Smith theory and the functor T

William G. Dwyer and Clarence W. Wilkerson

University of Notre Dame Purdue University

§1. Introduction

J. Lannes has introduced and studied a remarkable functor T [L1] which takes an unstable module (or algebra) over the Steenrod algebra to another object of the same type. This functor has played an important role in several proofs of the generalized Sullivan Conjecture [L1]

[L2] [DMN] and has led to homotopical rigidity theorems for classifying spaces [DMW1] [DMW2]. In this paper we will use techniques of Smith theory [DW] to calculate the functor T explicitly in certain key special situations (see 1.1 and 1.3). On the one hand, our calculation gives general structural information (1.4) about T itself. On the other hand, up to a convergence question which we will not discuss here our calculation produces a direct analogue of Smith theory (1.2) for actions of elementary abelian p-groups on certain infinite-dimensional complexes; this analogue differs from Smith theory only in that “homotopy fixed point set” is substituted for “fixed point set”.

We will now state the main results, which are completely algebraic in nature although they have a geometric motivation. Fix a prime p; the field F_{p} with p elements will be the coefficient ring for all cohomology. Let A_{p} denote the mod p Steenrod algebra, and U (resp. K) the category of unstable modules (resp. unstable algebras) over A_{p} (see [L1]). If R is an object of K, an unstable A_{p} ⊙ R module M is by definition an object of U which is also an R module in such a way that the multiplication map R⊗M → M obeys the Cartan formula; we will denote the category of A_{p} ⊙ R modules by U(R). An object of U(R) typically arises from a map q : E → B of spaces; in this case the induced cohomology map q^{∗} makes H^{∗}E an object of U(R) for R = H^{∗}B.

Let V be an elementary abelian p-group, ie., a finite-dimensional vector space over F_{p}, and H^{V} the classifying space cohomology H^{∗}BV. Lannes [L1] has constructed a functor T^{V} : U → U which is left adjoint to the functor given by tensor product (over F_{p}) with H^{V} and has shown that T^{V} lifts to a functor K → K which is also left adjoint to tensoring

Typeset by AMS-TEX

Both authors were supported in part by the National Science Foundation.

with H^{V}. The adjointness property of T^{V} produces for any space X a natural map

λ_{X} : T^{V} (H^{∗}X) → H^{∗} Hom(BV, X)

which is often an isomorphism [L1] [L2] [DS]. Given an object R of K there is a simple way (see §2) of using a particular K-map f : R → H^{V} to single out a quotient T_{f}^{V} (R) of T^{V} (R) or for M ∈ U(R) a quotient T_{f}^{V} (M) of T^{V} (M). These quotients correspond via λ to subspaces of function spaces; more precisely, if q : E → B is a map of spaces and f : H^{∗}B → H^{V} is a map in K, then λ_{E} induces a quotient map λ_{E,f} from T_{f}^{V} (H^{∗}E) to the cohomology of the subspace of Hom(BV, E) consisting of maps r : BV → E with r^{∗}q^{∗} = f.

For any K-map R → H^{V}, we will let S_{f} ⊂ R denote the multiplicative subset of R generated by the Bockstein images in R^{2} of the elements of R^{1} which map non-trivially under f. Recall from [W1] [Si] that if M is an object of U(R) any localization of the form S_{f}^{−1}M inherits an action of A_{p}, although this action is not necessarily unstable. Denote the largest unstable A_{p}-submodule [DW, 2.2] of such a localization by Un(S_{f}^{−1}M).

Theorem 1.1. Let W be an elementary abelian p-group, V a subgroup of W, and f : H^{W} → H^{V} the map induced by subgroup inclusion. Suppose that M is an object of U(H^{W}) which is finitely-generated as a module over H^{W}. Then there is a natural isomorphism

T_{f}^{V} (M) ≅ Un S_{f}^{−1}(M) .

This theorem has a geometric background. Let V and W be as in 1.1. Suppose that X is a finite CW-complex with a cellular action of W and let M be the cohomology of the Borel construction E_{W} X = EW ×_{W} X on this action. According to [L2] there is an isomorphism

T_{f}^{V} (M) ≅ H^{∗}E_{W} (X^{V})

where X^{V} is the fixed point set of the action of V on X. Similarly, by Smith theory [DW, 2.3] there is an isomorphism

H^{∗}E_{W} (X^{V}) ≅ Un S_{f}^{−1}(M) .

The composite of these two maps is the isomorphism of 1.1; the present paper sprang in part from a desire to produce this isomorphism in a purely algebraic way without assuming that M ≅ E_{W} X for a finite complex X.

With Theorem 1.1 in hand, though, it is possible to work backwards through the above geometric example. Suppose that X is a space with an action of W but instead of assuming that X is finite assume only that H^{∗}X is finite. Again let M = H^{∗}(E_{W} X); a spectral sequence argument shows that M is finitely generated as a module over H^{W}. Let X^{hV} denote the homotopy fixed point set [M, p. 581] of the action of V on X. It is not hard to see that the Borel construction E_{W} (X^{hV}) is homotopy equivalent to the space of maps BV → E_{W} X which up to cohomology cover the map BV → BW induced by V ⊂ W, so that λ_{E_{W} X,f} produces a map T_{f}^{V} (H^{∗}E_{W} X) → H^{∗}E_{W} (X^{hV}). Theorem 1.1 now computes the domain of λ_{E_{W} X,f}; this gives the following corollary.

Corollary 1.2. Let W be an elementary abelian p-group, V a subgroup of W and f : H^{W} → H^{V} the map induced by subgroup inclusion. Let X a space on which W acts and assume that H^{∗}X is finite and that λ_{E_{W} X,f} is an isomorphism. Then there is a natural isomorphism

H^{∗}E_{W} (X^{hV}) ≅ Un S_{f}^{−1}H^{∗}(E_{W} X) .

Remark: The conclusion of Corollary 1.2 implies that the localized restriction map

S_{f}^{−1}H^{∗}(E_{W} X) → S_{f}^{−1}H^{∗}E_{W} (X^{hV})

is an isomorphism. This is the promised extension of Smith theory [H, Chap. III] to infinite dimensional complexes (the only finiteness condition on X is a cohomological one). The question of whether λ_{E_{W} X,f} is an isomorphism is essentially a spectral sequence convergence problem [DS] [B] and we intend to consider it in a future note. Remark: As in [DW, 2.5], for V = W the conclusion of Corollary 1.2 gives an isomorphism H^{∗}(X^{hW}) ≅ F_{p} ⊗_{H^{W}} Un S_{f}^{−1}H^{∗}(E_{W} X) .

An unstable A_{p} ⊙ H^{V} algebra is an object R of K together with a K-map H^{V} → R. Denote the category of A_{p} ⊙ H^{V} algebras by K(H^{V}). (We will also occasionally consider the category K(R) for other objects R of K.) If M is an R module and I ⊂ R is an ideal, let M_{I}^{∧} stand for the completion of M with respect to powers of I, i.e. for the inverse limit lim_{←}_{s}M/I^{s}M.

Theorem 1.3. Let V be an elementary abelian p-group and f : R → H^{V} a map in K(H^{V}) with kernel I. Assume that R is finitely generated as a ring and that M is an object of U(R) which is finitely-generated as a module over R. Then there is a natural isomorphism

T_{f}^{V} (M) ≅ Un((S_{f}^{−1}M)_{I}^{∧}) .

Remark: Example 4.6 shows that the appearance of something like a completion is necessary in Theorem 1.3. It is easy to see (4.4) that in the statement of Theorem 1.3 the localization S_{f}^{−1}M can be replaced by the localization S^{−1}M of M with respect to the multiplicative set S generated by the Bockstein images in H^{2}BV of the non-zero elements of H^{1}BV.

The restriction in Theorem 1.3 that R be an algebra over H^{V} is not too serious, since it is possible to make any object of K into an object of K(H^{V}) by tensoring with H^{V}. If g : R → H^{V} is a map of K and f : H^{V} ⊗_{F_{p}} R → H^{V} is the map of K(H^{V}) which extends g, then (see 2.2 and the proof of 1.4) there is a natural isomorphism T_{f}^{V} (H^{V} ⊗_{F_{p}} R) ≅ H^{V} ⊗_{F_{p}} T_{g}^{V} (R). In conjunction with Theorem 1.3 this calculation leads to the following result.

Theorem 1.4. Let V be an elementary abelian p-group, R an object of K which is finitely generated as a ring and M an object of U(R) which is finitely generated as a module over R. Then T^{V} (R) is finitely generated as a ring and T^{V} (M) is finitely generated as a module over T^{V} (R).

Organization of the paper. In sections 2 through 4 we will prove the above theorems in the special case in which V is the rank one elementary abelian p-group Z/p; for this particular V, T^{V} is simply denoted by T and H^{V} by H. Section 5 describes the argument that extends the results to general V. Section 6 contains some auxiliary algebraic material on K.

§2. Some properties of the functor T

The purpose of this section is to set up some machinery involving the functor T.

Let f : R → H be a K-map. The adjoint of f is a K-map T (R) → F_{p}, which amounts to a ring homomorphism f̂ : T (R)^{0} → F_{p}. For M ∈ U(R), let T_{f}(M) denote the tensor product T (M) ⊗_{T (R)^{0}} F_{p}, where the action of T (R)^{0} on F_{p} is given by f̂. Note that T_{f}(R) ∈ K.

Proposition 2.1. For any K-map f : R → H the construction T_{f}(−) induces functors U(R) → U(T_{f}(R)) and K(R) → K(T_{f}(R)). Moreover, T_{f} is exact, and preserves tensor products in the sense that if M and N are objects of U(R) there is a natural isomorphism

T_{f}(M ⊗_{R} N) ≅ T_{f}(M) ⊗_{T_{f} (R)} T_{f}(N) .

Proof: Most of what is asserted follows from the fact that T is exact and preserves tensor products [L1]. To see that T_{f} is exact, note in addition that f̂ makes F_{p} into a flat module over T (R)^{0}; this flatness is an algebraic consequence of the fact that T (R)^{0} is a p-boolean ring [L1, 3.5] (that is, each element x in T (R)^{0} satisfies the equation x^{p} = x). In fact, if T (R)^{0} is finite, then T_{f}(R) is a summand of T (R).

Proposition 2.2. Let f_{i} : R_{i} → H, i = 1, 2 be K-maps with M_{i} ∈ U(R_{i}) and let f be the product map f_{1} · f_{2} : R_{1} ⊗_{F_{p}} R_{2} → H. Then there is a natural isomorphism

T_{f}(M_{1} ⊗_{F_{p}} M_{2}) → T_{f_{1}}(M_{1}) ⊗_{F_{p}} T_{f_{2}}(M_{2}) .

Proof: This is again a consequence of the fact that T preserves tensor products.

Proposition 2.3. Suppose that R ∈ K, M ∈ U(R), and that x ∈ R is an element such that x · M = 0. Let f : R → H be a K-map such that f(x) ≠ 0. Then T_{f}(M) = 0.

Proof: By the Cartan formula the annihilator ideal I of M in R is invariant under A_{p} and hence the quotient map R → R/I is a morphism of K. By Proposition 2.1 T_{f}(M) ≅ T_{f}(R/I) ⊗_{T_{f} (R)} T_{f}(M) so we will be done if we can show that T_{f}(R/I) vanishes in dimension 0. Now T_{f}(R) ≅ F_{p} and the map T_{f}(R)^{0} → T_{f}(R/I)^{0} is surjective, so non-vanishing of T_{f}(R/I)^{0} implies that T_{f}(R)^{0} → T_{f}(R/I)^{0} is an isomorphism and therefore that f̂: T (R)^{0} → T_{f}(R)^{0} ≅ F_{p} extends to a map T (R/I)^{0} → F_{p}. This is impossible, because by assumption the adjoint map f : R → H does not extend to a map R/I → H.

For any object M of U the adjunction map M → H ⊗_{F_{p}} T (M) can be combined with the unique algebra map H → F_{p} to give a map M → T (M); call this map ϵ. (If M = H^{∗}X for some space X, then ϵ corresponds via λ_{X} to the cohomology homomorphism induced by the basepoint evaluation map Hom(BZ/p, X) → X.) If R ∈ K, M ∈ U(R) and f : R → H is a K-map, we will denote the composite

M →^{ϵ} T (M) → T_{f}(M) by ϵ_{f}.

If f : H → H is the identity map then ϵ_{f} : H → T_{f}(H) is an isomorphism [L1, 4.2], so that by (2.1) T_{f} lifts to a functor U(H) → U(H) (or K(H) → K(H).

Proposition 2.4 (Splitting Property). Let φ : H → H be the identity map and I ⊂ H the kernel of the unique K-map H → F_{p}. Then for any object R of K(H) there is a natural K(H)-isomorphism

T_{φ}(R) ≅ H ⊗_{F_{p}} (T_{φ}(R)/I · T_{φ}(R))

where the action of H on the tensor product is by multiplication on the left-hand factor.

Remark: The proof of Proposition 2.4 also shows that if R ∈ K(H) and M ∈ U(R) there is an isomorphism T_{φ}(M) ≅ H ⊗_{F_{p}} (T_{φ}(M)/I · T_{φ}(M)). The action of T_{φ}(R) on T_{φ}(M) then respects the tensor product splittings of both objects. Proof of 2.4: The adjunction map R → H ⊗_{F_{p}} T (R) can be combined with the Hopf algebra coproduct map H → H ⊗_{F_{p}} H to give a map R → H⊗_{F_{p}}H⊗_{F_{p}}T (R); the adjoint to this is a map T (R) → H⊗_{F_{p}}T (R) which has as quotient a K(H)-map T_{φ}(R) → H ⊗_{F_{p}} T_{φ}(R). The desired isomorphism σ is the composite of this map with the projection

H ⊗_{F_{p}} T_{φ}(R) → H ⊗_{F_{p}} (T_{φ}(R)/I · T_{φ}(R)) .

The fact that φ is the identity map insures that the action of H on the target tensor product is the desired one. Note that σ induces an isomorphism on Tor_{0}^{H}(H/I, –) and an epimorphism on Tor_{1}^{H}(H/I, –) (the latter because Tor_{1}^{H}(H/I, –) vanishes on the free H-module which is the target of σ). Let C be the cokernel of σ and K the kernel. The fact that Tor_{0}^{H}(H/I, C) = C/I · C = 0 implies that C = 0, since I is a connected ideal. The long exact sequence for Tor_{∗}^{H}(H/I, –) then shows that Tor_{0}^{H}(H/I, K) vanishes, which similarly implies K = 0.

§3. Spherical elements

The purpose of this section is to prove Proposition 3.1, which is the algebraic basis of all of the results in this paper.

Let S ⊂ H be the multiplicative subset generated by the non-zero elements of degree 2. The closure of an object M of U(H) is defined to be Un(S^{−1}M); M is closed if the natural map from M to the closure of M is an isomorphism. An element x of an unstable A_{p} module is spherical if αx = 0 for each strictly positive-dimensional element α ∈ A_{p}.

Proposition 3.1. If M ∈ U(H) is non-zero, closed and finitely generated as a module over H then M contains a non-zero spherical element.

Let H_{+} be the subalgebra of H generated by the elements of S; H_{+} is an unstable A_{p} algebra which is isomorphic to the cohomology ring of CP^{∞}. The multiplicative set S is contained in H_{+} and so it is possible to speak of the closure of an object M of U(H_{+}); as above, M is closed if the natural map from M to the closure of M is an isomorphism. An object of U is even-dimensional if it vanishes in odd degrees. Proposition 3.1 is a consequence of the following result.

Proposition 3.2. If M ∈ U(H_{+}) is non-trivial, closed, even-dimension-al, and finitely generated as a module over H_{+}, then M contains a nonzero spherical element.

Proof of 3.1 (given 3.2): We will assume that p is an odd prime since the case p = 2 is a little simpler. Consider the map V : M → M given by

V(x) = \{ \begin{matrix}& P^{k}(x)  |x| = 2k \\ β & P^{k}(x)  |x| = 2k + 1 \end{matrix} .

The map V is not an A_{p}-map, but the image of V is an A_{p}-submodule of M [Li]. Let N be the closure of the H_{+}-submodule of M generated by the image of V. It is clear that N is an A_{p} ⊙ H_{+} submodule of M which is closed, even-dimensional and finitely generated over H_{+}. If N is non-trivial we are done, since by Proposition 3.2 N contains a nonzero spherical element. If N is trivial, then V is the zero map and M is the suspension [Li] of a closed A_{p} ⊙ H module M′. We can argue by induction on the largest degree in which M ⊗_{H_{+}} F_{p} fails to vanish that M′ contains a non-zero spherical element; the suspension of this element is then a non-zero spherical element of M. The induction begins with the fact that any zero-dimensional class is spherical.

In order to prove Proposition 3.2 we will need some additional notation. Let R be an even-dimensional object of K and M an evendimensional object of U(R). Given a degree 2 element a of R and a degree 2k element x of M, define Φ_{a}(x) by the formula

Φ_{a}(x) = ∑_{i=0}^{k}(−1)^{i}a^{i(p−1)}P^{k−i}(x)

(cf. [DW, 3.2]). The Cartan formula shows that Φ_{a}(x) is multiplicative in x whenever this makes sense; in particular, Φ_{a}(a) = 0 implies that Φ_{a}(x) = 0 if x is a-decomposable in M. We will be particularly interested in the case R = H_{+} and will let c denote a chosen fixed degree-two generator of H_{+}.

The following lemma is essentially a reformulation of the Adem relations for the reduced p-th powers.

Lemma 3.3. Let R be an even-dimensional object of K, M an evendimensional object of U(R), and a, b degree 2 elements of R. Then, for any x ∈ M, Φ_{a}Φ_{b}(x) = Φ_{b}Φ_{a}(x).

Proof: Suppose that |x| = 2k. Let F_{2k} be the free object in U on a single generator ι_{2k} of dimension 2k; F_{2k} is isomorphic to the submodule of H^{∗}K(Z/p, 2k) generated by the fundamental class. By a universality argument we can assume that x is the element 1 ⊗ ι_{2k} of R ⊗_{F_{p}} F_{2k}. By [AW, 2.7] we can assume that x is the element 1⊗(c^{⊗k}) of R⊗_{F_{p}}(H_{+})^{⊗k}. In fact, by multiplicativity of Φ we can even prove the lemma by checking the desired relation on the element x = 1 ⊗ c of R ⊗ H_{+}. In this case explicit calculation gives

Φ_{a}(x) = x^{p} − a^{p−1}x = ∏_{i=0}^{p−1}(x + ia)

and hence

Φ_{a}Φ_{b}(x) = ∏_{i=0}^{p−1} ∏_{j=0}^{p−1}(x + ia + jb) .

The lemma follows from the fact that this expression is symmetric in a and b.

Lemma 3.4. Let M ∈ U(H_{+}) be closed and even-dimensional, and let x be an element of M. Then there exists an element y of M such that x = c · y if and only if Φ_{c}(x) = 0.

Proof: If such a y exists, then Φ_{c}(x) = Φ_{c}(c)Φ_{c}(y) = 0. On the other hand, suppose that Φ_{c}(x) = 0 and let |x| = 2k. The Cartan formula shows that P^{k+i}(x/c) = (−1)^{i}c^{i−1}Φ_{c}(x), so that the vanishing of Φ_{c}(x) implies that P^{j}(x/c) vanishes for j > k − 1. Since M is evendimensional, this easily ([Li] [AW, §2]) leads to the conclusion that y = x/c ∈ Un(S^{−1}M) = M. Then x = c · (x/c) = c · y.

Lemma 3.5. Let M be an even-dimensional object of U(H_{+}) and x ∈ M an element of degree 2k with the property that P^{i}x is c-decomposable for each i > 0. Then the action of A_{p} on Φ_{c}(x) is given by the formula

P^{i}Φ_{c}(x) = \binom{k(p − 1)}{i}c^{i(p−1)}Φ_{c}(x) .

Proof: Identify M with the submodule 1 ⊗ M of N = H_{+} ⊗_{F_{p}} M; it is clear that N is an object of U(H_{+} ⊗_{F_{p}} H_{+}). Let a denote the element c⊗1 of H_{+} ⊗H_{+} and b the element 1⊗c. Since P^{i}(x) is b-decomposable for i > 0, it follows from the multiplicativity of Φ that Φ_{b}Φ_{a}(x) depends only on the leading term of Φ_{a}(x), in particular,

Φ_{b}Φ_{a}(x) = Φ_{b}((−1)^{k}a^{k(p−1)}x) = (−1)^{k}(a^{p} − b^{p−1}a)^{k(p−1)}Φ_{b}(x) .

By definition, however,

Φ_{a}Φ_{b}(x) = ∑_{i=1}^{pk}(−1)^{i}a^{i(p−1)}P^{pk−i}Φ_{b}(x) .

The proof is finished by equating these two expressions (Lemma 3.3) and matching up the coefficients of corresponding powers of a. Proof of 3.2: Let 2k be the largest dimension in which M/(c · M) is not zero, and choose an element x ∈ M of dimension 2k which is not divisible by c. It is clear that P^{i}(x) is c-decomposable for each i > 0, so the action of A_{p} on Φ_{c}(x) is given by the formula of Lemma 3.5. A check with the Cartan formula shows that y = c^{−k(p−1)}Φ_{c}(x) is a spherical element of M. The element y is non-zero by Lemma 3.4.

§4 The rank one case

This section contains the proofs of Theorems 1.1, 1.3 and 1.4 in the special case in which the elementary abelian p-group involved is Z/p. Recall from §3 that S denotes the multiplicative subset of H generated by the non-trivial elements of degree 2. Throughout this section φ : H → H will denote the identity map.

Proposition 4.1. If M ∈ U(H) is finitely generated as a module over H there is a natural isomorphism

T_{φ}(M) ≅ Un(S^{−1}M).

Remark: An unstable A_{p} module F is finite if F^{j} is finite-dimensional for all j and zero for almost all j. Suppose that M ∈ U(H) is finitely generated as a module over H. Propositions 2.4 and 4.1 combine to give the surprising fact that Un(S^{−1}M) splits as a tensor product H ⊗_{F_{p}} F for some finite A_{p} module F.

Recall from §2 that T_{φ}(H) ≅ H so that T_{φ}(M) ∈ U(H) if M ∈ U(H). It is easy to see that the map ϵ_{φ} : M → T_{φ}(M) is a map of H-modules.

Lemma 4.2. Let M ∈ U(H) be the tensor product H ⊗_{F_{p}} F, where F ∈ U is finite. Then ϵ_{φ} : M → T_{φ}(M) is an isomorphism.

Proof: Combine Proposition 2.2 and the fact [L1, 4.1] that ϵ : F → T (F) is an isomorphism.

Lemma 4.3. If M ∈ U(H) is finitely generated as a module over H, then the map S^{−1}ϵ_{φ} : S^{−1}M → S^{−1}T_{φ}(M) is an isomorphism.

Proof: Work by induction on the rank rk(S^{−1}M) of S^{−1}M as a module over S^{−1}(H_{+}). Let M′ = Un(S^{−1}M), so that rk(S^{−1}M′) = rk(S^{−1}M) and M′ is closed. If M′ = 0 then T_{φ}(M) vanishes by Proposition 2.3; this case begins the induction.

Suppose then that M′ is not trivial. By Proposition 3.1 M′ contains a non-zero spherical element x. The annihilator ideal I of x in H is closed under the action of the Steenrod algebra, but, since M′ embeds in S^{−1}M′, I contains no element of S. It follows from Proposition 6.4 that I is trivial and thus that the cyclic A_{p} ⊙ H submodule <x> of M′ generated by x is a free H module of rank one. This shows both that rk(S^{−1}M′) > 0 and, by Lemma 4.2, that the map <x> → T_{φ}<x> is an isomorphism. Let M′′ = M/<x>. By induction the map S^{−1}M′′ → S^{−1}T_{φ}(M′′) is an isomorphism. Exactness of T_{φ} and exactness of localization now together imply that S^{−1}M′ → S^{−1}T_{φ}(M′) is an isomorphism. The inductive step is completed by observing that the map M → M′ induces isomorphisms S^{−1}M ≅ S^{−1}M′ and T_{φ}(M) ≅ T_{φ}(M′) (the last by Proposition 2.3).

Proof of 4.1: By Lemma 4.3 the map S^{−1}ϵ_{φ} : S^{−1}M → S^{−1}T_{φ}(M) is an isomorphism. By Proposition 2.4 T_{φ}(M) is a tensor product H ⊗_{F_{p}} F for some F ∈ U, so [DW, 3.6] guarantees that the map T_{φ}(M) → Un(S^{−1}T_{φ}(M)) is an isomorphism. The proposition follows.

Let f : R → H be a map in K(H). Any object M of U(R) is an H module as well as an R module, so it is possible to form T_{φ}(M) as well as T_{f}(M). There is a natural surjection T_{φ}(M) → T_{f}(M).

Lemma 4.4. Let f : R → H be a map in K(H) with kernel I, and let M be an object of U(R). Then for each s ≥ 0

(1) the map T_{f}(M) → T_{f}(M/I^{s}M) is an isomorphism up through dimension s − 1 and

(2) the map T_{φ}(M/I^{s}M) → T_{f}(M/I^{s}M) is an isomorphism. Suppose moreover that R is finitely generated as a ring and M is finitely generated as an R module. Then for each s ≥ 0

(3) there is a natural isomorphism T_{f}(M/I^{s}M) ≅ Un S_{f}^{−1}(M/I^{s}M).

Proof: To prove (1), observe that the map

F_{p} ≅ T_{f}(R)^{0} → T_{f}(R/I)^{0} ≅ F_{p}

is an isomorphism, so by exactness T_{f}(I) vanishes in dimension 0. By Proposition 2.1, T_{f}(I ⊗_{R} I ⊗_{R} ⋯ ⊗_{R} I) (s factors) vanishes through dimension s − 1. The rest follows from exactness of T_{f}. To prove (2), argue from the fact that I/I^{s} is nilpotent to conclude [L1, 4.3.2] that the projection map R/I^{s} → R/I ≅ H induces an isomorphism hom_{K}(H, H) → hom_{K}(R/I^{s}, H) and thus by adjointness an isomorphism T (R/I^{s})^{0} ≅ T (H)^{0}. It follows that there are isomorphisms

T_{f}(M/I^{s}M) = T (M/I^{s}M) ⊗_{T (R)^{0}} F_{p}
≅ T (M/I^{s}M) ⊗_{T (R/I^{s})^{0}} F_{p}
≅ T (M/I^{s}M) ⊗_{T (H)^{0}} F_{p} = T_{φ}(M/I^{s}M) .

Statement (3) now follows easily from (2) and Proposition 4.1. The hypotheses imply that M/I^{s}M is finitely generated as a module over H. It is clear that given an element x of S_{f} there is an element y of S such that the image of x in R/I^{s} differs from the image of y by a nilpotent element; this implies that the natural map S^{−1}(M/I^{s}M) → S_{f}^{−1}(M/I^{s}M) is an isomorphism.

Proof of 1.3 (for V = Z/p): Lemma 4.4(1) implies that there is an isomorphism T_{f}(M) ≅ lim_{←}_{s}T_{f}(M/I^{s}M) and it is clear by inspection

that there is an isomorphism Un((S_{f}^{−1}M)_{I}^{∧}) ≅ lim_{←}_{s} Un S_{f}^{−1}(M/I^{s}M). The theorem follows from Lemma 4.4(3). Proof of 1.4 (for V = Z/p): We will prove only the statement about T (R). Suppose first that R ∈ K(H) and that f : R → H is a K(H) map. By Lemma 4.4(1) and exactness there are isomorphisms

T_{f}(R) ≅ lim_{←}_{s}T_{f}(R/I^{s}) ≅ lim_{←}_{s}T_{f}(R)/T_{f}(I^{s})

and the proof of 4.4(1) shows that the associated graded ring gr T_{f}(R) = ∑_{s} T_{f}(I^{s})/T_{f}(I^{s+1}) is generated as an H algebra by T_{f}(I)/T_{f}(I^{2}) = T_{f}(I/I^{2}). Lemma 4.3(3) and exactness of T_{f} imply that T_{f}(I/I^{2}) is finitely generated as an H module. It follows that T_{f}(R) is finitely generated as a ring.

Now choose R ∈ K, pick a K-map g : R → H, and let f : H ⊗_{F_{p}} R → H be the product of g with the identity map φ of H. The above considerations show that T_{f}(H ⊗_{F_{p}} R) is finitely generated as a ring, and Proposition 2.2 shows that T_{f}(H ⊗_{F_{p}} R) is isomorphic to T_{φ}(H) ⊗_{F_{p}} T_{g}(R) ≅ H ⊗_{F_{p}} T_{g}(R). This proves that T_{g}(R) is finitely generated. There are only a finite number of choices for the map g (since R is finitely generated as a ring) and T (R) is isomorphic to a product ∏_{g} T_{g}(R) indexed by these choices (cf. [L1, 3.5]). This implies that T_{g}(R) is finitely generated.

The next lemma requires an additional bit of notation. For any Kmap f : R → H^{V} let Σ_{f} ⊂ R denote the multiplicative subset of R consisting of all elements x such that f(x) is not a zero-divisor in H^{V}.

Lemma 4.5. Let f : R → H be a map in K(H) and M an object of U(R). Suppose that R is finitely generated as a ring, that M is finitely generated as an R module, and that ϵ_{f} : R → T_{f}(R) is an isomorphism. Then there is a natural isomorphism T_{f}(M) ≅ Un(Σ_{f}^{−1}M).

Proof: Calculating with Lemma 4.4 shows that the map ϵ_{f} : M → T_{f}(M) induces an isomorphism T_{f}(M) → T_{f}T_{f}(M). It follows that T_{f}(N) = 0 if N is either the kernel or cokernel of M → T_{f}(M). Theorem 1.4 guarantees that N is finitely-generated as an R module, so by Lemma 4.4(3) the localization Σ_{f}^{−1}(N/IN) vanishes, and hence by Nakayama’s lemma [AM, p. 21] the localization Σ_{f}^{−1}N itself vanishes. In other words, the map ϵ_{f} induces an isomorphism Σ_{f}^{−1}M → Σ_{f}^{−1}T_{f}(M). Now Proposition 2.3 (together with a finite generation argument) shows that there is an isomorphism T_{f}(M) → T_{f} Un Σ_{f}^{−1}(M) and hence that there is a map ϵ_{f} from Un Σ_{f}^{−1}M to T_{f}(Un Σ_{f}^{−1}M) ≅ T_{f}(M). The above considerations produce a map in the other direction from T_{f}M to Un Σ_{f}^{−1}T_{f}(M) ≅ Un Σ_{f}^{−1}M. The lemma follows easily.

Proof of 1.1 (for V = Z/p): Any subgroup of W is a summand, so the map f : H^{W} → H is split and can be treated as a map in K(H). Since ϵ_{f} : H^{W} → T_{f}(H^{W}) is an isomorphism [L1, 4.2], Lemma 4.5 applies and reduces the proof of the theorem to showing that the natural map Un S_{f}^{−1}(M) → Un Σ_{f}^{−1}(M) is an isomorphism. Let N denote either the kernel or the cokernel of the map ϵ_{f} : M → T_{f}(M) ≅ Un Σ_{f}^{−1}(M). It is clear that N ∈ U(H^{W}) is a finitely generated module over H^{W} with the property that Σ_{f}^{−1}(N) = 0; to finish the proof it is enough to show that any such module N has S_{f}^{−1}(N) = 0, or in other words, it is enough to find an element w of S_{f} which annihilates N. By the finite generation of N there exists an element x of Σ_{f} which annihilates N. Write x = u + v where u belongs to the polynomial subalgebra H_{+}^{W} of H^{W} generated by the image of the Bockstein β : H^{1}BW → H^{2}BW and v is nilpotent. By replacing x with x^{p^{L}} for large L we can in fact assume that x = u ∈ H_{+}^{W}; note that x ≠ 0 because the image of x in H is non-nilpotent. Let I ⊂ H_{+}^{W} be the annihilator ideal of N, and write the radical of I in H_{+}^{W} as an intersection ⋂_{i} ρ_{i} of prime ideals ρ_{i} which are closed under the action of the Steenrod algebra (Proposition 6.1). By Proposition 6.3 each ρ_{i} is generated as an ideal by two-dimensional classes. If any ρ_{i} has all of its two-dimensional generators contained in the kernel of f then I ⊂ ker(f), which is impossible because x ∈ I. Consequently, it is possible to choose from each ρ_{i} a two-dimensional generator w_{i} such that the image of w_{i} in H is non-zero. It is clear that w_{i} ∈ S_{f}. If w is set equal to a sufficiently high power of the product ∏_{i} w_{i}, then w is the desired element in S_{f} ∩ I. Example 4.6: The following example from Smith theory illustrates that it is necessary to include a completion of some type in the statement of Theorem 1.3 . Let p = 2, let R = H^{∗}BO(2) and let f : R → H be the map induced by an homomorphism Z/2 → O(2) sending the non-trivial element of Z/2 to a matrix of determinant −1. Note that the determinant map O(2) → Z/2 lifts R to an H algebra and f to a morphism in K(H). It follows from [L2], say, that T_{f}(R) is isomorphic to H^{∗}(RP^{∞} × RP^{∞}), but there is only one unstable one-dimensional generator in S_{f}^{−1}R. Let w_{1} and w_{2} be the Stiefel-Whitney classes which generate H^{∗}BO(2) as a polynomial algebra. The additional necessary one-dimensional generator appears in the completion of the localization of R as the infinite sum

∑_{i=0}^{∞} w_{1}(w_{2}/w_{1}^{2})^{2^{i}}

(a formula which can be derived by the a posteriori knowledge that this generator α satisfies the equation α^{2} + w_{1}α + w_{2} = 0).

§5 The general case

In this section we will describe the argument which is used to prove Theorems 1.1, 1.3 and 1.4 for a general elementary abelian p-group V. We will use the fact that for V = Z/p these theorems have already been proven in §4.

Let S^{V} ⊂ H^{V} be the multiplicative subset generated by the nonzero elements in the image of the Bockstein map β : H^{1}BV → H^{2}BV and H_{+}^{V} the subalgebra of H^{V} generated by the elements of S^{V}. It is possible to check that the proofs in §2 and §4 hold almost verbatim with H replaced by H^{V}, H_{+} by H_{+}^{V}, T by T^{V} and S by S^{V}; the only exception is the proof of Lemma 4.3 to the extent to which it relies on Lemma 3.1. To complete the proof of the desired results for V, then, it is enough to give an appropriate generalization of Lemma 3.1.

The closure of an object M ∈ U(H^{V}) is defined to be Un(S^{V})^{−1}M; M is closed if the natural map from M to the closure of M is an isomorphism.

Lemma 5.1. If M ∈ U(H^{V}) is non-zero, closed and finitely generated as a module over H^{V} then M contains a non-zero spherical element.

Proof: The argument is by induction on the dimension of V as an F_{p} vector space. By Lemma 3.1, we can assume that this dimension is greater than one. Write V as a direct sum Z/p ⊕ W for some W; this gives a map g : H → H^{V} which lifts H^{V} to K(H), as well as a map f : H^{V} → H such that f · g is the identity map of H. By the special case of Theorem 1.3 proven in §4 there is an isomorphism T_{f}(M) ≅ Un S_{f}^{−1}(M). It follows from Proposition 2.4 that there is a tensor product splitting T_{f}(M) ≅ H ⊗_{F_{p}} N which is compatible with the evident [L1, 4.2] splitting T_{f}(H^{V}) ≅ H^{V} ≅ H ⊗_{F_{p}} H^{W}. Since M is closed, Un S_{f}^{−1}(M) is isomorphic to M. The conclusion is that M splits as a tensor product H ⊗_{F_{p}} N where N is an object of U(H^{W}) which is necessarily non-zero, closed and finitely generated as a module over H^{W}. By induction N contains a non-zero spherical element x. The element 1 ⊗ x ∈ H ⊗_{F_{p}} N ≅ M is the desired spherical element of M.

§6 Some algebraic facts

The purpose of this section is to gather together some standard algebraic results. Proposition 6.3 is used in the proof of Theorem 1.1 (see §4) and Proposition 6.4 in the proof of Lemma 4.3 (particularly in the inductive setting described in §5).

Proposition 6.1. Suppose that R ∈ K is evenly graded and finitely generated as a ring, and that I ⊂ R is a homogeneous ideal which is closed under the action of A_{p}. Then the radical of I in R can be written as an intersection ⋂_{i} ρ_{i} of homogeneous prime ideals ρ_{i} closed under the action of A_{p}.

Proof: Let J be the radical of I (that is, the ideal of all elements x ∈ R such that some power of x lies in I). It is easy to prove by induction on j and the Cartan formula that P^{j}x ∈ J if x ∈ J, in other words, that J is closed under the action of A_{p}. Write J as an intersection ⋂_{i} σ_{i} of (homogeneous) prime ideals (see [AM, Chap. 7]). For each i let σ_{i}^{(1)} be the set of elements x ∈ σ_{i} such that P^{j}x ∈ σ_{i} for all j ≥ 0. The argument of [AW, p. 138] shows that σ_{i}^{(1)} is a prime ideal contained in σ_{i}, although it is not evident that σ_{i}^{(1)} is closed under the action of A_{p}. Iterate the procedure of passing from σ_{i} to σ_{i}^{1} to obtain a descending chain

σ_{i} ⊇ σ_{i}^{(1)} ⊇ σ_{i}^{(2)} ⊇ ⋯

of prime ideals in R. Such a chain must eventually become constant [ZS, p. 241]; let ρ_{i} denote the limiting constant value. It is clear that ρ_{i} is a prime ideal of R which is closed under the action of A_{p} and that J = ⋂_{i} ρ_{i}.

The following propositions use some of the notation of §5.

Proposition 6.2. Let V be an elementary abelian p-group, and suppose that I ⊂ H^{V} is a non-trivial homogeneous ideal closed under the action of A_{p}. Then I contains a non-zero element of H_{+}^{V}.

Proof: Let e_{1}, …, e_{k} be a collection of generators for H^{1}BV and t_{1}, …, t_{k} their Bockstein images in H^{2}BV, so that H_{+}^{V} is the polynomial algebra F_{p}[t_{1}, …, t_{k}] and H^{V} is isomorphic as an algebra to the tensor product of H_{+}^{V} and an exterior algebra on e_{1}, …, e_{k}. Pick a non-zero element x in I. By multiplying x, if necessary, by a suitable product of e_{i}’s we can assume that x has the form te_{1}e_{2} ⋯ e_{k} where t ∈ H_{+}^{V} is non-zero. Define a sequence d_{1}, …, d_{k} of integers inductively by setting d_{1} = |t|/2 + 1 and d_{i+1} = pd_{i} + 1 (i ≥ 1). Define elements α_{0}, …, α_{k−1} of A_{p} by setting α_{0} = β and α_{i+1} = βP^{d_{i+1}}α_{i} (i ≥ 0). A short calculation then shows that

α_{k−1}x = t^{p^{k−1}} ∑_{σ} sgn(σ)t_{σ(1)}^{p^{k−1}}t_{σ(2)}^{p^{k−2}} ⋯ t_{σ(k)}

where σ runs through the permuation group on k letters. It is clear that α_{k−1}x is a non-zero element of I ∩ H_{+}^{V}.

Proposition 6.3. Suppose that V is an elementary abelian p-group and that I ⊂ H_{+}^{V} is a homogeneous prime ideal which is closed under the action of A_{p}. Then I is generated as an ideal by its elements of dimension 2.

Proof: This result is due to Serre ([Se], cf. [AW, 1.11]).

Proposition 6.4. Suppose that V is an elementary abelian p-group and that I ⊂ H^{V} is a non-trivial homogeneous ideal closed under the action of A_{p}. Then I contains an element of S^{V}.

Proof: Let J be the intersection of the radical of I with H_{+}^{V}. By Propositions 6.1 and 6.3, J can be written as an intersection ⋂_{i} ρ_{i} of prime ideals in H_{+}^{V} each of which is generated by elements of dimension 2. Proposition 6.3 guarantees that J is not trivial, so each of the ρ_{i} is also a non-trivial ideal. Pick non-zero two-dimensional elements x_{i} ∈ ρ_{i} and let x be the product of the x_{i}’s. It is clear that some power of x lies in I ∩ S^{V}.

Remark: The top Dickson invariant c_{0} [W2] in H^{V} is by definition the product of the two-dimensional elements of S^{V}. It follows from Proposition 6.4 that any ideal I of the indicated type actually contains some power of c_{0}.

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