I_{n}-Local Johnson-Wilson Spectra

and their Hopf Algebroids

Andrew Baker

Received: January 30, 2000 Revised: June 25, 2000

Communicated by Max Karoubi

Abstract. We consider a generalization E(n) of the Johnson-Wilson spectrum E(n) for which E(n)_{∗} is a local ring with maximal ideal I_{n}. We prove that the spectra E(n), E(n) and { \atop E(n)} are Bousfield equivalent. We also show that the Hopf algebroid E(n)_{∗}E(n) is a free E(n)_{∗}-module, generalizing a result of Adams and Clarke for KU_{∗}KU.

1991 Mathematics Subject Classification: Primary 55N20 55N22 Keywords and Phrases: Johnson-Wilson spectrum, Hopf algebroid, localization, free module

Introduction

For each prime p and n > 0, the Johnson-Wilson ring spectrum E(n) provides an important example of a p-local periodic ring spectrum. The associated Hopf algebroid E(n)_{∗}E(n) is well known to be flat over E(n)_{∗}, but as far as we are aware there is no proof in the literature that it is a free module for every n. Of course, after passage to the I_{n}-adic completion { \atop E(n)}, and more drastically the I_{n}-adic completion of E(n)_{∗}E(n) (see [4, 8]), such problems disappear. On the other hand, for the ring spectrum KU, the associated Hopf algebroid KU_{∗}KU was shown to be free over KU_{∗} by Frank Adams and Francis Clarke [3, 2, 6]. Actually their approach has two parallel interpretations: one purely algebraic involving stably numerical polynomials [5]; the other topological in that it makes use of the cofibre sequence

Σ^{2}kU ⟶^{t} kU ⟶ Hℤ

induced by the Bott map t: S^{2} ⟶ kU in connective K-theory. In this paper we demonstrate an analogous result by constructing an E(n)_{∗}- basis for E(n)_{∗}E(n) for a generalized Johnson-Wilson spectrum E(n) whose homotopy ring is the (graded) local ring

E(n)_{∗} = (E(n)_{∗})_{I_{n}}.

For completeness, in Section 1 we discuss even more general generalized Johnson-Wilson spectra to which appropriate analogues of our results apply, however we only describe the E(n) case explicitly. Our main result is the following which has some immediate consequences stated in the Corollary.

Theorem. E(n)_{∗}E(n) is a free E(n)_{∗}-module on a countably infinite basis.

Corollary. A) For every E(n)_{∗}-module M_{∗} and s > 0,

Ext_{E(n)_{∗}}^{s,∗}(E(n)_{∗}E(n), M_{∗}) = 0.

In particular,

E(n)^{∗}E(n) = Hom_{E(n)_{∗}}^{∗}(E(n)_{∗}E(n), E(n)_{∗}),

and this is a free E(n)_{∗}-module on an uncountably infinite basis. B) The E(n)-module spectrum E(n) ∧ E(n) is a countable wedge

E(n) ∧ E(n) ≃ ⋁_{α} Σ^{2ℓ(α)}E(n),

where ℓ is some integer valued function of the index α.

Actually, when s ⩾ 2, Ext_{E(n)_{∗}}^{s,∗}(E(n)_{∗}E(n), M_{∗}) = 0 for formal reasons. The statement about E(n)^{∗}E(n) follows from a version of the Universal Coefficient Spectral Sequence of Adams [1]. Our approach to constructing a basis follows a line of argument suggested by that of Adams [2] which also has a purely algebraic interpretation in Adams and Clarke [3, 6]. Although the technology of brave new ring spectra applies to generalized Johnson-Wilson spectra [7, 15], we have no need of such structure, except perhaps to ensure the existence of the relevant Universal Coefficient Spectral Sequence mentioned above; alternatively, M. Hopkins has shown that such spectral sequences exist for all multiplicative cohomology theories constructed using the Landweber Exact Functor Theorem.

I would like to thank Francis Clarke, Neil Strickland and the referee for their helpful comments.

1. Generalized Johnson-Wilson spectra

Given a prime p and n ⩾ 1 we define generalized Johnson-Wilson spectra as follows. Begin with a regular sequence u: u_{0} = p, u_{1}, …, u_{k}, … in BP_{∗} satisfying

u_{k} ∈ BP_{2(p^{k}−1)}, (p, u_{1}, …, u_{k−1}) = I_{k} ◁ BP_{∗},

where I_{k} is actually independent of the choice of generators for BP_{∗}. Of course we have

I_{n}-Local Johnson-Wilson Spectra 353

I_{k} = (p, v_{1}, …, v_{k−1}) = (p, w_{1}, …, w_{k−1}), where v_{j} and w_{j} are the Hazewinkel and Araki generators respectively. There is a commutative ring spectrum BP ⟨n; u⟩ for which

BP ⟨n; u⟩_{∗} = π_{∗}BP ⟨n; u⟩ = BP_{∗}/(u_{j} : j ⩾ n + 1).

We will denote by I_{n} ◁ BP ⟨n; u⟩_{∗} the image of the ideal I_{n} ◁ BP_{∗} under the natural ring homomorphism BP_{∗} ⟶ BP ⟨n; u⟩_{∗}. For any multiplicative set S ⊆ BP ⟨n; u⟩_{∗} containing u_{n} and having I_{n} ∩S = ∅, we can form the localization

E(n; u; S)_{∗} = BP ⟨n; u⟩_{∗} [S^{−1}].

There is a commutative ring spectrum E(n; u; S) with

E(n; u; S)_{∗} = π_{∗}E(n; u; S) = BP_{∗}/(u_{j} : j ⩾ n + 1)[S^{−1}].

Example 1.1. a) When S = {u_{n}^{r} : r ⩾ 1},

E(n; u; {u_{n}^{r} : r ⩾ 1})_{∗} = BP ⟨n; u⟩_{∗} [u_{n}^{−1}].

This ring contains a maximal ideal I_{n} generated by the image of I_{n}◁BP ⟨n; u⟩_{∗}, whose quotient ring is

E(n; u; {u_{n}^{r} : r ⩾ 1})_{∗}/I_{n} = K(n)_{∗}.

This is a mild generalization of the original notion of a Johnson-Wilson spectrum. There is also an I_{n}-adic completion E(n; u; {u_{n}^{r} : r ⩾ 1})_{I_{n}} with homotopy ring (E(n; u; {u_{n}^{r} : r ⩾ 1})_{∗})_{I_{n}}.

b) When S = BP ⟨n; u⟩_{∗} − I_{n},

E(n; u; BP ⟨n; u⟩_{∗} − I_{n})_{∗} = (BP ⟨n; u⟩_{∗})_{I_{n}}.

This is a (graded) local ring with residue (graded) field

E(n; u; BP ⟨n; u⟩_{∗} − I_{n})_{∗}/I_{n} = K(n)_{∗}.

In all cases we have the following which is a consequence of modified versions of standard arguments based on the Landweber Exact Functor Theorem.

Theorem 1.2. For each spectrum E(n; u; S) the following hold.

a) On the category of BP_{∗}BP -comodules, tensoring with the BP_{∗}-module E(n; u; S)_{∗} preserves exactness.

b) E(n; u; S)_{∗}E(n; u; S) is a flat E(n; u; S)_{∗}-module.

c) (E(n; u; S)_{∗}, E(n; u; S)_{∗}E(n; u; S)) is a Hopf algebroid over ℤ_{(p)}.

Setting u_{k} = v_{k}, the Hazewinkel generator, for all k, we obtain the standard connective spectrum BP ⟨n⟩ and the Johnson-Wilson spectra E(n), E(n) for which

π_{∗}E(n) = E(n)_{∗} = BP ⟨n⟩_{∗} [v_{n}^{−1}],
π_{∗}E(n) = E(n)_{∗} = (BP ⟨n⟩_{∗})_{I_{n}}.

Notice that every unit u ∈ E(n)_{∗} has the form

u = av_{n}^{r} + w, (1.1) where a ∈ ℤ_{(p)}^{×} and w ∈ I_{n}; in particular, u ∈ E(n)_{2(p^{n}−1)r}. Of course, unlike the case of E(n), the multiplicative set inverted to form E(n)_{∗} from BP ⟨n⟩_{∗} is infinitely generated. However, for every such unit u arising in BP ⟨n⟩_{∗}, multiplication by U = η_{R}(u) ∈ E(n)_{∗}BP ⟨n⟩ preserves E(n)_{∗}-linearly independent sets by courtesy of the following algebraic result (see for example theorem 7.10 of [12]) and Corollary 2.3 which shows that E(n)_{∗}BP ⟨n⟩ is a free E(n)_{∗}-module.

Proposition 1.3. Let A be a commutative unital local ring with maximal ideal 𝔪. Let M be a flat A-module and (m_{i} : i ⩾ 1) be a collection of elements in M. Suppose that under the reduction map

q: M ⟶ \overline{M} = A/𝔪 ⊗_{A} M,

the resulting collection (q(m_{i}) : i ⩾ 1) of elements in \overline{M} is A/𝔪-linearly independent. Then (m_{i} : i ⩾ 1) is A-linearly independent in M.

We end this section with some remarks intended to justify working with E(n) rather than E(n). For algebraic reasons, our proof of E_{∗}-freeness for E_{∗}E only appears to work for E = E(n) although we conjecture that the result is true for E = E(n). However, there are sound topological reasons for viewing E(n) as a substitute for E(n). Notice that

E(n)_{∗}/I_{n} = E(n)_{∗}/I_{n} = { \atop E(n)}_{∗}/I_{n} = K(n)_{∗}.

Theorem 1.4. The spectra

E(n), E(n), { \atop E(n)}

are Bousfield equivalent. More generally, the spectra

E(n; u; {u_{n}^{r} : r ⩾ 1}), E(n; u; BP ⟨n; u⟩_{∗} − I_{n}), E(n; u; {u_{n}^{r} : r ⩾ 1})_{I_{n}}

are Bousfield equivalent.

Remark 1.5. It is claimed in proposition 5.3 of [10] that E(n) and { \atop E(n)} are Bousfield equivalent. The proof given there is not correct since the extension E(n)_{∗} ⟶ { \atop E(n)}_{∗} is not faithfully flat because I_{n} is not contained in the radical of E(n)_{∗}. We refer the reader to Matsumura [12], especially theorem 8.14(3), for standard algebraic facts concerning faithful flatness. In the following proof, we provide an alternative argument based on the Landweber Filtration Theorem [11].

Proof. For simplicity we only give the proof for the classical case. Since

{ \atop E(n)}_{∗}(X) = { \atop E(n)}_{∗} ⊗_{E(n)_{∗}} E(n)_{∗}(X),

we need only show that { \atop E(n)}_{∗}(X) = 0 implies E(n)_{∗}(X) = 0. Let M_{∗} a BP_{∗}BP -comodule which is finitely generated as a BP_{∗}-module. Then M_{∗} admits a Landweber filtration by subcomodules

0 = M_{∗}^{[0]} ⊆ M_{∗}^{[1]} ⊆ ⋯ ⊆ M_{∗}^{[k]} = M_{∗} such that for each j = 0, …, k,

M_{∗}^{[j]}/M_{∗}^{[j−1]} ≅ BP_{∗}/I_{d_{j}}

for some d_{j} ⩾ 0. The E(n)_{∗}E(n)-comodule

\overline{M}_{∗} = E(n)_{∗} ⊗_{BP_{∗}} M_{∗}

inherits a filtration by subcomodules

0 = \overline{M}_{∗}^{[0]} ⊆ \overline{M}_{∗}^{[1]} ⊆ ⋯ ⊆ \overline{M}_{∗}^{[k]} = \overline{M}_{∗}

satisfying

\overline{M}_{∗}^{[j]}/\overline{M}_{∗}^{[j−1]} ≅ E(n)_{∗}/I_{d_{j}},

where E(n)_{∗}/I_{d_{j}} = 0 if d_{j} > n. For a BP_{∗}-module N_{∗},

{ \atop E(n)}_{∗} ⊗_{E(n)_{∗}} E(n)_{∗} ⊗_{BP_{∗}} N_{∗} ≅ { \atop E(n)}_{∗} ⊗_{BP_{∗}} N_{∗}.

Then writing N̂_{∗} = { \atop E(n)}_{∗} ⊗_{BP_{∗}} N_{∗} we have

M̂_{∗}^{[j]}/M̂_{∗}^{[j−1]} ≅ { \atop E(n)}_{∗}/I_{d_{j}}.

From this it follows that \overline{M}_{∗} = 0 if and only if M̂_{∗}. So { \atop E(n)}_{∗} is faithfully flat in this sense on E(n)_{∗}-comodules of the form \overline{M}_{∗} for some finitely generated BP_{∗}BP -comodule. We can extend this to faithful flatness on all BP_{∗}BP -comodules. Such a comodule N_{∗} is the union of its finitely generated subcomodules, by corollary 2.13 of [13]. For each finitely generated subcomodule M_{∗} ⊆ N_{∗}, the short exact sequence

0 → M_{∗} ⟶ N_{∗} ⟶ N_{∗}/M_{∗} → 0

gives rise to the sequences

0 → \overline{M}_{∗} ⟶ \overline{N}_{∗} ⟶ \overline{N_{∗}/M_{∗}} → 0,

0 → M̂_{∗} ⟶ N̂_{∗} ⟶ { \atop N_{∗}/M_{∗}} → 0.

Each of these is short exact since by the Landweber Exact Functor Theorem, tensor product over BP_{∗} with either of E(n)_{∗} or { \atop E(n)}_{∗} is an exact functor on BP_{∗}-comodules. Suppose that N̂_{∗} = 0; then M̂_{∗} = 0, which implies \overline{M}_{∗} = 0. Since

\overline{N}_{∗} = \underrightarrow{lim}_{M_{∗}⊆N_{∗}} \overline{M}_{∗},

this gives \overline{N}_{∗} = 0. Applying this to the case of N_{∗} = BP_{∗}(X) we obtain the Bousfield equivalence of E(n) with { \atop E(n)}. In the chain of rings E(n)_{∗} ⊆ E(n)_{∗} ⊆ { \atop E(n)}_{∗}, the extension E(n)_{∗} ⟶ { \atop E(n)}_{∗} is faithfully flat, hence E(n) and { \atop E(n)} are also Bousfield equivalent. Alternatively, by the Landweber Exact Functor Theorem, tensoring with E(n)_{∗} is exact on BP_{∗}BP -comodules, so the above proof works as well with E(n) in place of E(n).

This result implies that the stable world as seen through the eyes of each of the homology theories E(n)_{∗}( ), E(n)_{∗}( ) and { \atop E(n)}_{∗}( ) looks the same; indeed this is true for any generalized Johnson-Wilson spectrum between BP ⟨n⟩ and E(n). The proof of the p-local part of the result of Adams and Clarke [3, 2, 6] also involves working over a (graded) local ring (KU_{∗})_{(p)} = ℤ_{(p)}[t, t^{−1}]; of course their result holds over the arithmetically global ring KU_{∗} = ℤ[t, t^{−1}].

2. Some bases for E(n)_{∗}BP and E(n)_{∗}BP ⟨n⟩

We first define a useful basis for E(n)_{∗}BP which projects to a basis for E(n)_{∗}BP ⟨n⟩ under the natural surjective homomorphism of E(n)_{∗}-algebras

q_{n}: E(n)_{∗}BP ⟶ E(n)_{∗}BP ⟨n⟩ .

E(n)_{∗}BP is the polynomial E(n)_{∗}-algebra with the standard generators

t_{k} ∈ E(n)_{2(p^{k}−1)}BP

induced from those for BP_{∗}BP described by Adams [1], where

E(n)_{∗}BP = E(n)_{∗}[t_{k} : k ⩾ 1].

Hence the latter has an E(n)_{∗}-basis consisting of the monomials

t_{1}^{r_{1}} ⋯ t_{ℓ}^{r_{ℓ}} (0 ⩽ r_{k}).

The kernel of q_{n} is the ideal generated by the elements V_{n+k} = η_{R}(v_{n+k}) (k ⩾ 1), where η_{R} is the right unit obtained from the right unit in BP_{∗}BP as the composite

BP_{∗} ⟶^{η_{R}} BP_{∗}BP ⟶ E(n)_{∗}BP.

By well known formulæ for the right unit of BP_{∗}BP, in the ring E(n)_{∗}BP we have

(2.1a)  η_{R}(v_{n+k}) = v_{n}t_{k}^{p^{n}} − v_{n}^{p^{k}}t_{k} + ⋯ + pt_{n+k}

≡ v_{n}t_{k}^{p^{n}} − v_{n}^{p^{k}}t_{k} mod I_{n}. (2.1b)

Here the undisplayed terms are polynomials over BP_{∗} in t_{1}, …, t_{k−1}.

Remark 2.1. The main source of difficulty in working with E(n) itself in place of E(n) seems to arise from the fact that the coefficient of t_{j}^{p^{n}} in Equation (2.1) is then only a unit modulo I_{n}, so we can only use monomials involving the η_{R}(v_{n+k}) as part of a basis when working over E(n)_{∗} rather than just E(n)_{∗}. This is used crucially in the proof of Proposition 2.2. Perhaps a careful choice of generators in place of the Hazewinkel or Araki generators would overcome this problem.

We will also require an expression for the right unit on v_{n}:

η_{R}(v_{n}) = v_{n} + ∑_{1⩽j⩽n} v_{j}θ_{j} ∈ E(n)_{∗}BP, (2.2)

where θ_{j} ∈ E(n)_{2(p^{n}−p^{j})}BP has the form

θ_{j} = t_{n−j}^{p^{j}} mod I_{n}.

In particular θ_{0} = t_{n} mod I_{n}. Although the θ_{j} are not unique, the terms v_{j}θ_{j} mod I_{n}^{2} are well defined. Notice that if u ∈ E(n)_{∗} has the form of Equation (1.1), then for the right unit η_{R}(u) on u,

η_{R}(u) ≡ av_{n}^{r} mod I_{n}.

Now we will define some elements that will eventually be seen to form a basis for E(n)_{∗}BP. First we introduce the following elements of ker q_{n}:

κ_{r_{1},... ,r_{k};s_{1},... ,s_{ℓ}} = v_{n}^{−(s_{1}+···+s_{ℓ})}t_{1}^{r_{1}} ⋯ t_{k}^{r_{k}}V_{n+1}^{s_{1}} ⋯ V_{n+ℓ}^{s_{ℓ}}, (2.3a)

where 0 ⩽ r_{j} ⩽ p^{n} − 1 with r_{k} ≠ 0 and ℓ > 0, s_{j} ⩾ 0 and s_{ℓ} ≠ 0. We also have the elements

κ_{r_{1},... ,r_{k}} = t_{1}^{r_{1}} ⋯ t_{k}^{r_{k}}, (2.3b)

where 0 ⩽ r_{j} ⩽ p^{n} − 1 with r_{k} ≠ 0. The empty sequence corresponds to the element κ_{∅} = 1. There are also elements

\overline{κ}_{r_{1},... ,r_{k}} = q_{n}(κ_{r_{1},... ,r_{k}}) ∈ E(n)_{∗}BP ⟨n⟩ . (2.4)

Next we introduce an increasing multiplicative filtration on E(n)_{∗}BP (apart from a factor of 2 in the indexing, this is the filtration associated with the Atiyah-Hirzebruch spectral sequence for E(n)_{∗}BP),

E(n)_{∗} = E(n)_{∗}BP^{[0]} ⊆ ⋯ ⊆ E(n)_{∗}BP^{[k]} ⊆ ⋯ ⊆ ⋃_{0⩽j} E(n)_{∗}BP^{[j]} = E(n)_{∗}BP.

Here the monomial t_{1}^{r_{1}} ⋯ t_{ℓ}^{r_{ℓ}} has exact filtration ∑_{j} r_{j}(p^{j} − 1). Of course each E(n)_{∗}BP^{[k]} is a finite rank free E(n)_{∗}-module with the basis consisting of all the elements κ_{r_{1},... ,r_{k}} it contains. There are also compatible filtrations ker q_{n}^{[k]}, E(n)_{∗}BP ⟨n⟩^{[k]} and K(n)_{∗}BP^{[k]} on ker q_{n}, E(n)_{∗}BP ⟨n⟩ and K(n)_{∗}BP. Notice that for j ⩾ 0, V_{n+j} has exact filtration (p^{n+j} −1); more generally, the elements defined in Equations (2.3) satisfy

κ_{r_{1},... ,r_{k};s_{1},... ,s_{ℓ}} ∈ E(n)_{∗}BP^{[d]} (2.5)

whenever

d ⩾ ∑_{i} r_{i}(p^{i} − 1) + ∑_{j} s_{j}(p^{n+j} − 1).

Proposition 2.2. The elements

\begin{cases}κ_{r_{1},... ,r_{k}} & for 0 ⩽ r_{j} ⩽ p^{n} − 1, r_{k} ≠ 0, \\ κ_{r_{1},... ,r_{k};s_{1},... ,s_{ℓ}} & for 0 ⩽ r_{j} ⩽ p^{n} − 1, r_{k} ≠ 0, 0 ⩽ s_{j}, s_{ℓ} ≠ 0, ℓ > 0, \end{cases} (2.6)

form an E(n)_{∗}-basis for E(n)_{∗}BP.

Proof. Since

E(n)_{∗}BP = ⋃_{j⩾0} E(n)_{∗}BP^{[m]}

it suffices to show that for each m ⩾ 0, the κ elements specified in Equation (2.6) and also contained in E(n)_{∗}BP^{[m]} actually form a basis for E(n)_{∗}BP^{[m]}. E(n)_{∗}BP^{[m]} has a natural basis consisting of all the t monomials t_{1}^{r_{1}} ⋯ t_{k}^{r_{k}} (r_{j} ⩾ 0) it contains. Notice that the number of κ elements in E(n)_{∗}BP^{[m]} is the same as the number of such monomials, hence is equal to the rank of E(n)_{∗}BP^{[m]}. Let M(m) be the Gram matrix over E(n)_{∗} expressing the κ elements in terms of the t monomial basis, with suitable orderings on these elements. It suffices to show that M(m) is invertible, and for this we need to show that det M(m) is a unit in E(n)_{∗}. As E(n)_{∗} is local, this is true if det M(m) mod I_{n} is a unit. We have

κ_{r_{1},... ,r_{k};s_{1},... ,s_{ℓ}} ≡ t_{1}^{r_{1}} ⋯ t_{k}^{r_{k}}(t_{1}^{p^{n}} − v_{n}^{p−1}t_{1})^{s_{1}} ⋯ (t_{ℓ}^{p^{n}} − v_{n}^{p^{ℓ}−1}t_{ℓ})^{s_{ℓ}} mod I_{n}

(2.7) ≡ t_{1}^{r_{1}+p^{n}s_{1}} ⋯ t_{ℓ}^{r_{ℓ}+p^{n}s_{ℓ}} + (terms of lower filtration) mod I_{n}.

Working modulo I_{n} in terms of the basis of t monomials, the Gram matrix for the κ elements is lower triangular with all diagonal terms being 1, therefore det M(m) ≡ 1 mod I_{n}. So det M(m) is a unit and M(m) is invertible. Thus the κ elements of E(n)_{∗}BP^{[m]} form a basis.

Corollary 2.3. The short exact sequence of E(n)_{∗}-modules

0 → ker q_{n} ⟶ E(n)_{∗}BP ⟶^{q_{n}} E(n)_{∗}BP ⟨n⟩ → 0

splits so there is an isomorphism of E(n)_{∗}-modules

E(n)_{∗}BP ≅ ker q_{n} ⊕ E(n)_{∗}BP ⟨n⟩ .

Also, E(n)_{∗}BP ⟨n⟩ and ker q_{n} are free E(n)_{∗}-modules.

3. E(n)_{∗}E(n) as a limit

In this section we will give a description of E(n)_{∗}E(n) as a colimit. Although we proceed algebraically, we note that this limit has topological origins since for each u ∈ BP ⟨n⟩_{2(p^{n}−1)r} with r > 0 and which is a unit in E(n)_{∗}, there is a cofibre sequence

Σ^{2(p^{n}−1)r}BP ⟨n⟩ ⟶^{u} BP ⟨n⟩ ⟶ BP ⟨n − 1; u⟩

and E(n) is the telescope

E(n) = Tel_{u} BP ⟨n⟩ .

On applying the functor E(n)_{∗}( ), there is a short exact sequence

0 → E(n)_{∗}BP ⟨n⟩ ⟶^{U} E(n)_{∗}BP ⟨n⟩ ⟶ E(n)_{∗}BP ⟨n − 1; u⟩ → 0,

and limit

E(n)_{∗}E(n) ≅ \underrightarrow{lim}_{U} E(n)_{∗}BP ⟨n⟩ ,

in which U denotes multiplication by the right unit on u. Since u ≡ av_{n}^{r} mod I_{n} in the notation of Equation (1.1), application of the functor K(n)_{∗}( ) induces another exact sequence and limit

0 → K(n)_{∗}BP ⟨n⟩ ⟶^{U} K(n)_{∗}BP ⟨n⟩ ⟶ K(n)_{∗}BP ⟨n − 1; u⟩ = 0,
K(n)_{∗}E(n) ≅ \underrightarrow{lim}_{U} K(n)_{∗}BP ⟨n⟩ .

There are also algebraic identities

E(n)_{∗}E(n) ≅ E(n)_{∗} ⊗_{BP_{∗}} BP_{∗}BP ⊗_{BP_{∗}} E(n)_{∗},
E(n)_{∗}BP ⟨n⟩ ≅ E(n)_{∗}BP/ ker q_{n},
K(n)_{∗}BP ⟨n⟩ ≅ K(n)_{∗} ⊗_{E(n)_{∗}} E(n)_{∗}BP ⟨n⟩ ≅ K(n)_{∗} ⊗_{BP_{∗}} BP_{∗}BP ⟨n⟩ ,

which allow us to work without direct reference to the underlying topology. First we describe a directed system (Λ, ≼). Recall that BP ⟨n⟩_{∗} is a graded unique factorization domain, with group of units BP ⟨n⟩_{∗}^{×} = ℤ_{(p)}^{×}. Define the sets

Λ_{r} = {(u) ◁ BP ⟨n⟩_{∗} : u ∈ BP ⟨n⟩_{2(p^{n}−1)r} , u ∈ E(n)_{∗} is a unit} (r ⩾ 0),

Λ = ⋃_{r⩾0} Λ_{r}.

We will often abuse notation and identify (u) with a generator u; this can be made precise by specifying a choice function to select a generator of each such principal ideal. Of course, (u) = (v) if and only if there is a unit a ∈ ℤ_{(p)}^{×} for which u = av, i.e., if u | v and v | u in BP ⟨n⟩_{∗}. We will write u ≼ v if (v) ⊆ (u), i.e., if u | v. We will also write u ≺ v if u ≼ v and (u) ≠ (v). The directed system (Λ, ≼) is filtered since for u, v ∈ Λ, u ≼ uv and v ≼ uv.

Remark 3.1. For later use we will need a cofinal subset of Λ and we now describe some obvious examples. Since BP ⟨n⟩_{∗} is a countable unique factorization domain, we may list the distinct prime ideals lying in Λ as (w_{1}), (w_{2}), (w_{3}), … say. Now inductively define

u_{0} = 1, u_{k} = u_{k−1}^{k}w_{k}.

Then u_{k−1} | u_{k} and indeed u_{k−1} ≺ u_{k}. Also, for every element (u) ∈ Λ there is a k such that u | u_{k}, hence u ≼ u_{k}. So the u_{k} form a cofinal sequence in Λ.

Now form the directed system consisting of pairs of the form (BP ⟨n⟩_{∗} , u) with u ∈ Λ. If u, v ∈ Λ, the morphism (BP ⟨n⟩_{∗} , u) ⟶ (BP ⟨n⟩_{∗} , uv) is multiplication by v,

BP ⟨n⟩_{∗} ⟶^{v} BP ⟨n⟩_{∗} .

On setting V = η_{R}(v), there is also a homomorphism

E(n)_{∗}BP ⟨n⟩ ⟶^{V} E(n)_{∗}BP ⟨n⟩ .

These give rise to limits

(3.1)  E(n)_{∗} = \underrightarrow{lim}_{u∈Λ} BP ⟨n⟩_{∗} = (BP ⟨n⟩_{∗})_{I_{n}},
(3.2)  E(n)_{∗}E(n) = \underrightarrow{lim}_{u∈Λ} E(n)_{∗}BP ⟨n⟩ = (E(n)_{∗}BP ⟨n⟩)_{η_{R}I_{n}}.

Remark 3.2. In describing E(n)_{∗}E(n) as a limit, it suffices to replace each map V by

E(n)_{∗}BP ⟨n⟩ ⟶^{v^{−1}V} E(n)_{∗}BP ⟨n⟩ ,

which is of degree 0 and satisfies

v^{−1}V ≡ 1 mod I_{n}. (3.3)

This will simplify the description of our basis. Notice that if (v) = (w) ◁ BP ⟨n⟩_{∗}, then

v^{−1}V = w^{−1}W,

providing another reason for using v^{−1}V in place of V. From now on we will consider E(n)_{∗}E(n) as the limit over such maps v^{−1}V rather than the limit of Equation (3.2).

4. Some bases for E(n)_{∗}BP ⟨n⟩ and E(n)_{∗}E(n)

For each pair (u, s) with u ∈ Λ_{r} and s a non-negative integer, set

M(u; s)_{∗} = E(n)_{∗}BP ⟨n⟩^{[s+r(p^{n}−1)]} .

By Corollary 2.3, M(u; s)_{∗} is free on the images under q_{n} of the κ_{r_{1},... ,r_{k}} defined in Proposition 2.2 and we refer to this as the q_{n}κ-basis. There are inclusion maps

inc: M(u; s)_{∗} ⟶ M(u; s + 1)_{∗}.

For v ∈ Λ_{t} and V = η_{R}(v), there is a multiplication by v^{−1}V map

v^{−1}V : M(u; s)_{∗} ⟶ M(uv; s)_{∗}.

By Equation (2.2), v^{−1}V raises filtration by t(p^{n} − 1). Equation (3.3) and Proposition 1.3 imply that v^{−1}V is also injective; indeed we have the following result.

Proposition 4.1. Let s ⩾ 0 and u, v ∈ Λ. The E(n)_{∗}-submodule

v^{−1}V M(u; s)_{∗} ⊆ M(uv; s)_{∗}

is a summand. Furthermore, if B is a basis for M(u; s)_{∗} then M(uv; s)_{∗} has a basis consisting of the elements

v^{−1}V b (b ∈ B), \overline{κ}_{r_{1},... ,r_{k}} ∈ M(uv; s)_{∗} − v^{−1}V M(u; s)_{∗}.

Proof. M(u; s)_{∗} and M(uv; s)_{∗} each have the q_{n}κ-bases. After reduction modulo I_{n}, the stated elements in K(n)_{∗}BP ⟨n⟩ satisfy

v^{−1}V b = b ∈ K(n)_{∗}BP ⟨n⟩^{[d+s]} ,
\overline{κ}_{r_{1},... ,r_{k}} ∈ K(n)_{∗}BP ⟨n⟩^{[d+h+s]} − K(n)_{∗}BP ⟨n⟩^{[d+s]} ,

where u and v have exact filtrations d and h. These elements are clearly K(n)_{∗}-linearly independent, so by Equation (3.3) and Proposition 1.3 they are E(n)_{∗}-linearly independent. Thus they form a basis, so the exact sequence

0 → M(u; s)_{∗} ⟶^{v^{−1}V} M(uv; s)_{∗} ⟶ M(uv; s)_{∗}/v^{−1}V M(u; s)_{∗} → 0

splits and there is a direct sum decomposition

M(uv; s)_{∗} = v^{−1}V M(u; s)_{∗} ⊕ M(uv; s)_{∗}/v^{−1}V M(u; s)_{∗}.

The E(n)_{∗}-linear maps v^{−1}V and inc commute and together form a doubly directed system. Then we have

E(n)_{∗}E(n) = \underrightarrow{lim}_{(u,s)} M(u; s)_{∗}

= \underrightarrow{lim}_{u} \underrightarrow{lim}_{s} M(u; s)_{∗}

= \underrightarrow{lim}_{s} \underrightarrow{lim}_{u} M(u; s)_{∗}.

Each M(u; s)_{∗} is a finitely generated free E(n)_{∗}-module, with a basis consisting of the \overline{κ} elements it contains; we will refer to this as its \overline{κ}-basis. M(u; s)_{∗} also has another useful basis which we will now define. Choose a cofinal sequence u_{k} in Λ, for example by the process described in Remark 3.1. For convenience we will assume that u_{0} = 1. Of course

E(n)_{∗}E(n) = \underrightarrow{lim}_{(r,s)} M(u_{r}; s)_{∗}

= \underrightarrow{lim}_{r} \underrightarrow{lim}_{s} M(u_{r}; s)_{∗}

= \underrightarrow{lim}_{s} \underrightarrow{lim}_{r} M(u_{r}; s)_{∗}.

When r = 0, we take the \overline{κ}-basis for M(1; s)_{∗}, denoting its elements by \overline{κ}_{r_{1},... ,r_{k}}^{1;s}. Now for r ⩾ 1, suppose that we have defined a basis \overline{κ}_{r_{1},... ,r_{k}}^{u_{r−1};s} for M(u_{r−1}; s)_{∗}.

For M(u_{r}; s)_{∗}, replace each \overline{κ}_{r_{1},... ,r_{k}}^{r−1;s} of this basis by

(4.1)  \overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s} = w_{r}^{−1}W_{r}\overline{κ}_{r_{1},... ,r_{k}}^{u_{r−1};s}
≡ \overline{κ}_{r_{1},... ,r_{k}}^{u_{r−1};s} mod I_{n}

whenever this element is also in M(u_{r}; s)_{∗}. For w_{r}^{−1}W_{r}\overline{κ}_{r_{1},... ,r_{k}}^{u_{r−1};s} ∉ M(u_{r}; s)_{∗}, set

\overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s} = \overline{κ}_{r_{1},... ,r_{k}}^{u_{r−1};s}. (4.2)

Notice that by repeated applications of Equation (3.3), we have for all basis elements,

\overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s} ≡ \overline{κ}_{r_{1},... ,r_{k}} mod I_{n}. (4.3)

Next we consider the effect of raising s by considering the extension

M(u_{r}; s)_{∗} ⊆ M(u_{r}; s + 1)_{∗}.

Clearly M(u_{r}; s + 1)_{∗} contains all the elements \overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s} together with its \overline{κ}- basis elements of exact filtration d_{r} + s + 1 where d_{r} is the exact filtration of u_{r}. Reducing modulo I_{n} these elements are K(n)_{∗}-linearly independent, so by Equation (4.3) and Proposition 1.3 these are E(n)_{∗}-linearly independent and hence form a basis, showing that this extension splits. We have demonstrated the following.

Proposition 4.2. For r, s ⩾ 0, the E(n)_{∗}-module M(u_{r}; s)_{∗} is free with the following two bases:

• B_{1}^{u_{r};s} consisting of the elements \overline{κ}_{r_{1},... ,r_{k}} contained in M(u_{r}; s)_{∗};

• B_{2}^{u_{r};s} consisting of the elements \overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s}.

Now we can state our main result.

Theorem 4.3. E(n)_{∗}E(n) is E(n)_{∗}-free with a basis consisting of the images of the non-zero elements of the form

\overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s} ∈ M(u_{r}; s)_{∗} − w_{r}^{−1}W_{r}M(u_{r−1}; s)_{∗} (r, s ⩾ 0)

under the natural map M(u_{r}; s)_{∗} ⟶ E(n)_{∗}E(n).

Proof. We begin by showing that these elements span E(n)_{∗}E(n). Let z ∈ E(n)_{∗}E(n) and suppose that t is the image of z_{r} ∈ M(u_{r}; s)_{∗} under the natural map

M(u_{r}; s)_{∗} ⟶ E(n)_{∗}E(n).

Then z_{r} can be uniquely expressed as an E(n)_{∗}-linear combination

z_{r} = ∑_{r_{1},... ,r_{k}} λ_{r_{1},... ,r_{k}} \overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s}.

We can split up this sum as

z_{r} = ( ∑_{r_{1},... ,r_{ℓ}} λ_{r_{1},... ,r_{ℓ}} \overline{κ}_{r_{1},... ,r_{ℓ}}^{u_{r−1};s}) + w_{r}^{−1}W_{r} ( ∑_{s_{1},... ,s_{k}} λ_{s_{1},... ,s_{k}} \overline{κ}_{s_{1},... ,s_{k}}^{u_{r−1};s}) .

Since

∑_{r_{1},... ,r_{ℓ}} λ_{r_{1},... ,r_{ℓ}} \overline{κ}_{r_{1},... ,r_{ℓ}}^{u_{r−1};s} ∈ M(u_{r−1}; s)_{∗}, ∑_{s_{1},... ,s_{k}} λ_{s_{1},... ,s_{k}} \overline{κ}_{s_{1},... ,s_{k}}^{u_{r−1};s} ∈ M(u_{r}; s)_{∗}

map to linear combinations of the asserted basis elements in the images of M(u_{r−1}; s)_{∗} and M(u_{r−1}; s)_{∗} in E(n)_{∗}E(n), z is also a linear combination of those basis elements. Now we show that these elements are linearly independent over E(n)_{∗}E(n). We know that E(n)_{∗}E(n) is E(n)_{∗}-flat, and also that

K(n)_{∗} ⊗_{E(n)_{∗}}E(n)_{∗}E(n) = K(n)_{∗}E(n)
( = K(n)_{∗}K(n) in the standard but misleading notation)

which has a K(n)_{∗}-basis consisting of the reductions of the elements

t_{1}^{r_{1}} ⋯ t_{k}^{r_{k}} (0 ⩽ r_{j} ⩽ p^{n} − 1).

Now t_{1}^{r_{1}} ⋯ t_{k}^{r_{k}} is the image of \overline{κ}_{r_{1},... ,r_{k}}^{u_{r};s} ∈ M(u_{r}; s) under the natural map. Careful book keeping shows that the asserted basis elements do indeed account for all the t_{j}-monomials in this basis of K(n)_{∗}E(n). These are linearly independent in E(n)_{∗}E(n) by Proposition 1.3.

The following useful consequence of our construction is immediate on taking

E(n)_{∗}BP ⟨n⟩ = \underrightarrow{lim}_{s} M(1; s)_{∗}.

Corollary 4.4. The natural map

E(n)_{∗}BP ⟨n⟩ ⟶ E(n)_{∗}E(n)

is a split monomorphism of E(n)_{∗}-modules.

References

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Andrew Baker

Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland. a.baker@maths.gla.ac.uk www.maths.gla.ac.uk/∼ajb