DIRECT SUM DECOMPOSITIONS OF INFINITELY GENERATED MODULES

D. J. BENSON AND WAYNE W. WHEELER

Abstract. Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull–Schmidt Theorem. Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally. In this paper we present an example showing that Green’s Indecomposability Theorem fails for infinitely generated modules. By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails.

1. Introduction

The representation theory of finite groups has traditionally concentrated on finitely generated modules over the group algebra. Recent work [1, 3, 4, 5] has made it clear, however, that infinitely generated modules play an important role even if one’s real goal is a better understanding of finitely generated modules. For this reason it should be useful to try to extend known results in representation theory by omitting the finiteness conditions on the modules. One of the main objectives of this paper is to study the analogue of Green’s Indecomposability Theorem in this greater generality.

Suppose that G is a finite group and k is a field of characteristic p > 0. Green’s Indecomposability Theorem [10] states that if H is a normal subgroup of index p in G and M is a finitely generated absolutely indecomposable kH-module, then the induced module M↑^{G} is also absolutely indecomposable. The usual proof of this result depends upon the Krull–Schmidt Theorem. Work of Brenner and Ringel [7] shows, however, that for infinitely generated modules the Krull–Schmidt Theorem fails spectacularly whenever G has noncyclic Sylow p-subgroups. In Section 2 we present a counterexample showing that Green’s result also fails without the assumption that M is finitely generated.

Sections 3 and 4 are devoted to studying the failure of the Krull–Schmidt property in somewhat more detail. This work begins by considering endomorphism rings of certain types of infinitely generated modules. The arguments of Section 4 then show, in effect, that

Date: October 1997.

Both authors are partially supported by the NSF.

2 D. J. BENSON AND WAYNE W. WHEELER

every complete intersection is isomorphic to End_{kG}(M)/I for some elementary abelian pgroup G, some countably generated module M, and some nilpotent ideal I of End_{kG}(M). If n ≥ 5 and G is elementary abelian of rank 2n + 1, then Raynaud’s examples of modules that are stably free but not free [12] allow us to exhibit modules M and N such that M is indecomposable and

M^{n} ≅ M ⊕ N

but M^{n−1} ≅ N̸. Such examples also exist for certain smaller values of n. If kG has infinite representation type, then the work of Brenner and Ringel proves the existence of modules for which the cancellation property fails in this way. Their methods leave it far from obvious what such modules might look like, however, and we believe that it is a valuable exercise to construct explicit examples as we have done.

2. Green’s Indecomposability Theorem

The goal of this section is to produce an example showing that the statement of Green’s Indecomposability Theorem fails for infinitely generated modules. The example is based upon Rickard’s construction of idempotent modules in [13]. We begin with a brief discussion of the necessary definitions and preliminary results.

Let G be a finite group, and let k be an algebraically closed field of characteristic p > 0. Recall that the objects of the stable category kG-\underline{Mod} consist of all left kG-modules, and the morphisms are equivalence classes of kG-homomorphisms; two homomorphisms f : M → N and g : M → N are equivalent if f − g factors through a projective module. The collection of morphisms from M to N in kG-\underline{Mod} is denoted by \underline{Hom}_{kG}(M, N). Because kG is a self-injective algebra, kG-\underline{Mod} is a triangulated category (Theorem I.2.6 of [11]), and the translation functor is given by Ω^{−1}. The full subcategory of kG-\underline{Mod} consisting of all finitely generated kG-modules is denoted by kG-\underline{mod}.

A triangulated subcategory C of kG-\underline{mod} is thick if it is closed under taking direct summands of objects. If C is a thick subcategory of kG-\underline{mod} such that M ⊗ N is in C whenever M is in C and N is in kG-\underline{mod}, then we say that C is an ideal subcategory of kG-\underline{mod}. Let C^{⊕} denote the smallest full triangulated subcategory of kG-\underline{Mod} that contains C and is closed under arbitrary direct sums, and write C^{⊥} for the subcategory of kG-\underline{Mod} consisting of all modules M such that \underline{Hom}_{kG}(C, M) = 0 for all C ∈ C. The objects of C^{⊥} are said to be C-local. With this notation and terminology we can now state the following result of Rickard [13].

DIRECT SUM DECOMPOSITIONS 3

Proposition 2.1. Let C be an ideal subcategory of kG-\underline{mod}. For any object M in kG-\underline{Mod} there is a triangle

E_{C}(M) → M → F_{C}(M) → Ω^{−1}(E_{C}(M))

such that E_{C}(M) is in C^{⊕} and F_{C}(M) is in C^{⊥}, and such a triangle is unique up to isomorphism. The morphism E_{C}(M) → M is the universal map in kG-\underline{Mod} from an object of C^{⊕} to M, and M → F_{C}(M) is the universal map from M to an object of C^{⊥}. The modules E_{C}(k) and F_{C}(k) are idempotent in the sense that E_{C}(k) ⊗ E_{C}(k) ≅ E_{C}(k) and F_{C}(k) ⊗ F_{C}(k) ≅ F_{C}(k) in kG-\underline{Mod}. Moreover, M ⊗ E_{C}(k) ≅ E_{C}(M) and M ⊗ F_{C}(k) ≅ F_{C}(M) in kG-\underline{Mod} for any module M.

For the remainder of this section k denotes an algebraically closed field of characteristic two. Let H be an elementary abelian group of order four generated by commuting involutions h_{1} and h_{2}, and let G be the group generated by H and an involution g satisfying gh_{1} = h_{1}h_{2}g and gh_{2} = h_{2}g. Then H is a normal subgroup of G of index two, and G is isomorphic to the dihedral group D_{8}. Set X_{i} = h_{i} − 1 ∈ kH for i = 1, 2. If we identify H^{1}(H, k) with the vector space Hom(H, k) of group homomorphisms into k, then there are elements ζ_{1}, ζ_{2} ∈ H^{1}(H, k) satisfying ζ_{1}(h_{1}) = ζ_{2}(h_{2}) = 1 and ζ_{1}(h_{2}) = ζ_{2}(h_{1}) = 0, and H^{∗}(H, k) is the polynomial ring generated by ζ_{1} and ζ_{2}. Let C be the subcategory of kH-\underline{mod} consisting of all finitely generated kH-modules N whose variety V_{H}(N) is contained in V_{H}(ζ_{2}), and let M = F_{C}(k) be the corresponding idempotent kH-module. More explicitly, M can be described as the module with generators x_{0}, x_{1}, x_{2}, … and relations

X_{1}x_{i} + X_{2}x_{i+1} = 0 for all i ≥ 0.  (2.1)

This module has a unique kH-endomorphism α satisfying α(x_{i}) = x_{i+1}. The relations (2.1) can be rewritten in terms of the endomorphism α as

(X_{1} + X_{2}α)m = 0 for all m ∈ M.  (2.2)

Using Proposition 7.3 of [9] and the techniques of [8], one can show that \underline{End}_{kH}(M) = k[α]. This fact is also easy to prove directly.

If the commutative ring kH[α] is regarded as a free left kH-module, then there is a kHhomomorphism onto M sending α^{i} to x_{i} for i ≥ 0. Thus by (2.2) there is an isomorphism of kH-modules

M ≅ kH[α]/(X_{1} + X_{2}α).  (2.3)

The conjugation action of g on kH extends to an action on the ring kH[α] satisfying g^{−1}αg = α + 1, and it is easy to check that g induces an action on the quotient in (2.3).

4 D. J. BENSON AND WAYNE W. WHEELER

Under the kH-isomorphism (2.3) this action of g determines a k-linear map ϕ : M → M satisfying

ϕ(xm) = (g^{−1}xg)ϕ(m)

for all x ∈ kH and all m ∈ M. The map ϕ is given on the generators of M by

ϕ(x_{i}) = ϕ(α^{i}(x_{0})) = (α + 1)^{i}(x_{0}) = ∑_{j=0}^{i} \binom{i}{j}x_{j}.

Now consider the induced module M↑^{G} = M ⊕ (g ⊗ M). The map m ↦ g ⊗ ϕ(m) is a kH-isomorphism from M to g ⊗ M; its inverse is given by g ⊗ m ↦ ϕ(m) because g is an involution so that ϕ^{2} = 1_{M}. Corresponding to this isomorphism is a kG-endomorphism ψ : M↑^{G} → M↑^{G} given by

ψ(m_{1}, g ⊗ m_{2}) = (ϕ(m_{2}), g ⊗ ϕ(m_{1})).

In addition, there is an endomorphism a of M↑^{G} given by

a(m_{1}, g ⊗ m_{2}) = (α(m_{1}), g ⊗ α(m_{2}));

this is the map induced from the kH-endomorphism α of M. If the isomorphism m ↦ g ⊗ ϕ(m) is used to identify the kH-modules M and g ⊗ M, then End_{kH}(M↑^{G}↓_{H}) is isomorphic to Mat_{2}(End_{kH}(M)). It is helpful to think of the endomorphisms a and ψ as matrices:

a = (\begin{matrix}α & 0 \\ 0 & α + 1 \end{matrix}) ψ = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) .

Lemma 2.2. The endomorphism ring \underline{End}_{kG}(M↑^{G}) is isomorphic to the k-algebra with generators a and ψ and relations

aψ = ψ(a + 1), ψ^{2} = 1.

Proof. Because End_{kG}(M↑^{G}) is contained in End_{kH}(M↑^{G}↓_{H}), it can be regarded as a subring of Mat_{2}(End_{kH}(M)). The composition of the isomorphisms

End_{kH}(M) ⊕ End_{kH}(M) ≅ End_{kH}(M) ⊕ Hom_{kH}(g ⊗ M, M)
≅ Hom_{kH}(M↑^{G}↓_{H}, M)
≅ End_{kG}(M↑^{G})

is given by the map

(β, β′) ↦ (\begin{matrix}β & β′ \\ ϕβ′ϕ & ϕβϕ \end{matrix}) .

Moreover, β and β′ are projective endomorphisms of M if and only if the corresponding matrix is a projective endomorphism of M↑^{G}. Because \underline{End}_{kH}(M) = k[α], we conclude

DIRECT SUM DECOMPOSITIONS 5

that \underline{End}_{kG}(M↑^{G}) is generated by the images of the maps a and ψ, and the desired result follows.

The kH-module M is indecomposable because \underline{End}_{kH}(M) = k[α], so the following result shows that Green’s criterion for indecomposability fails for M↑^{G}.

Theorem 2.3. The module M↑^{G} is decomposable.

Proof. Set e = a(1+ψ) ∈ End_{kG}(M↑^{G}). It is easy to check that e is a nontrivial idempotent of End_{kG}(M↑^{G}), so M↑^{G} is decomposable.

Our next objective in this section is to study direct sum decompositions of M↑^{G} by considering idempotents in the endomorphism ring \underline{End}_{kG}(M↑^{G}). Let P, Q ∈ k[x] be polynomials with Q ≠ 0. Then one can check that P (a) + Q(a)ψ is an idempotent in \underline{End}_{kG}(M↑^{G}) if and only if

P (x + 1) = P (x) + 1  (2.4a)
P (x)P (x + 1) = Q(x)Q(x + 1).  (2.4b)

We will describe how to find the solutions to these equations.

First consider equation (2.4a), and observe that P (x) = x is a solution. Moreover, any other solution is of the form P (x) = x + P_{0}(x), where P_{0} ∈ k[x] is a polynomial satisfying

P_{0}(x + 1) = P_{0}(x).

Such polynomials P_{0} form a subspace V of the vector space k[x], and the solutions of (2.4a) are the elements of the coset x + V. The next result describes a basis for V.

Proposition 2.4. Let ℤ/2 = ⟨g⟩ act on k[x] by (g · P)(x) = P (x + 1) for all P ∈ k[x]. For each n ≥ 0 there is a g-invariant polynomial of degree 2n in k[x], and there are no g-invariant polynomials of odd degree. Moreover, if B is a subset of k[x] consisting of precisely one g-invariant polynomial of degree 2n for all n ≥ 0, then B is a basis for V = soc k[x].

Proof. Set V_{−1} = 0, and for any integer n ≥ 0 let V_{n} be the k[ℤ/2]-submodule of k[x] consisting of all polynomials of degree at most n. Then V_{2n+1}/V_{2n−1} is a two-dimensional k[ℤ/2]-module. Because x^{2n+1} + (x + 1)^{2n+1} is a polynomial of degree 2n, it follows that V_{2n+1}/V_{2n−1} is not a trivial module and must be free of rank one. It is easy to see by induction on n that V_{2n+1} is free of rank n+1. Thus soc(V_{2n+1}) = soc(V_{2n}) has dimension n + 1 and soc(V_{2n−1}) has dimension n, so V_{2n+1} has a g-invariant polynomial of degree 2n but no g-invariant polynomial of degree 2n + 1. Moreover, if B is a set consisting of

6 D. J. BENSON AND WAYNE W. WHEELER

exactly one g-invariant polynomial of degree 2n for all n ≥ 0, then B is a basis for soc k[x], as desired.

Now suppose that P ∈ k[x] is a polynomial satisfying (2.4a). Write

P (x)P (x + 1) = λ(x + a_{1})^{e_{1}} ⋯ (x + a_{n})^{e_{n}},

where a_{1}, …, a_{n} ∈ k are distinct. Then for each index i there is a unique j ≠ i such that a_{j} = a_{i} + 1, and e_{j} = e_{i}. Thus n = 2s for some natural number s. By reordering the indices if necessary, we may assume that a_{2i} = a_{2i−1} + 1 for 1 ≤ i ≤ s. Let μ ∈ k be the element satisfying μ^{2} = λ. Then Q ∈ k[x] satisfies (2.4b) if and only if

Q(x) = μ(x + a_{1})^{m_{1}} ⋯ (x + a_{n})^{m_{n}}

for some nonnegative integers m_{1}, …, m_{n} such that m_{2i−1} + m_{2i} = e_{2i} for 1 ≤ i ≤ s.

Thus we obtain a description of the polynomials satisfying (2.4). In particular, it follows that the ring \underline{End}_{kG}(M↑^{G}) contains infinitely many nontrivial idempotents so that the module M↑^{G} has infinitely many different direct sum decompositions. We end this section by showing that all of these nontrivial idempotents are actually conjugate in \underline{End}_{kG}(M↑^{G}).

If \underline{End}_{kG}(M↑^{G}) is regarded as a subalgebra of Mat_{2}(k[α]) as before, then an arbitrary element P (a) + Q(a)ψ ∈ \underline{End}_{kG}(M↑^{G}) is given by the matrix

(\begin{matrix}P (α) & Q(α) \\ Q(α + 1) & P (α + 1) \end{matrix}) .

Thus P (a) + Q(a)ψ is a unit if and only if the determinant P (α)P (α + 1) + Q(α)Q(α + 1) lies in k − {0}. The units of determinant one form a subgroup U_{1} of the full group of units U in \underline{End}_{kG}(M↑^{G}). If k^{×} denotes the multiplicative group of the field k, then it is easy to see that U = U_{1} × k^{×}.

Proposition 2.5. The group U_{1} of units of determinant one in \underline{End}_{kG}(M↑^{G}) acts transitively on the collection of nontrivial idempotents in \underline{End}_{kG}(M↑^{G}). The stabilizer in U_{1} of the idempotent e = a(1 + ψ) is the subgroup Stab_{U_{1}}(e) = {c + (c + c^{−1})e | c ∈ k^{×}}.

Proof. Let P, Q ∈ k[x] be polynomials with Q ≠ 0 such that P (a)+Q(a)ψ is an idempotent in \underline{End}_{kG}(M↑^{G}), and let

e_{P,Q} = (\begin{matrix}P (α) & Q(α) \\ Q(α + 1) & P (α + 1) \end{matrix})

be the corresponding idempotent matrix in Mat_{2}(k[α]). By (2.4b) we can write Q(x) = Q_{0}(x)Q_{1}(x), where Q_{0}(x) divides P (x) and Q_{1}(x) divides P (x + 1). Because P (x + 1) = P (x)+1, the polynomials P (x) and P (x+1) have no common factors, so Q_{0}(x) and Q_{1}(x)

DIRECT SUM DECOMPOSITIONS 7

are unique up to scalar multiples. Moreover, P (x) = Q_{0}(x)Q_{1}(x + 1). Then one can check that

u_{P,Q} = (\begin{matrix}Q_{0}(α) & Q_{1}(α) \\ Q_{0}(α + 1) & Q_{1}(α + 1) \end{matrix})

is a matrix of determinant one in Mat_{2}(k[α]) such that

u−1_{P,Q}e_{P,Q}u_{P,Q} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) .

Now suppose that P′, Q′ ∈ k[x] are also polynomials with Q′ ≠ 0 such that P′(a) + Q′(a)ψ is an idempotent. Write Q′(x) = Q′_{0}(x)Q′_{1}(x) as above, and set

Q̂_{0}(x) = Q_{0}(x)Q′_{1}(x + 1) + Q_{1}(x)Q′_{0}(x + 1)
Q̂_{1}(x) = Q_{0}(x)Q′_{1}(x) + Q_{1}(x)Q′_{0}(x).

Then

u_{P,Q}u−1_{P′,Q′} = (\begin{matrix}Q̂_{0}(α) & Q̂_{1}(α) \\ Q̂_{1}(α + 1) & Q̂_{0}(α + 1) \end{matrix})

is a matrix of determinant one that conjugates e_{P,Q} to e_{P′,Q′}. Moreover, the matrix u_{P,Q}u−1_{P′,Q′} corresponds to the unit Q̂_{0}(a) + Q̂_{1}(a)ψ ∈ \underline{End}_{kG}(M↑^{G}). Thus U_{1} acts transitively on the nontrivial idempotents in \underline{End}_{kG}(M↑^{G}).

Finally, suppose that P (x) = Q(x) = x so that e_{P,Q} is the matrix corresponding to the idempotent e = a(1 + ψ). The stabilizer in SL_{2}(k[α]) of the matrix e_{P,Q} consists of all matrices of the form

u_{P,Q} (\begin{matrix}c^{−1} & 0 \\ 0 & c \end{matrix}) u−1_{P,Q} = (\begin{matrix}c + (c + c^{−1})α & (c + c^{−1})α \\ (c + c^{−1})(α + 1) & c + (c + c^{−1})(α + 1) \end{matrix})

for c ∈ k^{×}. It follows that Stab_{U_{1}}(e) consists of all elements of the form

c + (c + c^{−1})a + (c + c^{−1})aψ = c + (c + c^{−1})e,

and this completes the proof.

Even if the coefficient field k is not algebraically closed, one can define an absolutely indecomposable kH-module M with generators x_{0}, x_{1}, x_{2}, … and relations given by (2.1). In this case the solutions to (2.4b) are obtained by writing

P (x)P (x + 1) = λP_{1}(x)^{e_{1}} ⋯ P_{n}(x)^{e_{n}},

where P_{1}, …, P_{n} are distinct irreducible monic polynomials in k[x] and λ ∈ k. The possibilities for Q(x) are then determined by considering the cases P_{i}(x) = P_{i}(x + 1) and P_{i}(x) ≠ P_{i}(x + 1) for each factor P_{i}. With this minor change all of the results proven in this section for M or for M↑^{G} remain valid. In particular, when k = 𝔽_{2}, Proposition 2.5 shows that the group U_{1} of units of determinant one in \underline{End}_{𝔽_{2}G}(M↑^{G}) acts regularly on the nontrivial idempotents in \underline{End}_{𝔽_{2}G}(M↑^{G}).

3. Graded endomorphism rings of idempotent modules

Throughout this section G denotes an arbitrary finite group, k is an algebraically closed field of characteristic p > 0, and C is an ideal subcategory of kG-\underline{mod}. The objective is to study graded endomorphism rings for the idempotent module F = F_{C}(k) and certain related modules. The results are used in the next section to produce examples of modules for which the cancellation property fails.

For any kG-modules M and N write \underline{Hom}_{kG}^{∗}(M, N) for the ℤ-graded vector space with

\underline{Hom}_{kG}^{n}(M, N) = \underline{Hom}_{kG}(Ω^{n}M, N).

Assume that Ω^{0} is the identity functor on kG-\underline{Mod} so that \underline{Hom}_{kG}(M, N) is the space of elements of degree zero in \underline{Hom}_{kG}^{∗}(M, N). If ξ ∈ \underline{Hom}_{kG}^{m}(M, N) and ζ ∈ \underline{Hom}_{kG}^{n}(L, M), then we write ξ ∘ ζ for the element of \underline{Hom}_{kG}^{m+n}(L, N) given by ξ ∘ ζ = ξ(Ω^{m}ζ). For convenience we usually identify Ω^{n}M with Ω^{n}k ⊗ M when no confusion is likely to arise.

Let η : k → F denote the universal map in kG-\underline{Mod} from k to an object of C^{⊥}, as in Proposition 2.1. Suppose that M and N are in C^{⊥} so that η ⊗ 1_{M} : M → F ⊗ M and η ⊗1_{N} : N → F ⊗N are stable isomorphisms. If ζ ∈ \underline{End}_{kG}^{m}(F, F) and f ∈ \underline{Hom}_{kG}^{n}(M, N), then there is a unique map ζ ·f : Ω^{m+n}M → N such that the following diagram commutes:

\xymatrix{Ω^{m}k ⊗ Ω^{n}M \ar[r]^{ζ·f} \ar[d]_{Ω^{m}η⊗1} & N \ar[d]^{η⊗1} \\ Ω^{m}F ⊗ Ω^{n}M \ar[r]_{ζ⊗f} & F ⊗ N.}

One can check that this definition makes \underline{Hom}_{kG}^{∗}(M, N) into a graded left module for \underline{End}_{kG}^{∗}(F). It is easy to see that

ζ · f = (ζ · 1_{N}) ∘ (1_{Ω^{m}k} ⊗ f) = f ∘ (ζ · 1_{Ω^{n}M}).

Similarly, \underline{Hom}_{kG}^{∗}(M, N) has a right module structure given by setting f · ζ = f ∘ (ζ · 1_{M}), and this definition makes \underline{Hom}_{kG}^{∗}(M, N) into an \underline{End}_{kG}^{∗}(F)-bimodule. Moreover, if ζ ∈ \underline{End}_{kG}^{m}(F) and ξ ∈ \underline{End}_{kG}^{n}(F), then ζ ∘ ξ = ζ · ξ.

If C is the zero subcategory of kG-\underline{mod} so that F = k, then this definition produces the usual bimodule structure of Ĥ^{∗}(G, k) on Êxt_{kG}^{∗}(M, N). Thus the following proposition generalizes a standard result.

DIRECT SUM DECOMPOSITIONS 9

Proposition 3.1. Let M be a C-local kG-module, and let ζ be a homogeneous element of \underline{End}_{kG}^{∗}(F). Then ζ · 1_{M} lies in the graded center of \underline{End}_{kG}^{∗}(M) so that \underline{End}_{kG}^{∗}(M) is a graded \underline{End}_{kG}^{∗}(F)-algebra.

Proof. The proof is similar to that of Theorem 5.2.3 of [6]. The details are left to the reader.

If M and N are kG-modules, we write τ : M ⊗ N → N ⊗ M for the kG-homomorphism that interchanges factors. By abuse of notation the same symbol τ is used for all such modules M and N.

Lemma 3.2. Let M be a C-local kG-module. If α ∈ \underline{End}_{kG}^{m}(F) and n is any integer, then

(α · 1_{Ω^{n}k⊗M})(τ ⊗ 1_{M}) = 1_{Ω^{n}k} ⊗ (α · 1_{M}).

Proof. The commutativity of the diagram

\xymatrix{Ω^{m}k ⊗ Ω^{n}k ⊗ M \ar[rrr]^{α·1_{Ω^{n}k⊗M}} \ar[ddd]_{τ⊗1} \ar[dr]^{1⊗η⊗1⊗1} &  &  & Ω^{n}k ⊗ M \ar[dl]_{η⊗1⊗1} \ar[ddd]^{1⊗1} \\  & Ω^{m}k ⊗ F ⊗ Ω^{n}k ⊗ M \ar[d] \ar[r]^{α⊗1⊗1} & F ⊗ Ω^{n}k ⊗ M \ar[d]^{τ⊗1} \\  & Ω^{n}k ⊗ Ω^{m}k ⊗ F ⊗ M \ar[r]_{1⊗α⊗1} & Ω^{n}k ⊗ F ⊗ M \\ Ω^{n}k ⊗ Ω^{m}k ⊗ M \ar[ur]_{1⊗1⊗η⊗1} \ar[rrr]_{1⊗(α·1_{M})} &  &  & Ω^{n}k ⊗ M \ar[ul]^{1⊗η⊗1}}

shows that α · 1_{Ω^{n}k⊗M} = (1 ⊗ (α · 1_{M}))(τ ⊗ 1), as desired.

If ζ ∈ \underline{End}_{kG}^{n}(F), write F/(ζ) for the module defined by the triangle

Ω^{n}F ⟶^{ζ} F ⟶^{λ} F/(ζ) ⟶^{μ} Ω^{−1}k ⊗ Ω^{n}F.

Note that if F = k, then F/(ζ) = Ω^{−1}L_{ζ}.

Proposition 3.3. Suppose that α ∈ \underline{End}_{kG}^{m}(F) and ζ ∈ \underline{End}_{kG}^{n}(F), and let M be a C-local kG-module. Then there is a morphism of triangles of the form

\xymatrix{Ω^{m}k ⊗ Ω^{n}M \ar[rr]^{1⊗(ζ·1_{M})} \ar[dr]_{α·1_{Ω^{n}M}} &  & Ω^{m}k ⊗ M \ar[r] \ar[d]_{α·1_{M}} & Ω^{m}k ⊗ F/(ζ) ⊗ M \ar[r] \ar[d]^{α·1_{F/(ζ)⊗M}} & Ω^{−1}k ⊗ Ω^{m}k ⊗ Ω^{n}M \ar[d]^{1⊗(α·1_{Ω^{n}M})} \\  & Ω^{n}M \ar[r]_{ζ·1_{M}} & M \ar[r] & F/(ζ) ⊗ M \ar[r] & Ω^{−1}k ⊗ Ω^{n}M.}

Proof. The top and bottom rows of the commutative diagram

[figure: Ω^{m}k ⊗ Ω^{n}F; 1⊗ζ; Ω^{m}η⊗1; Ω^{m}k ⊗ F; 1⊗λ; Ω^{m}η⊗1; Ω^{m}k ⊗ F/(ζ); Ω^{m}η⊗1; (τ⊗1)(1⊗μ); Ω^{−1}k ⊗ Ω^{m}k ⊗ Ω^{n}F; (Ω^{m}η⊗1⊗1)(τ⊗1); Ω^{m}F ⊗ Ω^{n}F; 1⊗ζ; α⊗1; Ω^{m}F ⊗ F; 1⊗λ; α⊗1; Ω^{m}F ⊗ F/(ζ); 1⊗μ; α⊗1; Ω^{m}F ⊗ Ω^{−1}k ⊗ Ω^{n}F; α⊗1⊗1; F ⊗ Ω^{n}F; 1⊗ζ; F ⊗ F; 1⊗λ; F ⊗ F/(ζ); 1⊗μ; F ⊗ Ω^{−1}k ⊗ Ω^{n}F; Ω^{n}F; ζ; η⊗1; F; λ; η⊗1; F/(ζ); μ; η⊗1; Ω^{−1}k ⊗ Ω^{n}F; η⊗1⊗1]

are triangles in kG-\underline{Mod}. Computing the compositions of the maps in the four columns and using the fact that (α · 1_{Ω^{−1}k⊗Ω^{n}F})(τ ⊗ 1_{Ω^{n}F}) = 1_{Ω^{−1}k} ⊗ (α · 1_{Ω^{n}F}) shows that there is a morphism of triangles

\xymatrix{Ω^{m}k ⊗ Ω^{n}F \ar[r]^{1⊗ζ} \ar[d]_{α·1_{Ω^{n}F}} & Ω^{m}k ⊗ F \ar[r]^{1⊗λ} \ar[d]_{α} & Ω^{m}k ⊗ F/(ζ) \ar[rr]^{(τ⊗1)(1⊗μ)} \ar[d]^{α·1_{F/(ζ)}} &  & Ω^{−1}k ⊗ Ω^{m}k ⊗ Ω^{n}F \ar[dl]^{1⊗(α·1_{Ω^{n}F})} \\ Ω^{n}F \ar[r]_{ζ} & F \ar[r]_{λ} & F/(ζ) \ar[r]_{μ} & Ω^{−1}k ⊗ Ω^{n}F.}

Applying the functor − ⊗ M to this diagram and using the stable isomorphism η : M → F ⊗ M now gives the desired result.

Assume that ζ_{1}, …, ζ_{s} are homogeneous elements in \underline{End}_{kG}^{∗}(F), and set n_{i} = deg ζ_{i}. Define

F/(ζ_{1}, …, ζ_{s}) = F/(ζ_{1}) ⊗ ⋯ ⊗ F/(ζ_{s}).

For simplicity set F_{0} = F and F_{i} = F/(ζ_{1}, …, ζ_{i}) for 1 ≤ i ≤ s. Then F_{i} can also be defined by observing that there is a triangle

Ω^{n_{i}}F_{i−1} ⟶^{ζ_{i}·1_{F_{i−1}}} F_{i−1} ⟶^{λ_{i}} F_{i} ⟶^{μ_{i}} Ω^{−1}k ⊗ Ω^{n_{i}}F_{i−1}.  (3.1)

Note that the modules F_{0}, F_{1}, …, F_{s} are C-local by induction.

Proposition 3.4. Let ζ_{1}, …, ζ_{s} be a regular sequence in \underline{End}_{kG}^{∗}(F). Then for 1 ≤ i ≤ s the elements ζ_{i} · 1, …, ζ_{s} · 1 form a regular sequence in \underline{End}_{kG}^{∗}(F_{i−1}), and

\underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) ≅ \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1)

as graded \underline{End}_{kG}^{∗}(F)-bimodules. Moreover, the composition

\underline{End}_{kG}^{∗}(F_{i}) ⟶^{λ_{i}^{∗}} \underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) ≅ \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1)

is an epimorphism of rings.

Proof. Assume by induction that i ≥ 1 and that ζ_{i} · 1, …, ζ_{s} · 1 is a regular sequence in \underline{End}_{kG}^{∗}(F_{i−1}). Applying the functor \underline{Hom}_{kG}^{n}(F_{i−1}, −) to the triangle (3.1) gives a short exact sequence

0 → \underline{End}_{kG}^{n−n_{i}}(F_{i−1}) ⟶^{(ζ_{i}·1)_{∗}} \underline{End}_{kG}^{n}(F_{i−1}) ⟶^{λ_{i∗}} \underline{Hom}_{kG}^{n}(F_{i−1}, F_{i}) → 0

because ζ_{i} · 1 is a regular element of \underline{End}_{kG}^{∗}(F_{i−1}). It follows that there is an isomorphism

\underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) ≅ \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1)

of graded right \underline{End}_{kG}^{∗}(F)-modules. Proposition 3.3 implies that this is actually an isomorphism of graded \underline{End}_{kG}^{∗}(F)-bimodules. For example, if α ∈ \underline{End}_{kG}^{m}(F) and f ∈ \underline{End}_{kG}^{n}(F_{i−1}), then

λ_{i∗}(α · f) = λ_{i} ∘ (α · 1_{F_{i−1}}) ∘ (1_{Ω^{m}k} ⊗ f) = (α · 1_{F_{i}}) ∘ (1_{Ω^{m}k} ⊗ λ_{i∗}(f)) = α · λ_{i∗}(f).

Applying the functor \underline{Hom}_{kG}^{n}(−, F_{i}) to the triangle (3.1) gives a long exact sequence

⋯ → \underline{End}_{kG}^{n}(F_{i}) ⟶^{λ_{i}^{∗}} \underline{Hom}_{kG}^{n}(F_{i−1}, F_{i}) ⟶^{(ζ_{i}·1)^{∗}} \underline{Hom}_{kG}^{n+n_{i}}(F_{i−1}, F_{i}) → · · · .

Because \underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) ≅ \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1), it follows that there is a short exact sequence

0 → \underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) → \underline{End}_{kG}^{∗}(F_{i}) ⟶^{λ_{i}^{∗}} \underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) → 0

in which the first map has degree 1−n_{i}. Moreover, the maps in this sequence are homomorphisms of graded \underline{End}_{kG}^{∗}(F)-bimodules by Proposition 3.3. The elements ζ_{i}·1, …, ζ_{s}·1 form a regular sequence in \underline{End}_{kG}^{∗}(F_{i−1}), so ζ_{i+1}, …, ζ_{s} is a regular sequence on the \underline{End}_{kG}^{∗}(F)- bimodule \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1) ≅ \underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) and hence also on \underline{End}_{kG}^{∗}(F_{i}). Finally, it is easy to check that the composition

\underline{End}_{kG}^{∗}(F_{i}) ⟶^{λ_{i}^{∗}} \underline{Hom}_{kG}^{∗}(F_{i−1}, F_{i}) ≅ \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1)

is an epimorphism of rings. This completes the proof.

Theorem 3.5. Let ζ_{1}, …, ζ_{s} be a regular sequence in \underline{End}_{kG}^{∗}(F). Then there is an epimorphism

\underline{End}_{kG}^{∗}(F/(ζ_{1}, …, ζ_{s})) → \underline{End}_{kG}^{∗}(F)/(ζ_{1}, …, ζ_{s})

whose kernel is an ideal I satisfying I^{2^{s}} = 0.

12 D. J. BENSON AND WAYNE W. WHEELER

Proof. For 1 ≤ i ≤ s let n_{i} = deg ζ_{i}, and let ϕ_{i} : \underline{End}_{kG}^{∗}(F_{i}) → \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1) be the epimorphism given by Proposition 3.4. If ζ ∈ \underline{End}_{kG}^{∗}(F) is homogeneous, then ϕ_{i} sends ζ · 1_{F_{i}} to ζ · 1_{F_{i−1}} + (ζ_{i} · 1) by Proposition 3.3. Thus ϕ_{i} induces an epimorphism

θ_{i} : \underline{End}_{kG}^{∗}(F_{i})/(ζ_{i+1} · 1, …, ζ_{s} · 1) → \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i} · 1, …, ζ_{s} · 1),

and the composition θ = θ_{1} ∘ ⋯ ∘ θ_{s} maps \underline{End}_{kG}^{∗}(F_{s}) onto \underline{End}_{kG}^{∗}(F)/(ζ_{1}, …, ζ_{s}).

It remains to show that I = Ker θ satisfies I^{2^{s}} = 0. We will prove this result by showing that Ker θ_{i} is an ideal of square zero for 1 ≤ i ≤ s. Consider the triangle

Ω^{n}k ⊗ Ω^{n_{i}}F_{i−1} ⟶^{1⊗(ζ_{i}·1)} Ω^{n}k ⊗ F_{i−1} ⟶^{1⊗λ_{i}} Ω^{n}k ⊗ F_{i} ⟶^{(τ⊗1)(1⊗μ_{i})} Ω^{−1}k ⊗ Ω^{n}k ⊗ Ω^{n_{i}}F_{i−1}.

If β ∈ \underline{End}_{kG}^{n}(F_{i}) satisfies 0 = λ_{i}^{∗}(β) = β(1_{Ω^{n}k} ⊗λ_{i}), then there is a map β′ : Ω^{−1}k ⊗Ω^{n}k ⊗ Ω^{n_{i}}F_{i−1} → F_{i} with β′(τ ⊗ 1_{Ω^{n_{i}}F_{i−1}})(1_{Ω^{n}k} ⊗ μ_{i}) = β. Because ζ_{i} · 1 is a regular element of \underline{End}_{kG}^{∗}(F_{i−1}) and μ_{i}β′ : Ω^{−1}k ⊗ Ω^{n}k ⊗ Ω^{n_{i}}F_{i−1} → Ω^{−1}k ⊗ Ω^{n_{i}}F_{i−1} satisfies

(1_{Ω^{−1}k} ⊗ (ζ_{i} · 1))(μ_{i}β′) = 0,

it follows that μ_{i}β′ = 0. This fact implies that Ker λ_{i}^{∗} = Ker ϕ_{i} has square zero. Because the vertical maps in the commutative diagram

[figure: 0; (ζ_{i+1}·1,...,ζ_{s}·1); \underline{End}_{kG}^{∗}(F_{i}); ϕ_{i}; \underline{End}_{kG}^{∗}(F_{i})/(ζ_{i+1}·1,...,ζ_{s}·1); θ_{i}; 0; 0; (ζ_{i}·1,...,ζ_{s}·1)/(ζ_{i}·1); \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i}·1); \underline{End}_{kG}^{∗}(F_{i−1})/(ζ_{i}·1,...,ζ_{s}·1); 0]

are all epimorphisms, it follows that Ker θ_{i} is an ideal of square zero for all i. This completes the proof.

4. Failure of cancellation

In this section we apply the previous results to construct a class of examples of modules for which the cancellation property fails. In particular, when p is an odd prime, we exhibit countably generated modules M and N for the group (ℤ/p)^{7} such that

M ⊕ M ⊕ M ≅ M ⊕ N

but M ⊕ M ≅ N̸. When p = 2, we give a similar example in which the cancellation property fails for (ℤ/2)^{9}. The idea of the construction is to begin by producing a module such that the cancellation property fails for finitely generated projective modules over its endomorphism ring. This step is based upon some related examples of Raynaud [12], and we begin by describing these examples.

DIRECT SUM DECOMPOSITIONS 13

Let k be an algebraically closed field of characteristic p > 0. Let q and n be integers with 1 < q < n, and let δ_{jl} denote the Kronecker delta function for 1 ≤ j, l ≤ n − q. Set

R = k[X_{ji}, Y_{ij}]_{1≤i≤n,1≤j≤n−q}
E = R/𝔄,

where 𝔄 is the ideal generated by the elements^{1}

(∑_{i=1}^{n} X_{ji}Y_{il}) − δ_{jl}  (4.1)

for 1 ≤ j, l ≤ n − q. We use the same symbols X_{ji} and Y_{ij} to denote the images in E of X_{ji}, Y_{ij} ∈ R. Let X denote the (n − q) × n matrix (X_{ji}) with entries in E, and let Y denote the n × (n − q) matrix (Y_{ij}). Then the relations (4.1) show that the product XY is the (n − q) × (n − q) identity matrix, so e = Y X satisfies e^{2} = e. If e is regarded as an idempotent endomorphism of a free module E^{n} of rank n, then

E^{n} = eE^{n} ⊕ (1 − e)E^{n}.

Regarding X and Y as homomorphisms X : E^{n} → E^{n−q} and Y : E^{n−q} → E^{n} of right E-modules, we see that

eE^{n} ≅ E^{n−q}

so that (1 − e)E^{n} is a finitely generated projective E-module that is stably free of rank q. In Corollaire 6.3 of [12] Raynaud uses Steenrod operations in étale cohomology to show that the modules (1−e)E^{n} are not free, except possibly when p = q = 2 or when p = q = 3.

Let G be an elementary abelian p-group of rank 2n + 1. If p = 2, then H^{∗}(G, k) is a polynomial ring k[ζ_{1}, …, ζ_{2n+1}] in 2n + 1 generators of degree one. If p is odd, then H^{∗}(G, k) is the tensor product of a polynomial ring k[ζ_{1}, …, ζ_{2n+1}] and an exterior algebra Λ(η_{1}, …, η_{2n+1}); the elements ζ_{1}, …, ζ_{2n+1} have degree two, and the elements η_{1}, …, η_{2n+1} have degree one. Let C be the subcategory of kG-\underline{mod} consisting of all finitely generated modules M such that V_{G}(M) ⊆ V_{G}(ζ_{2n+1}), and set F = F_{C}(k). Then \underline{End}_{kG}^{∗}(F) is isomorphic to Ĥ^{∗}(G, k)[ζ_{2n+1}^{−1}]. Set α_{i} = ζ_{2n+1}^{−1}ζ_{i} for 1 ≤ i ≤ 2n. We claim that every nonzero element of k[α_{1}, …, α_{2n}] is regular in \underline{End}_{kG}^{∗}(F). Indeed, Theorem 3.3 of [2] shows that the product of any two elements of negative degree is zero in Ĥ^{∗}(G, k). Lemma 2.1 of [2] then implies that if ξ ∈ Ĥ^{−d}(G, k) for some d > 0, then ζ_{2n+1}^{d}ξ = 0 so that ξ = 0 in Ĥ^{∗}(G, k)[ζ_{2n+1}^{−1}]. Thus

\underline{End}_{kG}^{∗}(F) = Ĥ^{∗}(G, k)[ζ_{2n+1}^{−1}] = H^{∗}(G, k)[ζ_{2n+1}^{−1}],

1There is a misprint in [12] at this point: the term −δ_{jl} is omitted.

14 D. J. BENSON AND WAYNE W. WHEELER

and it follows that every nonzero element of k[α_{1}, …, α_{2n}] is regular in \underline{End}_{kG}^{∗}(F).

If q = n − 1, then the relations (4.1) reduce to a single equation in E of the form ∑_{i=1}^{n} X_{1i}Y_{i1} − 1 = 0. Let ϕ : k[α_{1}, …, α_{2n}] → E be the ring epimorphism satisfying ϕ(α_{i}) = X_{1i} and ϕ(α_{n+i}) = Y_{i1} for 1 ≤ i ≤ n. Then

ξ = ∑_{i=1}^{n} α_{i}α_{n+i} − 1

is in Ker ϕ, and ϕ induces an isomorphism of \underline{End}_{kG}(F)/(ξ) modulo its nilradical onto E. Because ξ is regular in \underline{End}_{kG}^{∗}(F), Theorem 3.5 implies that if 𝔑_{0} is the nilradical of E_{0} = End_{kG}(F/(ξ)), then E_{0}/𝔑_{0} ≅ E. We use this isomorphism to identify E_{0}/𝔑_{0} with E.

Let X′ and Y′ be matrices with entries in E_{0} such that the reduction modulo 𝔑_{0} of X′ is X and of Y′ is Y. Then X′Y′ − 1 ∈ 𝔑_{0}, so there is an integer t > 0 such that (X′Y′ − 1)^{t} = 0. Set X_{0} = X′ and

Y_{0} = Y′ ∑_{i=1}^{t}(−1)^{i−1}\binom{t}{i}(X′Y′)^{i−1}.

Then X_{0} and Y_{0} are matrices with entries in E_{0} such that X_{0}Y_{0} = 1 and the reduction modulo 𝔑_{0} of X_{0} is X and of Y_{0} is Y. In particular, e_{0} = Y_{0}X_{0} is an idempotent endomorphism of E_{0}^{n} lifting the idempotent e. Thus E_{0} satisfies

E_{0}^{n} = e_{0}E_{0}^{n} ⊕ (1 − e_{0})E_{0}^{n}

and e_{0}E_{0}^{n} ≅ E_{0}. Moreover, if (1 − e)E^{n} is not a free E-module, then (1 − e_{0})E_{0}^{n} is not a free E_{0}-module.

The following proposition provides the connection between the above discussion and modules over a group algebra.

Proposition 4.1. Let G be a finite group, and let M be a kG-module with endomorphism ring E_{0}. Let e_{0} and f_{0} be idempotent elements of Mat_{n}(E_{0}). Then e_{0}E_{0}^{n} ≅ f_{0}E_{0}^{n} as right E_{0}-modules if and only if e_{0}M^{n} ≅ f_{0}M^{n} as left kG-modules.

Proof. Observe that there are isomorphisms

Hom_{E_{0}}(e_{0}E_{0}^{n}, f_{0}E_{0}^{n}) ≅ f_{0}Mat_{n}(E_{0})e_{0} ≅ Hom_{kG}(e_{0}M^{n}, f_{0}M^{n})

and

Hom_{E_{0}}(f_{0}E_{0}^{n}, e_{0}E_{0}^{n}) ≅ e_{0}Mat_{n}(E_{0})f_{0} ≅ Hom_{kG}(f_{0}M^{n}, e_{0}M^{n}).

Thus e_{0}E_{0}^{n} ≅ f_{0}E_{0}^{n} if and only if there are matrices A ∈ f_{0}Mat_{n}(E_{0})e_{0} and B ∈ e_{0}Mat_{n}(E_{0})f_{0} with AB = f_{0} and BA = e_{0}, which occurs precisely when e_{0}M^{n} ≅ f_{0}M^{n}.

Proposition 4.1 implies that there is an isomorphism

(F/(ξ)) ⊕ (F/(ξ))^{n−1} = (F/(ξ))^{n} ≅ (F/(ξ)) ⊕ (1 − e_{0})(F/(ξ))^{n},  (4.2)

and (1 − e_{0})(F/(ξ))^{n} is not isomorphic to (F/(ξ))^{n−1} as long as (1 − e)E^{n} is not a free E-module. For p = 2 Raynaud’s result implies, therefore, that (1 − e_{0})(F/(ξ))^{n} is not isomorphic to (F/(ξ))^{n−1} when n = 4 and G = (ℤ/2)^{9}; similarly, for p > 2 these modules are not isomorphic when n = 3 and G = (ℤ/p)^{7}. Thus (4.2) shows that the cancellation property fails in these cases. Brenner and Ringel [7] show by more abstract reasoning that the cancellation property fails as long as the p-rank of G is at least two.

More generally, if 1 < q < n and G is an elementary abelian p-group of rank 2n(n−q)+1, then Raynaud’s examples show that the cancellation property fails for a module of the form F/(ξ_{1}, …, ξ_{(n−q)^{2}}). The details are left to the reader.

References

[1] D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196–205.

[2] D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107–129.

[3] D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, I, Math. Proc. Cambridge Phil. Soc. 118 (1995), 223–243.

[4] D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Phil. Soc. 120 (1996), 597–615.

[5] D. J. Benson, J. F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.

[6] D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, preprint.

[7] S. Brenner and C. Ringel, Pathological modules over tame rings, J. London Math. Soc. (2) 14 (1976), 207–215.

[8] J. F. Carlson, P. W. Donovan, and W. W. Wheeler, Complexity and quotient categories for group algebras, J. Pure Appl. Algebra 93 (1994), 147–167.

[9] J. F. Carlson and W. W. Wheeler, Homomorphisms in higher complexity quotient categories, Seattle Conference on Representations of Finite Groups, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, to appear.

[10] J. A. Green, On the indecomposable representations of a finite group, Math. Z. 70 (1959), 430–445.

[11] D. Happel, “Triangulated Categories in the Representation Theory of Finite Dimensional Algebras”, Cambridge Univ. Press, Cambridge, 1988.

[12] M. Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1–26.

[13] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc., to appear.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

E-mail address: www@sloth.math.uga.edu