Self-maps of modules over the Steenrod algebra

John H. Palmieri

April 1992 Journal of Pure and Applied Algebra 79 (1992) 281–291

Abstract

For a finite module M over the Steenrod algebra A, we prove the existence of a non-nilpotent element in Ext_{A}^{s,t}(M, M) “parallel to the vanishing line.” We use this result to give a proof of Margolis’ construction to kill P_{t}^{s}-homology groups, at all primes.

Let A be the mod p Steenrod algebra. It is well-known (see [8], for example) that its dual A_{∗} is isomorphic (as an algebra) to F_{2}[ξ_{1}, ξ_{2}, …] when p = 2, and to F_{p}[ξ_{1}, ξ_{2}, …] ⊗ E[τ_{0}, τ_{1}, …] when p is odd. Define the Milnor basis of A to be the dual to the monomial basis of A_{∗}; let P_{t}^{s} be the Milnor basis element dual to ξ_{t}^{p^{s}}, and, when p is odd, let Q_{t} be dual to τ_{t}. It is easy to check that when s < t then (P_{t}^{s})^{p} = 0, and for p odd, (Q_{t})^{2} = 0 for all t. Therefore we call these elements differentials, and given any differential x and any A-module M we can define the homology of M with respect to x by:

H(M, P_{t}^{s}) = \frac{ker P_{t}^{s} : M → M}{im (P_{t}^{s})^{p−1} : M → M} ,

and for p odd

H(M, Q_{t}) = \frac{ker Q_{t} : M → M}{im Q_{t} : M → M} .

Given a differential x, define its slope, slope(x), by

slope(P_{t}^{s}) = \frac{p|P_{t}^{s}|}{2},

and

slope(Q_{t}) = |Q_{t}|.

(This notation is motivated by Theorem 1.4.) The differentials are linearly ordered by slope; say that a module M is type ⟨m, n⟩ (also written M = M⟨m, n⟩) if and only if H(M, x) = 0 whenever slope(x) < m or slope(x) > n. We prove the following theorem.

Theorem A Fix a differential x with slope(x) = m. Given a finite module M = M⟨m, ∞⟩ with H(M, x) ≠ 0, there is an element v ∈ Ext_{A}^{k,km}(M, M) for some k which is non-nilpotent under Yoneda composition.

This is a Steenrod algebra analog of the theorem of Hopkins and Smith that any finite p-local spectrum X with K(n−1)_{∗}(X) = 0 and K(n)_{∗}(X) ≠ 0 has a v_{n}-map, a non-nilpotent self-map that induces an isomorphism on K(n)- homology (see [4]). (In the appropriate setting, the map v in Theorem A induces an isomorphism on x-homology.) Theorem A is a generalization of a result used by Hopkins and Smith to prove the theorem for spectra. We actually prove Theorem A for a slightly larger collection of modules than finite ones, namely for “stably finite” modules (see Section 1). In Section 3, for each differential x we construct a stably finite module M = M⟨slope(x), ∞⟩ with H(M, x) ≠ 0, solving the algebraic analog of the problem of constructing a finite spectrum with a v_{n}-map. Note that for each x one can also construct a finite module M of this sort, by taking a tensor product of “pth powers” of sub-Hopf algebras A(n) of A for particular n’s, using Mitchell’s A-module structure. At the prime 2, for example, the module A(1) ⊗ ΦA(1) (where Φ is the doubling functor) is a module of type ⟨7, ∞⟩. See [10].

Using Theorem A, we give a simple proof of

Theorem B ([6] for p = 2) Given any bounded below module M and any integer m ≥ 1,

(a) there exists a bounded below module N = N⟨m, ∞⟩ and a map f : M → N so that H(f, x) is an isomorphism if slope(x) ≥ m; (b) there exists a bounded below module L = L⟨1, m − 1⟩ and a map g :

L → M so that H(g, y) is an isomorphism if slope(y) < m.

In other words, we describe how to kill off homology groups for initial and terminal intervals of differentials in the linear ordering by slope. Theorem B is an important step in developing a chromatic picture for modules over the Steenrod algebra (see [10]).

All of the main results in this paper hold when A is a sub-Hopf algebra of the Steenrod algebra, as long as one restricts attention to the differentials contained in A. For convenience, we state and prove everything in terms of the full Steenrod algebra. For the more general case some changes are needed; in particular, Proposition 1.9 will not hold, so Proposition 1.10 will only be valid for s ≫ 0. The reader can work out the necessary changes in the proofs of the main results.

In Section 1 we review some properties of the x-homology groups and the stable category of modules over the Steenrod algebra; we also introduce the category of stably finite modules. In Section 2 we prove Theorem A, and in Section 3 we prove Theorem B.

I would like to extend my thanks to my thesis advisor Haynes Miller, for teaching me about the Steenrod algebra, and to Mike Hopkins, for teaching me about self-maps, and how to use them to prove results like Theorem B.

1 Preliminaries

We need a few facts about the P_{t}^{s}- and Q_{t}-homology groups defined above. For the differentials x with x^{2} = 0, given a short exact sequence of modules, there is a long exact sequence in homology; there is a variant (which we will not use explicitly) when x^{p} = 0, for p > 2. We have the following two basic results; in [6] Margolis proves both of these at the prime 2, and his proofs extend easily to the general case. Given two A-modules, their tensor product is an A-module via the diagonal action. There is not a Künneth formula for x-homology in general, but the following weaker result does hold.

Proposition 1.1 (19.18 in [6], for p = 2) Let L, M, and N be bounded below A-modules, and let x ∈ A be a differential.

(a) If H(M, x) = 0 or H(N, x) = 0, then H(M ⊗ N, x) = 0.

(b) If f : M → N induces an isomorphism on x-homology, then so does f ⊗ 1_{L} : M ⊗ L → N ⊗ L.

Proposition 1.2 (19.16 in [6], for p = 2) Given a differential x ∈ A and an inverse system of modules M_{α}, with {|M_{α}|} bounded below, lim^{1}M_{α} = 0, and lim^{1}H(M_{α}, x) = 0, then

H(lim M_{α}, x) = lim H(M_{α}, x).

We also recall these two much deeper results:

Theorem 1.3 ([1], [9]) Let M be a bounded below A-module. Then M is free if and only if H(M, x) = 0 for all differentials x ∈ A.

Theorem 1.4 ([2], [7]) If M = M⟨m, ∞⟩ is a bounded below finite type module, then Ext_{A}^{s,t}(M, F_{p}) = 0 if t < ms − c, where c = e − |M| with e independent of M.

Remark 1.5 The statement of the main theorem in [7] is not this precise regarding the form of the intercept c; the proof is, though.

Henceforth, the slope of a line in Ext^{s,t} will refer to its slope in (s, t)- coordinates; for example, the line t = ms − c in Theorem 1.4 has slope m. Similarly, when we speak about phenomena “above” or “below” such a line, we mean in these coordinates; hence, Theorem 1.4 says that Ext_{A}^{s,t}(M, F_{p}) is 0 below a line of slope m. Note that this language conflicts with the usual pictures of Ext^{s,t}, in (t − s, s)-coordinates, where a line with slope m in our sense would be written s = \frac{1}{m−1}(t − s) + d, and Ext_{A}^{s,t}(M, F_{p}) would be 0 above this line.

It is convenient to work in the stable category of A-modules, as defined in [6]: the objects of this category are bounded below A-modules, and the morphisms are given by a bigraded group, written {M, N}_{A}^{∗,∗}. Define

{M, N}_{A}^{0,t} = Hom_{A}^{t}(M, N)/ ≃
= Hom_{A}^{0}(M, Σ^{t}N)/ ≃ ,

where f ≃ 0 (“f is stably trivial”) if and only if f factors through a projective module. To define {M, N}_{A}^{s,t} with s ≠ 0, we need more notation: given M, define an A-module ΩM by the short exact sequence

0 → ΩM → A ⊗ M → M → 0,

and let Ω^{s}M = Ω(Ω^{s−1}M) (and Ω^{0}M = M by convention). Then define

\{M, N\}_{A}^{s,t} = \{ \begin{matrix}\{Ω^{s}M, N\}_{A}^{0,t} & if s ≥ 0, \\ \{M, Ω^{−s}N\}_{A}^{0,t} & if s ≤ 0. \end{matrix}

We say that M is stably equivalent to N, written M ≃ N, if there exist f : M → N and g : N → M with f ∘ g ≃ 1_{N} and g ∘ f ≃ 1_{M}.

Remark 1.6 (1) Note that if there is a short exact sequence of A-modules

0 → K → P → M → 0

with P projective, then K ≃ ΩM. We will use ΩM to denote any of these equivalent modules.

(2) For a differential x ∈ A with x^{2} = 0, then using the long exact sequence in homology and Theorem 1.3, for any k we have

H(Ω^{k}M, x) ≅ Σ^{k·slope(x)}H(M, x);

for a general differential x, we have

H(Ω^{2k}M, x) ≅ Σ^{2k·slope(x)}H(M, x).

(3) To build Ω^{s}M for s = 1, 2, 3, …, one constructs a projective resolution for M. So the next result should not be surprising.

Proposition 1.7 ([6], 14.8) If M and N are bounded below A-modules and s > 0, then Ext_{A}^{s,t}(M, N) ≅ {M, N}_{A}^{s,t}.

In fact, the collection of stable maps satisfies most of the nice homological properties of Ext. We will make free use of these without statement or proof; see Chapter 14 in [6] for a reference.

We need a generalization of “finite module” suitable for use with the stable category. We certainly want to include all finite modules; we also need to include modules which are stably equivalent to finite ones. We also want to be able to take kernels and cokernels. Motivated by work related to the nilpotence theorem (see [3], for example), we make the following definitions.

Call a property P on the category of bounded below A-modules stably generic if the following conditions are satisfied:

(a) If M ⊕ N satisfies P, then so does M.

(b) If 0 → L → M → N → 0 is a short exact sequence of A-modules, and two of L, M, and N satisfy P, then so does the third.

(c) If M is stably equivalent to N and N satisfies P, then M satisfies P.

(d) If M satisfies P, then Σ^{k}M satisfies P for all k ∈ Z.

Let F be the smallest sub-category of the stable category of A-modules which contains all finite A-modules, so that the property “M ∈ F” is stably generic. If M ∈ F, we say that M is stably finite. By definition of F, we have

Proposition 1.8 If P is a stably generic property which holds for all finite A-modules, then P holds for all modules in F.

We combine these definitions with the earlier results on the Steenrod algebra to get a few simple corollaries; first, we need this proposition.

Proposition 1.9 If M is finite and N is any A-module, then for s < 0, {M, N}_{A}^{s,t} = 0.

Proof: We show that if M is a finite module and N is any module, then Hom_{A}(M, ΩN) = 0; this is certainly sufficient. For every x ∈ M, there is an n such that if r > n then Sq^{r}x = 0. On the other hand, since ΩN ↪ P for some projective P, then for every y ∈ ΩN and for every n, there is an r > n such that Sq^{r}y ≠ 0. Therefore there can be no non-trivial A-maps from M to ΩN. □

This gives an immediate generalization of the vanishing line theorem (1.4).

Proposition 1.10 If M and N are stably finite with M = M⟨m, ∞⟩, then {M, N}_{A}^{s,t} = 0 if t < ms − c, for some c depending on |M|, m, and N.

Using stable module language, Theorem 1.4 can be “realized” at the module level; for an A-module M, let ||M|| be the stable connectivity of M, defined by ||M|| = |{M, F_{p}}_{A}^{0,∗}|. (This is also the connectivity of a module stably equivalent to M which contains no free summands—see [6].)

Proposition 1.11 (22.6 in [6], for p = 2) For each m there is an integer e so that for any bounded below module M, we have M = M⟨m, ∞⟩ if and only if ||Ω^{s}M|| ≥ |M| − e + sm for all s ≥ 0.

Proof: First, assume M = M⟨m, ∞⟩. By Theorem 1.4, for all s ≥ 1 we have Ext_{A}^{s,t}(M, F_{p}) = {Ω^{s}M, F_{p}}_{A}^{0,t} = 0 when t < sm−e+|M|, for some e = e(m). Therefore, ||Ω^{s}M|| ≥ |M| − e + sm.

Conversely, assume that H(M, x) ≠ 0 for some x with slope(x) < m. Then H(Ω^{2s}M, x) = Σ^{2s·slope(x)}H(M, x). So for any e, for s ≫ 0, we have ||Ω^{2s}M|| ≤ |H(Ω^{2s}M, x)| = |H(M, x)| + 2s · slope(x) < |M| − e + 2sm. □

2 Constructing self-maps

In this section we prove Theorem A, generalized to stably finite modules (Theorem 2.2). First, we need to describe some particular self-maps. Let A(n) be the sub-Hopf algebra of A generated (as an algebra) by {P_{t}^{s} : s+t ≤ n + 1} for p = 2, and by {P_{t}^{s} : s + t ≤ n} ∪ {Q_{t} : t ≤ n} for p odd. Given a differential x in A(n), let C(x) be the subalgebra of A(n) generated by x (either F_{p}[x]/x^{p} for x = P_{t}^{s}, or E[x] for x = Q_{t}). Then Ext_{C(x)}^{∗∗}(F_{p}, F_{p}) contains as a subalgebra a polynomial algebra on a generator v(x) in bidegree (2, 2 · slope(x)) (see [7], for example). We need this result, due to Lin at the prime 2 and to Wilkerson in the general case.

Lemma 2.1 ([5],[11]) Let x be a differential in A(n). Then we can lift some power of v(x) through the restriction map Ext_{A(n)}^{s,t}(F_{p}, F_{p}) → Ext_{C(x)}^{s,t}(F_{p}, F_{p}).

Now we come to our main result, that modules in F have self-maps parallel to their vanishing lines. Part (a) is a slight generalization of a result of Hopkins and Smith ([4]).

Theorem 2.2 Fix a differential x in A with slope(x) = m. Assume that M ≃ M⟨m, ∞⟩ is stably finite with H(M, x) ≠ 0 (i.e., M ≠ M⟨m + 1, ∞⟩).

(a) Then for some k there is a map v : Ω^{k}M → Σ^{km}M which is nonnilpotent in {M, M}_{A}^{∗,∗}; H(v, x) is an isomorphism, and H(v, y) = 0 when y ≠ x.

(b) Let K = lim_{⟵}(⋯ ⟶^{v}Σ^{−2km}Ω^{2k}M⟶^{v}Σ^{−km}Ω^{k}M⟶^{v}M). Then K

is bounded below, H(K, y) = 0 when y ≠ x, and the map K → M induces an isomorphism in H(−, x). We denote K by M⟨m⟩.

(c) M⟨m⟩ is loop-periodic of period k, via v : Σ^{−km}Ω^{k}M⟨m⟩⟶^{≃}M⟨m⟩.

(d) For stably finite N, we have {M⟨m⟩, N}_{A}^{∗,∗} ≅ v^{−1}{M, N}_{A}^{∗,∗}.

Proof: First we construct the element v ∈ {M, M}_{A}^{k,km}. By Proposition 1.10, {M, M}_{A}^{∗,∗} has a vanishing line of slope m. If n is large enough so that x ∈ A(n), we have an isomorphism in a band parallel to the vanishing line:

{M, M}_{A}^{s,t} ≅ {M, M}_{A(n)}^{s,t}

for t < ms − k(n), where k(n) is some constant depending on n (and on m and M). Indeed, as n → ∞ we have k(n) → ∞, so that for n large enough, the line t = ms is in this band.

If x ∈ A(n), then by Lemma 2.1 we can construct a non-nilpotent element w ∈ {F_{p}, F_{p}}_{A(n)}^{k,km}.

Now choose n ≫ 0. Let C(x) be the subalgebra of A(n) generated by x. Consider this diagram:

{M, M}_{A}^{s,t} ⟶^{ρ} {M, M}_{A(n)}^{s,t} ⟶ {M, M}_{C(x)}^{s,t}
↑^{−⊗M}  ↑^{−⊗M}
{F_{p}, F_{p}}_{A(n)}^{s,t} ⟶ {F_{p}, F_{p}}_{C(x)}^{s,t}

Since H(M, x) ≠ 0, then {F_{p}, F_{p}}_{C(x)}^{∗,∗}⟶{M, M}_{C(x)}^{∗,∗} is injective; the diagram clearly commutes, so the element w ∈ {F_{p}, F_{p}}_{A(n)}^{k,km} maps to a nonnilpotent element v ∈ {M, M}_{A(n)}^{k,km}. Since the restriction map ρ is an isomorphism in that bidegree (and in all bidegrees (j, jm)), we have a non-nilpotent element v ∈ {M, M}_{A}^{k,km}.

We want our self-map v to satisfy certain homology properties, so we may have to use a power of v: we claim that we can choose j so that v^{j} induces an isomorphism in x-homology, and H(v^{j}, y) = 0 when y ≠ x. The first of these is easy—multiplication by v induces an isomorphism on {M, M}_{C(x)}^{∗,∗}, so H(v, x) must be an isomorphism (so any choice of j will work). Now choose j so that v^{j} lies below the line with slope m + 1 and intercept given by the minimum degree of a (vector-space) map from H(M, y) to itself, for all y ≠ x. (By genericity, (max deg H(M, y)−min deg H(M, y)) is uniformly bounded, so this minimum exists.) Then for all y with slope(y) > slope(x), the restriction of v^{j} to {M, M}_{C(y)}^{∗,∗} will be zero; hence H(v^{j}, y) = 0. This finishes (a).

Let v ∈ {M, M}_{A}^{k,km} be the map just constructed. Let K = lim_{⟵}(⋯ → Σ^{−2km}Ω^{2k}M → Σ^{−km}Ω^{k}M → M). By part (a) and Proposition 1.2 we have

H(K, y) = \{ \begin{matrix}H(M, x) & if y = x \\ 0 & if y ≠ x \end{matrix}

There is obviously a map from K to M which is an isomorphism on H(−, x); thus to finish part (b) we only need to show that K is bounded below. This follows from Proposition 1.11: there exists an e independent of M and a choice of Ω^{r}M so that |Σ^{−rm}Ω^{r}M| ≥ |M| − e for all r; therefore the tower is uniformly bounded by |M| − e, so the inverse limit is bounded below. This finishes part (b).

Applying v to the tower that defines K clearly induces a stable equivalence of the inverse limits, proving (c).

Part (d) follows from another vanishing line argument—for any stably finite module N, there is a vanishing line in {M, N}_{A}^{∗,∗} of slope m; we have an isomorphism between {M, N}_{A}^{s,t} and {M⟨m⟩, N}_{A}^{s,t} below a line of slope at least m + 1. By part (c), we know that {M⟨m⟩, N}_{A}^{∗,∗} is v-periodic, and this isomorphism says that {M, N}_{A}^{∗,∗} is v-periodic below a line of slope at least m+1. Hence inverting v extends the periodicity to the whole plane, yielding the result. □

3 Margolis’ killing construction

Our goal here is to prove the following theorem:

Theorem 3.1 (21.1 in [6], for p = 2) Given any bounded below A-module M and any integer m ≥ 1,

(a) there exists a bounded below module N = N⟨m, ∞⟩ and a map f : M → N so that H(f, x) is an isomorphism if slope(x) ≥ m;

(b) there exists a bounded below module L = L⟨1, m − 1⟩ and a map g : L → M so that H(g, x) is an isomorphism if slope(x) < m.

Given a module M, if we have N and f : M → N as in part (a) of the theorem, we say that N is of type M⟨m, ∞⟩, and write N = M⟨m, ∞⟩. Similarly we write L = M⟨1, m − 1⟩ if we have L and g as in part (b). Given a module M and two integers k ≤ m, we can also construct a module of type M⟨k, m⟩, as

M → M⟨k, ∞⟩ ← M⟨k, m⟩

or

M ← M⟨1, m⟩ → M⟨k, m⟩,

where the maps are as in the theorem. We also write M⟨k⟩ for M⟨k, k⟩. One can check that for any integers 0 < k < m ≤ ∞, the modules M⟨k, m⟩ and M⟨k⟩ are unique up to stable equivalence.

Proof of Theorem 3.1: (a) First of all, it suffices to prove the theorem in the case M = F_{p}: if we have F_{p} → F_{p}⟨m, ∞⟩, then given any module M we have

M = M ⊗ F_{p} → M ⊗ F_{p}⟨m, ∞⟩.

By Proposition 1.1, M ⊗ F_{p}⟨m, ∞⟩ is of type ⟨m, ∞⟩, and the map is an isomorphism on H(−, y) when slope(y) ≥ m.

So we need to construct F_{p}⟨m, ∞⟩. Let {x_{0}, x_{1}, …, x_{j}} be the differentials with slope less than m; let s_{i} = slope(x_{i}), and assume that s_{0} < s_{1} < ⋯ < s_{j}.

Lemma 3.2 For 0 ≤ i ≤ j + 1, there are stably finite modules M_{i}, with M_{0} = F_{p} and M_{i} = M_{i}⟨s_{i}, ∞⟩, so that there are short exact sequences

0⟶Σ^{−k_{i}s_{i}}Ω^{k_{i}}M_{i}⟶^{f_{i}}M_{i}⟶M_{i+1}⟶0,

for some integers k_{i} > 0.

Proof: This is a simple consequence of Theorem 2.2—given M_{i} = M_{i}⟨s_{i}, ∞⟩, we construct a self-map f_{i} ∈ {M_{i}, M_{i}}_{A}^{k_{i},k_{i}s_{i}}; then we have f_{i} : Σ^{−k_{i}s_{i}}Ω^{k_{i}}M_{i} → M_{i}. Since Ω^{k_{i}}M_{i} is a sub-module of a projective module, we may arrange that f_{i} is injective by replacing M_{i} by a stably equivalent module. Let M_{i+1} be the cokernel of f_{i}. We know that H(f_{i}, x_{i}) is an isomorphism and H(f_{i}, y) = 0 for y ≠ x_{i}; thus by the long exact sequences in homology, H(M_{i+1}, x_{i}) = 0 and H(M_{i+1}, y) ≠ 0 when slope(y) > s_{i}. The lemma follows by induction. □

Of course, M_{j+1} depends on the (j + 1)-tuple \overline{k} = (k_{0}, k_{1}, …, k_{j}); we write M_{j+1}(\overline{k}) to distinguish them. Assume that at the ith stage one has self-maps f, g ∈ {M_{i}, M_{i}}_{A}^{∗,∗} of bidegrees (k, s_{i}k) and (ℓ, s_{i}ℓ) respectively, and let M_{i+1,k} and M_{i+1,ℓ} denote their cokernels. If k < ℓ, then one can check that there is a map M_{i+1,ℓ} → M_{i+1,k}; hence increasing the k_{i}’s gives us an inverse system of the M_{j+1}(\overline{k})’s. We claim that

F_{p}⟨m, ∞⟩ = lim_{⟵} M_{j+1}(\overline{k}).

Let N denote the inverse limit; we need to check that the effect of F_{p} → N in homology is right, and that the N is bounded below.

For homology, we give the argument for H(−, y) where y^{2} = 0; the other case follows similarly. Using the long exact sequence in homology and the fact that H(f_{i}, y) = 0, at the ith stage in the construction of M_{j+1}(\overline{k}), we have

0 → H(M_{i}, y) → H(M_{i+1}, y) → Σ^{−slope(y)}H(Σ^{−s_{i}k_{i}}Ω^{k_{i}}M_{i}, y) → 0.

If slope(y) < m, then by Proposition 1.2 we see that

H(lim M_{j+1}(\overline{k}), y) = 0.

If slope(y) ≥ m, then by induction we have a vector space isomorphism between the y-homology of M_{j+1}(\overline{k}) and an exterior algebra:

H(M_{j+1}(\overline{k}), y) ≅ ⨂_{i=0}^{j} E[z_{i}],

with |z_{i}| = k_{i}(slope(y) − s_{i}) − slope(y). We have inclusions

H(F_{p}, y) = H(M_{0}, y) ↪ H(M_{1}, y) ↪ ⋯ ↪ H(M_{j+1}, y),

so by Proposition 1.2 we have

H(F_{p}, y) ≅ H(lim M_{j+1}(\overline{k}), y).

Thus the homology of the inverse limit is what it should be.

By applying Proposition 1.11 to Ω^{k_{i}−1}M_{i}(\overline{k}), we can construct an inclusion Σ^{−s_{i}k_{i}}Ω^{k_{i}}M_{i}(\overline{k}) ↪ P, with P projective and |P | ≥ −e(s_{i})−s_{i} +|M_{i}(\overline{k})|; thus |M_{i+1}(\overline{k})| ≥ |P | ≥ −e(s_{i}) − s_{i} + |M_{i}(\overline{k})|. By induction, then, we have

|M_{j+1}(\overline{k})| ≥ − ∑_{i}(e(s_{i}) + s_{i}),

for any \overline{k}; therefore N is bounded below. This finishes part (a).

(b) Again, it suffices to construct F_{p}⟨1, m − 1⟩. By part (a) we have f : F_{p} → F_{p}⟨m, ∞⟩. By replacing F_{p} with a (bounded below) stably equivalent module, we may assume that f is surjective. Let L be the kernel of f; then it is easy to check that L is a module of type F_{p}⟨1, m − 1⟩. □

References

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Massachusetts Institute of Technology