G)}(R_{G}^{∗}) as required.

This finishes the proof in case E is complete.

For the non-complete case, let E^{∧} = lim_{⟵}_{k} E/v_{i}^{k}, denote its completion, and similarly let R^{∗} = E^{∗}(BG_{+}) and (R^{∧})^{∗} = (E^{∧})^{∗}(BG_{+}). Now we have a ring homomorphism R^{∗} ⟶ (R^{∧})^{∗}, and all (R^{∧})^{∗}-modules can be regarded as R^{∗}-modules by pullback. Next we claim that if J is the augmentation ideal of R, and J^{∧} of R^{∧}, then the image of J has the same radical as J^{∧}; this allows us to use the results for the complete case. If G is a p-group, this follows from [4, 1.4] since the ideal J contains all Euler classes, and in the general case it follows since J(G) = J(P) ∩ E^{∗}(BG_{+}). Thus local and Čech cohomology at J^{∧} of (R^{∧})^{∗}-modules can be calculated by regarding them as R^{∗}-modules and using J.

The main input is that the natural map

t(E) ⟶^{≃} t(E^{∧}) (6-4)

is an equivalence. Indeed, the cofibre is t(E′), where E′ is the cofibre of E ⟶ E^{∧}. This cofibre is lim_{⟵} (E, v_{i}), so by construction v_{i} is an isomorphism on E′. If we localize by inverting v_{i}, we get the cofibration

v_{i}^{−1}E → v_{i}^{−1}E^{∧} → E′. (6-5)

By [7, 1.5], t(v_{i}^{−1}E) ≃ t(v_{i}^{−1}E^{∧}) ≃ ∗, and we conclude that t(E′) ≃ ∗ as required.

Note that, since by the local cohomology theorem (2-2) E_{∗}(BG_{+}) is J-power torsion, the norm map E_{∗}(BG_{+}) ⟶ E^{∗}(BG_{+}) = R^{∗} factors through Γ_{J}R^{∗}. Consider then the map n : E_{∗}(BG_{+}) ⟶ Γ_{J}R^{∗}. We argue below that it is surjective, so that we have an exact sequence 0 ⟶ R^{∗}/Γ_{J}R^{∗} ⟶ t(E)_{G}^{∗} ⟶ Σ^{−1}ker(n) ⟶ 0. We also show that n becomes an isomorphism when we invert v_{i}. The result follows since v_{i} is invertible on t(E)_{G}^{∗} by [7, 1.5]. Alternatively, it can be deduced from (5.1) together with (4.2).

Consider the diagram

E ∧ EG_{+} ⟶ E^{∧} ∧ EG_{+} ⟶ E′ ∧ EG_{+}
↓ ↓ ↓
F (EG_{+}, E) ⟶ F (EG_{+}, E^{∧}) ⟶ F (EG_{+}, E′),

and apply homotopy groups to obtain a diagram

⋯ ⟶^{α} E_{∗}(BG_{+}) ⟶^{β} E_{∗}^{∧}(BG_{+}) ⟶^{γ} E′_{∗}(BG_{+}) ⟶^{Σ^{−1}α} ⋯
↓  ↓  ↓
⋯ ⟶^{θ}  R^{∗}  ⟶^{ϕ} (R^{∧})^{∗}  ⟶^{ψ} (R′)^{∗}  ⟶^{Σ^{−1}θ} · · · .

Suppose x ∈ Γ_{J}R^{∗}. Then x maps to 0 in t(E)_{∗}^{G} since it must map to 0 in t(E^{∧})_{∗}^{G} and these are isomorphic by (6-4). This establishes that n : E_{∗}(BG_{+}) ⟶ Γ_{J}R^{∗} is surjective.

Finally, we show that n is injective after inverting v_{i}. First note that v_{i}^{−1}(E_{∗}BG_{+}) = (v_{i}^{−1}E)_{∗}BG_{+}, and that there is a natural map v_{i}^{−1}(E^{∗}BG_{+}) ⟶ (v_{i}^{−1}E)^{∗}BG_{+}. Thus we have a sequence of maps

(v_{i}^{−1}E)_{∗}BG_{+} = v_{i}^{−1}(E_{∗}BG_{+}) → v_{i}^{−1}Γ_{J}R^{∗} → v_{i}^{−1}(E^{∗}BG_{+}) → (v_{i}^{−1}E)^{∗}BG_{+}.

The composite is an isomorphism by [7, 4.1], and hence the map v_{i}^{−1}(E_{∗}BG_{+}) → v_{i}^{−1}Γ_{J}R^{∗} must be an injection.

7. Examples for elementary abelian groups.

In this section we calculate t(E)_{G}^{∗} when G is elementary abelian in a more explicit form than given above. There are two classes of example that we consider. Firstly we consider the complete 2-periodic theories E = E_{n}/(p, u_{1}, …, û_{i}, …, u_{n−1}), where E_{n}^{∗} = ℤ_{p}^{∧}[[u_{1}, …, u_{n−1}]][u, u^{−1}], as in Section 3. Secondly, by way of contrast we include a calculation for the theory E = E(n)/(p, v_{1}, …, v̂_{i}, …, v_{n−1}). For convenience we refer to these as the 2-periodic and 2(p^{n} − 1)-periodic cases.

We remind the reader that in [10] Landweber calculates

MU^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{k}})_{+}) = MU^{∗}[[x_{1}, …, x_{k}]]/([r_{1}](x_{1}), …, [r_{k}](x_{k})).

where C_{r} is the cyclic group of order r, and [r](x) is the r-series of the formal group law over MU^{∗} on the indeterminate x. This calculuation is easily generalized to give L^{∗}(B(C_{r_{1}} × ⋯×C_{r_{k}})_{+}) where L is Landweber exact: in [10], it is shown that MU^{∗}(B(C_{r_{1}} ×⋯×C_{r_{k}})_{+}) is a flat MU^{∗} module, so one uses induction and observes that if L^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{j}})_{+}) is isomorphic to MU^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{j}})_{+}) ⊗_{MU^{∗}} L^{∗} then

L^{∗}(B(C_{r_{1}} × ⋯ × BC_{r_{j}})_{+} ∧ X) = MU^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{j}})_{+}) ⊗_{MU^{∗}} L^{∗}(X)

for X finite. It follows by taking inverse limits over the skeleta of BC_{r_{j+1}} that

L^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{j+1}})_{+}) ≃ MU^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{j+1}})_{+}) ⊗_{MU^{∗}} L^{∗}.

We wish to apply this knowledge to the theory E, where E is one of the two examples named above. To do this, we use the fact that MU^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{k}})_{+}) is a flat MU^{∗} module, so if I ⊆ E(n)^{∗} is generated by some of the v_{j}’s, then (E(n)/I)^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{k}})_{+}) = MU^{∗}(B(C_{r_{1}} × ⋯ × C_{r_{k}})_{+}) ⊗_{MU^{∗}} (E(n)/I)^{∗} (and similarly in the complete case).

Let F denote the formal group law associated to E where E is as above, and let [p](x) ∈ E^{∗}[[x]] denote the p-series for F. Since E is p-local we implicitly consider a complex orientation of E factoring through BP. Hence [p](x) = v_{i}x^{p^{i}} +_{F} v_{n}x^{p^{n}} [12, A2.2.4]. Now consider the case G = C_{p}, and let S = E^{∗}((BC_{p})_{+}). From the short exact sequence

1 ⟶ C_{p} ⟶ S^{1} ⟶^{<p>} S^{1} ⟶ 1 we have S = E^{∗}[[x]]/[p](x).

Lemma 7.1. In S we have x^{p^{n}} = u · v_{i}x^{p^{i}}/v_{n} where u is a unit of degree zero in S.

Proof. In E^{∗}[[x]] we find [p](x) = v_{i}x^{p^{i}} + v_{n}x^{p^{n}} mod v_{i}v_{n}x^{p^{i}+p^{n}}, and hence v_{n}x^{p^{n}} = −v_{i}x^{p^{i}}(1 + x · α) for some α.

Thus any multiple of x^{p^{n}} may be expressed as a multiple of a lower power of x by a power series in v_{i}, and as modules over E^{∗}

S = E^{∗}{1, x, …, x^{p^{i}−1}} ⊕ (E^{∗})_{v_{i}}^{∧}{x^{p^{i}}, …, x^{p^{n}−1}}. (7-2)

Here the completion at v_{i} arises since power series in x have been converted to power series in v_{i} using the lemma.

It is worth noting some general features of this expression. Firstly S is concentrated in even degrees. Next observe that it is not usually finite over E^{∗} unless E^{∗} is v_{i}-complete. Finally notice that the degrees of the generators are in different congruence classes modulo 2(p^{n} − 1) except that 1 and x^{p^{n}−1} both lie in the congruence class of 0. The expression also shows that the first summand does not generally disappear if we replace BG_{+} by BG, unless i = 0. This should provide a nonequivariant cure for any temptation the reader may have to delete the basepoint; working equivariantly provides another cure.

Now since C_{p} acts freely on S(η) where η is a faithful one dimensional representation, and since x is the Euler class of any faithful one dimensional representation, we see that t(E)_{C_{p}}^{∗} = S[1/x]. If we now let A denote the first summand in (7-2), and B the second then x^{p^{i}}A ⊆ B and hence (S/B)[1/x] = 0. Thus there is an isomorphism B[1/x] ⟶^{≅} S[1/x], which completes the calculation for G = C_{p}. Note that the ideal Γ_{J}R^{∗} is generated by the elements x^{p^{n}−1} − uv_{i}x^{p^{i}−1}/v_{n}, x^{p^{n}−2} − uv_{i}x^{p^{i}−2}/v_{n}, …, x^{p^{n}−p^{i}} − uv_{i}/v_{n}.

If G is of rank r we may choose an isomorphism G ≅ H_{1} × H_{2} × ⋯ × H_{r} where H_{i} ≅ C_{p}. Consider R^{∗} = E^{∗}(BG_{+}), and note from above that R^{∗} ≅ S_{1} ˆ⊗S_{2} ˆ⊗ ⋯ ˆ⊗S_{r} [10]. Now since H_{i} acts freely on S(η_{i}) where η_{i} is a faithful one dimensional representation, G acts freely on S(η_{1}) × S(η_{2}) × ⋯ × S(η_{r}) and so t(E)_{G}^{∗} may be calculated from Čech complex for x_{1}, x_{2}, …, x_{r}. Since every term has at least one x_{i} inverted (which kills A_{i}) the result is the same as if E was replaced by its v_{i}-completion at the start. Since R is finite over E^{0} in the complete case, and hence of dimension 1, the cohomology of the Čech complex is concentrated in degree zero and we find an exact sequence

0 ⟶ t(E)_{G}^{∗} ⟶ ⨁_{0≤j≤r} R^{∗}[1/x_{j}] ⟶ ⨁_{1≤k<l≤r} R^{∗}[1/x_{k}x_{l}].

On the other hand, purely as a matter of combinatorics we know there is an exact sequence

0 ⟶ A_{1} ⊗ A_{2} ⊗ ⋯ ⊗ A_{r} ⟶ R^{∗} ⟶ ⨁_{0≤j≤r} R_{j}^{∗} ⟶ ⨁_{0≤k<l≤r} R_{k,l}^{∗},

where R_{j}^{∗} is the part of R^{∗} with the j coordinate constrained to be B_{j} and similarly for R_{k,l}^{∗}. This discussion leads to the following calculation.

Theorem 7.2. The coefficient ring of E-Tate theory is

t(E)_{G}^{∗} = {R^{∗}/(A_{1} ⊗ A_{2} ⊗ ⋯ ⊗ A_{r})}[1/v_{i}].

In particular it is concentrated in even degrees and it is free of finite rank p^{nr} − p^{ir} over (E^{∗})_{v_{i}}^{∧}[1/v_{i}] on generators of degree 0.

Proof. The only outstanding point is to observe that since G is abelian, all representations are one dimensional. Therefore the Čech complex consists of 2-periodic modules, and the Tate theory is 2-periodic.

Corollary 7.3. The homology of BG_{+} is as follows

E_{even}(BG_{+}) = A_{1} ⊗ A_{2} ⊗ ⋯ ⊗ A_{r}

and

E_{odd}(BG_{+}) = Σ{R/(A_{1} ⊗ A_{2} ⊗ ⋯ ⊗ A_{r})}/v_{i}^{∞}.

The main difference between the 2-periodic and 2(p^{n} − 1)-periodic cases comes in translating these results into specific statements about ranks. In the 2-periodic case the answer is easy to state.

Corollary 7.4. If E is the 2-periodic complete theory then t(E)_{G}^{∗} is concentrated in even degrees, and in each even degree it is free of rank p^{nr} − p^{ir} over E^{0}[1/u_{i}].

For example when n = 1 and E is p-adic K-theory we obtain a sum of (|G| − 1) copies of the p-adic rationals in each even degree and zero in each odd degree. For n = 1 this applies to any p-group.

Corollary 7.5. If E is the 2-periodic complete theory then E_{∗}(BG_{+}) is a sum of p^{ir} copies of E^{0} in each even degree and p^{nr} − p^{ir} copies of E^{0}/u_{i}^{∞} in each odd degree.

For example when n = 1 and E is p-adic K-theory we obtain ℤ_{p}^{∧} in each even degree and a sum of (|G| − 1) copies of ℤ/p^{∞} in each odd degree. For n = 1 this applies to any p-group.

By way of contrast consider the 2(p^{n} − 1)-periodic case.

If n = 1 the coefficients of the Tate theory has a sum of (|G| − 1)/(p − 1) copies of the p-adic rationals in each even degree and zero in odd degrees. Correspondingly E_{∗}(BG_{+}) is ℤ_{(p)} in each even degree, and a sum of (|G| − 1)/(p − 1) copies of ℤ/p^{∞} in each odd degree.

8. Appendix: an algebraic proof in the 0 dimensional case.

We give a more algebraic proof of Theorem 1.1 of [7] using the algebra of [3]. Since E^{0}(BG_{+}) is a 0-dimensional Noetherian ring if E is Morava K-theory, it is enough to prove the following.

Theorem 8.1. If G is a finite group and E is a v_{n}-periodic, complex oriented theory with E^{0}(BG_{+}) Noetherian of Krull dimension 0, then t(E)_{G}^{∗} = 0.

It is enough to deal with the case when G is a p-group by using the transfer argument 6.3.

Suppose then that G is a p-group. From the local Tate spectral sequence (2-5) it is enough to show that Ȟ^{∗}(R^{∗}) = 0. We shall prove a more general algebraic statement; in fact the ring R^{∗} = E^{∗}(BG_{+}) arising in our topological applications has certain completeness properties [3, 3.5] which show that Ȟ_{J}^{∗}(R^{∗}) = Ĥ_{J}^{∗}(R^{∗}), where Ĥ_{J}^{∗}(·) denotes local Tate cohomology. To obtain an algebraic result valid without completeness assumptions on R it is necessary to use local Tate cohomology.

Proposition 8.2. If R is a 0 dimensional Noetherian ring then the local Tate cohomology at any ideal vanishes: Ĥ_{J}^{∗}(R; M) = 0

Proof. Since Ĥ_{J}^{∗}(R; M) is a module over Ĥ_{J}^{∗}(R), it is sufficient to deal with the case M = R. We use the spectral sequence of the second avatar [3, 4.2]:

E_{2}^{∗,∗} = Ȟ_{∗}^{J}(H_{J}^{∗}(ΣR)) ⟹ Ĥ_{J}^{∗}(R).

Now if R is of dimension 0 the local cohomology is concentrated in degree 0, so that H_{J}^{∗}(R) = Γ_{J}(R), which is also finitely generated. Hence J^{e}H_{J}^{∗}(R) = 0 for some e. The following lemma completes the proof of the proposition.

Lemma 8.3. If R is a 0-dimensional Noetherian ring and N is a finitely generated module with J^{e}N = 0 for some e then Ȟ_{∗}^{J}(N) = 0.

Proof. In the 0-dimensional case the only possible nonzero Čech homology groups are in degrees 0 and −1, which are the kernel and cokernel of the natural map M ⟶ H_{0}^{J}(M) for any M. If M is Noetherian H_{0}^{J}(M) = M_{J}^{∧}, and the hypotheses on N guarantee that N = N_{J}^{∧}.

We note that the Rationality Theorem of [3] could similarly be used to give a substitute for Theorem 5.1. Indeed, provided v_{i} is a regular element on E^{∗}(BG_{+}) and E^{0}(BG_{+}) is 1-dimensional, the Rationality Theorem shows that v_{i} is invertible on Ĥ_{J}^{∗}(R). Since this is the E_{2} term of a spectral sequence for calculating topological Tate cohomology, v_{i} is also invertible on t(E)_{G}^{∗}.

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School of Mathematics and Statistics, Hicks Building, Sheffield S3 7RH. UK. email: j.greenlees@sheffield.ac.uk

Department of Mathematics, Johns Hopkins University, Baltimore MD 21218,USA. email: hs@adams.mat.jhu.edu