THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES

D. J. BENSON AND WAYNE W. WHEELER

Abstract. If G is a finite group, then the usual version of the Green correspondence applies to finitely generated kG-modules when k is a field of characteristic p > 0 or a p-adic ring. This paper presents a categorical version of the Green correspondence and a version of the Burry–Carlson–Puig Theorem that remain valid for arbitrary modules and coefficient rings. In this generality, however, it is not clear whether the Green correspondent of a finitely generated module is always finitely generated.

1. Introduction

Although the chief concern in the representation theory of finite groups is usually to study finitely generated modules, in recent years interest in infinitely generated modules has increased considerably. It seems natural, therefore, to ask which of the classical results of modular representation theory extend to this larger class of representations. In [3] the authors showed that Green’s Indecomposability Theorem does not generalize. The purpose of this paper is to show that the module correspondence studied by Green in [9] and [10], on the other hand, does extend to arbitrary modules. One of the surprising consequences of this work is that the correspondence remains valid not only for infinitely generated modules but for arbitrary coefficient rings. Curiously, however, for arbitrary coefficient rings we do not know whether a finitely generated module necessarily has a finitely generated Green correspondent.

Suppose that G is a finite group and k is a commutative ring. In the usual treatment of the Green correspondence k is a field or a p-adic ring, and one can use the theory of vertices and sources, which in turn depends upon the Krull–Schmidt Theorem. For infinitely generated modules, however, the Krull–Schmidt property fails, and different techniques are needed. Section 2 is devoted to proving that relatively stable categories are triangulated and that idempotent endomorphisms split. This splitting is a weak substitute for the Krull–Schmidt property, but it is sufficient for the purpose of proving the Green correspondence in Section 3 and a variant of the Burry–Carlson–Puig Theorem [5] in Section 4.

The failure of the Krull–Schmidt property raises a number of interesting problems that do not arise in the classical Green correspondence. For example, if 𝔇 is a collection of subgroups of G, then

Date: December 1999.

Both authors thank the NSF for its support.

2 D. J. BENSON AND WAYNE W. WHEELER

it is possible for an indecomposable module to be relatively 𝔇-projective even if it is not relatively D-projective for any subgroup D ∈ 𝔇. Dress has studied relative projectivity in this generality in [8]; although he was concerned with finitely generated modules over arbitrary coefficient rings, most of his results remain valid for infinitely generated modules.

Our approach is to fix a subgroup H ⊆ G and a nonempty collection 𝔇 of subgroups of H. Set

𝔛 = {x^{−1}D_{1}x ∩ D_{2} | D_{1}, D_{2} ∈ 𝔇 and x ∈ G − H}
𝔜 = {x^{−1}Dx ∩ H | D ∈ 𝔇 and x ∈ G − H}.

Let Mod(kG) denote the category of right kG-modules, and for any nonempty collection ℨ of subgroups of G let ℨ-Proj(kG) denote the full subcategory of Mod(kG) consisting of ℨ-projective kG-modules. The version of the Green correspondence given in Section 3 states that there are equivalences of additive categories

\xymatrix{\frac{𝔇-Proj(kG)}{𝔛-Proj(kG)} \ar[r]^{f} & \frac{𝔇-Proj(kH)}{𝔛-Proj(kH)} \ar[l]^{g}} .

This result extends the main theorem of Green in [10], where the correspondence is first described as an equivalence of categories.

In the usual setting for the Green correspondence k is a field of characteristic p > 0, the set 𝔇 consists of a single p-subgroup D, and H is a subgroup containing N_{G}(D). If M is a finitely generated D-projective kG-module, then the restriction M↓_{H} is the direct sum of a D-projective kH-module N and a 𝔜-projective kH-module, and one can define the functor f on objects by taking f(M) = N. If M is infinitely generated, however, then we do not know whether M↓_{H} has such a direct sum decomposition; it is a summand of a module having such a decomposition, but without the Krull–Schmidt Theorem it does not seem to follow that M↓_{H} is itself the direct sum of a 𝔇-projective module and a 𝔜-projective module. Thus in general it is necessary to be significantly more careful in defining the functor f. Our hope is that the techniques used here to circumvent such difficulties will also be helpful in other situations where the Krull–Schmidt property fails.

2. Properties of Relatively Stable Categories

A well known property of group rings over fields is that the corresponding stable category is triangulated. What is not well known, however, is that relatively stable categories can be triangulated in a similar way. In this section we show that if G is a finite group, k is a commutative ring, and 𝔛 is a nonempty collection of subgroups of G, then the relatively 𝔛-stable category is triangulated. This fact has some consequences that will be crucial in the next section.

When k is a field, the triangulated structure on the 𝔛-stable category is described in [6]. Essentially the same construction works over an arbitrary coefficient ring k, but we have nevertheless

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 3

chosen to include a description here as well. This is done partly for the convenience of the reader and partly because the work of [6] is presented in a somewhat different context. The category studied in that paper is the W -stable category determined by a finitely generated kG-module W, and the proof that this category is triangulated remains valid over any coefficient ring k as long as W is also projective as a k-module. The case of interest here is obtained by taking W to be the direct sum of induced modules

W = ⨁_{X∈𝔛}k_{X}↑^{G} .

But the additional generality of [6] means that a number of the ideas become simpler for the 𝔛-stable category, even over an arbitrary coefficient ring.

Let Mod(kG) denote the category of all right kG-modules, and let mod(kG) denote the full subcategory of finitely generated modules. In particular, it should be noted that we do not restrict our attention to kG-modules that are k-free. If H is a subgroup of G, then the restriction to H of a kG-module M is denoted by M↓_{H}; if N is a kH-module, then N↑^{G} denotes the induced module N ⊗_{kH} kG. We write H\G to denote a fixed set of representatives for the right cosets of H in G, and we always assume that 1 ∈ H\G. If M and M′ are kG-modules such that M is isomorphic to a summand of M′, then we write M | M′. A short exact sequence

0 → M′ → M → M′′ → 0

of kG-modules is said to be 𝔛-split if the restriction

0 → M′↓_{X}→ M↓_{X}→ M′′↓_{X}→ 0

splits for all X ∈ 𝔛. A monomorphism α : M′ → M is said to be 𝔛-split if the short exact sequence

\xymatrix{0 \ar[r] & M′ \ar[r]^{α} & M \ar[r] & Coker α \ar[r] & 0}

is 𝔛-split, and 𝔛-split epimorphisms are defined similarly.

Definition 2.1. Let P be a kG-module. Assume that if α : M → M′′ is any 𝔛-split epimorphism and γ ∈ Hom_{kG}(P, M′′), then there is a map β ∈ Hom_{kG}(P, M) such that αβ = γ. Then P is said to be 𝔛-projective (or relatively 𝔛-projective). The idea of an X-injective module is defined similarly.

The following proposition shows that 𝔛-projective modules are always the same as 𝔛-injective modules, even though kG may not be self-injective. It is essentially a restatement of Theorem 1 of [8].

Proposition 2.2. If M is a kG-module, then the following statements are equivalent:

(1) M is 𝔛-projective;

4 D. J. BENSON AND WAYNE W. WHEELER

(2) M is 𝔛-injective;

(3) M is a summand of ⨁_{X∈𝔛} M↓_{X}↑^{G};

(4) for each X ∈ 𝔛 there is a kX-module N_{X} such that M is a summand of ⨁_{X∈𝔛} N_{X}↑^{G};

(5) for each X ∈ 𝔛 there is an endomorphism γ_{X} ∈ End_{kX}(M↓_{X}) such that

1_{M} = ∑_{X∈𝔛}Tr_{X}^{G}(γ_{X}).

Proof. The equivalence of most of the statements is well known. We prove only that (2) implies (3) and that (5) implies (2).

Suppose that M is 𝔛-injective. For each X ∈ 𝔛 there is a kG-homomorphism ϕ_{X} : M → M↓_{X}↑^{G} given by

ϕ_{X} (m) = ∑_{g∈X\G} mg^{−1} ⊗ g.

Define a kG-homomorphism Φ_{M} : M → ⨁_{X∈𝔛} M↓_{X}↑^{G} by setting Φ_{M}(m) = (ϕ_{X} (m))_{X∈𝔛}. It is easy to check that Φ_{M} is an 𝔛-split monomorphism. Because M is 𝔛-injective, it follows that M is a summand of ⨁_{X∈𝔛} M↓_{X}↑^{G}, so (3) holds.

Now assume that (5) holds. Suppose that α : M′ → M′′ is an 𝔛-split monomorphism, and let γ : M′ → M be any kG-homomorphism. Then for each X ∈ 𝔛 there is a kX-homomorphism β_{X} : M′′ → M′ with β_{X} α = 1_{M′}. If β = ∑_{X∈𝔛} Tr_{X}^{G}(γ_{X} γβ_{X}), then

βα = ∑_{X∈𝔛}Tr_{X}^{G}(γ_{X} γβ_{X} α) = ∑_{X∈𝔛}Tr_{X}^{G}(γ_{X} γ) = γ.

Thus M is 𝔛-injective, as desired.

It is useful to observe that the previous proof shows something extra: if M is any kG-module, then the homomorphism Φ_{M} : M → ⨁_{X∈𝔛} M↓_{X}↑^{G} given by Φ_{M}(m) = (ϕ_{X} (m))_{X∈𝔛} is an 𝔛-split monomorphism of M into an 𝔛-injective module. We set

Ω_{𝔛}^{−1}(M) = Coker Φ_{M}.

Similarly, the map Ψ_{M} : ⨁_{X∈𝔛} M↓_{X}↑^{G}→ M given by

Ψ_{M}( ∑_{X∈𝔛} ∑_{g∈X\G} m_{X,g} ⊗ g) = ∑_{X∈𝔛} ∑_{g∈X\G} m_{X,g}g

is an 𝔛-split epimorphism from an 𝔛-projective module onto M, and we set

Ω_{𝔛}(M) = Ker Ψ_{M}.

Suppose that C is an additive category and C′ is a full additive subcategory. Then C/C′ denotes the category in which the objects are the same as those of C, but Hom_{C/C′}(M, N) is the quotient of Hom_{C}(M, N) by the additive subgroup of maps that factor through an object in C′.

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 5

In this section our main interest will be the case in which C = Mod(kG) and C′ = 𝔛-Proj(kG) is the full subcategory of 𝔛-projective kG-modules. In this case we write Mod^{𝔛}(kG) for C/C′ and Hom_{kG}^{𝔛}(M, N) for Hom_{C/C′}(M, N), and we refer to the category Mod^{𝔛}(kG) as the 𝔛-stable category. The full subcategory of Mod^{𝔛}(kG) consisting of finitely generated modules is denoted by mod^{𝔛}(kG). The functors Ω_{𝔛} and Ω_{𝔛}^{−1} are adjoint self-equivalences of Mod^{𝔛}(kG), and they restrict to self-equivalences of mod^{𝔛}(kG). A homomorphism is said to be 𝔛-projective if it factors through an 𝔛-projective module.

For any module M set I_{𝔛}(M) = ⨁_{X∈𝔛} M↓_{X}↑^{G}. If α : M → M′ is a kG-homomorphism, then there is a commutative diagram of short exact sequences

[figure: 0; M; Φ_{M}; α; I_{𝔛}(M); Ω_{𝔛}^{−1}(M); 0; 0; M′; β; M′′; γ; Ω_{𝔛}^{−1}(M); 0]

in Mod(kG), and both sequences are 𝔛-split. Letting \underline{α} denote the image of α in Mod^{𝔛}(kG), we obtain a sequence of modules and maps

\xymatrix{M \ar[r]^{\underline{α}} & M′ \ar[r]^{\underline{β}} & M′′ \ar[r]^{\underline{γ}} & Ω_{𝔛}^{−1}(M′)}

in Mod^{𝔛}(kG), called a standard triangle. This standard triangle can also be written as a sextuple (M, M′, M′′, \underline{α}, \underline{β}, \underline{γ}). More generally, any sextuple isomorphic to a standard triangle will be called a distinguished triangle, or usually just a triangle for simplicity.

Let T denote the collection of all triangles in Mod^{𝔛}(kG). By the same argument used to prove Theorem I.2.6 of [11] one obtains the following result.

Proposition 2.3. The collection of sextuples in T is a triangulation of Mod^{𝔛}(kG), and mod^{𝔛}(kG) is a triangulated subcategory of Mod^{𝔛}(kG).

The next result describes the precise connection between short exact sequences in Mod(kG) and triangles in Mod^{𝔛}(kG). The proof is similar to that of Lemma I.2.7 of [11].

Proposition 2.4. Let 𝔛 be a nonempty collection of subgroups of G, and suppose that

\xymatrix{0 \ar[r] & M′ \ar[r]^{α} & M \ar[r]^{β} & M′′ \ar[r] & 0}

is an 𝔛-split short exact sequence in Mod(kG). Then there is a triangle in Mod^{𝔛}(kG) of the form

\xymatrix{M′ \ar[r]^{\underline{α}} & M \ar[r]^{\underline{β}} & M′′ \ar[r] & Ω_{𝔛}^{−1}(M′).}

Moreover, every triangle in Mod^{𝔛}(kG) is isomorphic to one that arises from an 𝔛-split short exact sequence in this way.

6 D. J. BENSON AND WAYNE W. WHEELER

The advantage of having a triangulated structure on Mod^{𝔛}(kG) is that it allows us to prove that idempotent endomorphisms split in this category. If M is an object in an additive category C and e ∈ End_{C}(M) is idempotent, then e is said to split if M ≅ M′ ⊕ M′′ in C in such a way that e is given by

e = (\begin{matrix}1_{M′} & 0 \\ 0 & 0 \end{matrix})

when End_{C}(M) is regarded as the ring of 2 × 2 matrices of the form

(\begin{matrix}α & β \\ γ & δ \end{matrix})

with α ∈ End_{C}(M′), β ∈ Hom_{C}(M′′, M′), γ ∈ Hom_{C}(M′, M′′), and δ ∈ End_{C}(M′′).

Proposition 2.5. Idempotent endomorphisms split in the category Mod^{𝔛}(kG).

Proof. The triangulated category Mod^{𝔛}(kG) has countable direct sums, and a countable direct sum of triangles is again a triangle. By Proposition 3.2 of [4] or Lemma 3.1 of [2] idempotent endomorphisms split in such a category.

In fact, the construction used in [2] and [4] is quite explicit. If M is a kG-module and e ∈ End_{kG}^{𝔛}(M) is idempotent, form a countable direct sum of copies of M with an endomorphism described by the following schema:

[figure: M; e; ⊕; M; ⊕; M; e; 1−e; ⊕; M; ⊕; ⋮; 1−e; ⋮]

Completing this endomorphism to a triangle defines a module M_{e}:

M_{e} → ⨁_{1}^{∞} M → ⨁_{1}^{∞} M → Ω_{𝔛}^{−1}(M_{e}).

Similarly, there is a module M_{1−e} determined by the endomorphism 1 − e. Then M ≅ M_{e} ⊕ M_{1−e} in Mod^{𝔛}(kG), and with respect to this direct sum decomposition the endomorphism e corresponds to projection onto the summand M_{e}. In other words, there is an 𝔛-projective kG-module X for which M ⊕ X has another direct sum decomposition M ⊕ X ≅ M′_{e} ⊕ M′_{1−e} in Mod(kG), and the composition of projections and inclusions given by

M → M ⊕ X → M′_{e} → M ⊕ X → M

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 7

is equal to e modulo 𝔛-projectives. As an illustration of the difficulties implicit in Proposition 2.5, the following example sketches a construction showing that it is indeed necessary to add the summand X to obtain such a result.

Let G = (ℤ/2)^{2} × D_{8}, and let k be an algebraically closed field of characteristic two. Let H be an elementary abelian 2-subgroup of order 16 in G, and set 𝔛 = {H}. The cohomology ring of G is given by

H^{∗}(G, k) = k[u, e_{1}, x_{1}, y_{1}, ζ]/(x_{1}y_{1}),

where ζ has degree two and the other generators are in degree one. We may assume that y_{1} is defined over 𝔽_{2} and corresponds to the group homomorphism with kernel H under the isomorphism Hom(G, ℤ/2) ≅ H^{1}(G, 𝔽_{2}). The restriction–transfer sequence

\xymatrix{⋯ \ar[r] & H^{n}(G, k) \ar[r]^{res} & H^{n}(H, k) \ar[r]^{Tr} & H^{n}(G, k) \ar[r]^{y_{1}} & H^{n+1}(G, k) \ar[r] & ⋯}

shows that the image of Tr_{H}^{G} is equal to the kernel of multiplication by y_{1}, which is the principal ideal (x_{1}).

The work of Rickard in [12] defines an idempotent module F = F (u) corresponding to u, and by [7] the endomorphism ring \underline{End}_{kG}(F) of F in the stable category is given by the elements of degree zero in H^{∗}(G, k)[u^{−1}]. Thus

\underline{End}_{kG}(F) = k[e_{0}, x_{0}, y_{0}, ζ_{0}]/(x_{0}y_{0}),

where e_{0} = e_{1}u^{−1}, x_{0} = x_{1}u^{−1}, y_{0} = y_{1}u^{−1}, and ζ_{0} = ζu^{−2}. Moreover, x_{0} generates the ideal of 𝔛-projective maps in \underline{End}_{kG}(F) because Tr_{H}^{G}(H^{∗}(H, k)) = (x_{1}). By [3] there is a module M = F/(e_{0}^{2} − e_{0} − x_{0}) such that \underline{End}_{kG}(M) has an ideal I of square zero with

\underline{End}_{kG}(M)/I ≅ \underline{End}_{kG}(F)/(e_{0}^{2} − e_{0} − x_{0}) = k[e_{0}, x_{0}, y_{0}, ζ_{0}]/(x_{0}y_{0}, e_{0}^{2} − e_{0} − x_{0}).

Using the ideas of [3], one can show that x_{0} lifts to a central element x ∈ \underline{End}_{kG}(M), and modulo I the element x generates the ideal of 𝔛-projective endomorphisms. Moreover,

\underline{End}_{kG}(M)/(I, x) ≅ k[e_{0}, y_{0}, ζ_{0}]/(e_{0}^{2} − e_{0}),

and e_{0} is idempotent in this ring. The ideal I is nilpotent, so e_{0} lifts to an endomorphism e ∈ \underline{End}_{kG}(M) such that e^{2} − e ∈ (x). Thus e determines an idempotent endomorphism in End_{kG}^{𝔛}(M). If there were a splitting of e corresponding to a decomposition M ≅ M_{e} ⊕ M_{1−e} in Mod(kG), then \underline{End}_{kG}(M) would contain an idempotent of the form e + xα + ξ for some α ∈ \underline{End}_{kG}(M) and some 𝔛-projective map ξ ∈ I. But if y ∈ \underline{End}_{kG}(M) lifts y_{0}, then

\underline{End}_{kG}(M)/(I, y) ≅ k[e_{0}, x_{0}, ζ_{0}]/(e_{0}^{2} − e_{0} − x_{0}).

8 D. J. BENSON AND WAYNE W. WHEELER

This ring contains no nontrivial idempotents because e_{0}^{2} − e_{0} − x_{0} is irreducible, so \underline{End}_{kG}(M) cannot contain an idempotent of the form e + xα + ξ. Thus it is necessary to add an 𝔛-projective summand to M in order to split the idempotent e ∈ End_{kG}^{𝔛}(M).

3. The Green Correspondence

In this section we apply the result of Proposition 2.5 to obtain a version of the Green correspondence for arbitrary modules and coefficient rings. Throughout this section k is a commutative ring, H is a subgroup of G, and 𝔇 is a collection of subgroups of H. Set

𝔛 = {x^{−1}D_{1}x ∩ D_{2} | D_{1}, D_{2} ∈ 𝔇 and x ∈ G − H}
𝔜 = {x^{−1}Dx ∩ H | D ∈ 𝔇 and x ∈ G − H}.

The statement of the Green correspondence proven here is that there are equivalences of categories

\xymatrix{\frac{𝔇-Proj(kG)}{𝔛-Proj(kG)} \ar[r] & \frac{𝔇-Proj(kH)}{𝔛-Proj(kH)} \ar[l]} .

Roughly speaking, these equivalences are induced by restriction and induction. In the case of the restriction functor, however, the absence of a Krull–Schmidt Theorem presents some subtle technical difficulties that do not arise for finitely generated kG-modules when k is a field.

In this section it will sometimes be useful to distinguish between internal and external direct sum decompositions. If M is a kG-module with submodules M_{1} and M_{2} such that M = M_{1} + M_{2} and M_{1} ∩ M_{2} = 0, then we write M = M_{1} ˙+M_{2} to emphasize that M is an internal direct sum of M_{1} and M_{2}.

Lemma 3.1. Let M and N be kH-modules, and assume that either M or N is 𝔇-projective. Then a kH-homomorphism α : M → N is 𝔛-projective if and only if it is 𝔜-projective.

Proof. If α is 𝔛-projective, then it is certainly 𝔜-projective. The converse follows from Lemma 12.4 of [1]. Although the proof of that lemma is given for finitely generated modules over an algebraically closed field, it is easy to modify so that these assumptions are unnecessary.

If M is a kG-module and N is a kH-module, let

Fr : Hom_{kH}(M↓_{H}, N) → Hom_{kG}(M, N↑^{G})

be the isomorphism given by

(Fr α)(m) = ∑_{g∈H\G} α(mg^{−1}) ⊗ g.

It is easy to see that this isomorphism is functorial in both M and N.

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 9

Proposition 3.2. Let M be a kG-module, and let N be a 𝔇-projective kH-module. If α : M↓_{H}→ N is a kH-homomorphism, then α is 𝔛-projective if and only if Fr α is 𝔛-projective. Moreover, Fr : Hom_{kH}(M↓_{H}, N) → Hom_{kG}(M, N↑^{G}) induces an isomorphism

Hom_{kH}^{𝔛}(M↓_{H}, N) ≅ Hom_{kG}^{𝔛}(M, N↑^{G}).

Proof. It suffices to show that α is 𝔛-projective if and only if Fr α is 𝔛-projective. Suppose that α : M↓_{H}→ N is 𝔛-projective. Then there exist an 𝔛-projective kH-module X and kH-homomorphisms β : M↓_{H}→ X and γ : X → N with γβ = α. Using the functoriality of the isomorphism Fr in the second variable, we see that Fr α = Fr (γβ) = γ↑^{G} ∘Fr β. Thus Fr α factors through the 𝔛-projective module X↑^{G}, as desired.

Now suppose that α : M↓_{H}→ N is a kH-homomorphism such that Fr α : M → N↑^{G} is 𝔛projective. Let X be an 𝔛-projective kG-module such that there are kG-homomorphisms β : M → X and γ : X → N↑^{G} with γβ = Fr α, and let π_{1} : N↑^{G}↓_{H}→ N be the kH-homomorphism given by

π_{1}( ∑_{g∈H\G} n_{g} ⊗ g) = n_{1}.

Then the composition

\xymatrix{M↓_{H} \ar[r]^{β↓_{H}} & X↓_{H} \ar[r]^{γ↓_{H}} & N↑^{G}↓_{H} \ar[r]^{π_{1}} & N}

is a kH-homomorphism satisfying

(π_{1} ∘ γ↓_{H} ∘β↓_{H})(m) = (π_{1} ∘ Fr α)(m) = π_{1}( ∑_{g∈H\G} α(mg^{−1}) ⊗ g) = α(m),

so α factors through X↓_{H}. But X↓_{H} is 𝔜-projective by the Mackey Decomposition Theorem, so Lemma 3.1 implies that α is 𝔛-projective, and this completes the proof.

If C is an additive category and C_{1} and C_{2} are full additive subcategories of C, write C_{1} + C_{2} for the full additive subcategory of C consisting of objects isomorphic to the direct sum of an object in C_{1} and an object in C_{2}. In particular, we want to study the category 𝔇-Proj(kH) + 𝔜-Proj(kH). Because every 𝔛-projective kH-module is also 𝔜-projective, there is a canonical functor

Ψ : \frac{𝔇-Proj(kH)}{𝔛-Proj(kH)} → \frac{𝔇-Proj(kH) + 𝔜-Proj(kH)}{𝔜-Proj(kH)}.

We will show that this functor provides an equivalence that is a categorical analogue of the Third Isomorphism Theorem for modules.

Each object in 𝔇-Proj(kH) + 𝔜-Proj(kH) has a decomposition as an internal direct sum M ˙+Y, where M is 𝔇-projective and Y is 𝔜-projective. If N is a 𝔇-projective kH-module, then for

10 D. J. BENSON AND WAYNE W. WHEELER

convenience we choose the decomposition of N↑^{G}↓_{H} given by

N↑^{G}↓_{H}= (N ⊗ 1) ˙+ ∑_{\begin{matrix}g∈H\G \\ g≠1 \end{matrix}}(N ⊗ g);

for other modules choose such a decomposition arbitrarily. If M ˙+Y and M′ ˙+Y′ are two objects of 𝔇-Proj(kH) + 𝔜-Proj(kH), then any kH-homomorphism f : M ˙+Y → M′ ˙+Y′ can be expressed as a 2 × 2 matrix

f = (\begin{matrix}α & β \\ γ & δ \end{matrix})

with α ∈ Hom_{kH}(M, M′), β ∈ Hom_{kH}(Y, M′), γ ∈ Hom_{kH}(M, Y′), and δ ∈ Hom_{kH}(Y, Y′). Define a functor

Φ : \frac{𝔇-Proj(kH) + 𝔜-Proj(kH)}{𝔜-Proj(kH)} → \frac{𝔇-Proj(kH)}{𝔛-Proj(kH)}

by setting Φ(M ˙+Y) = M and Φf = α : M → M′. If f′ : M′ ˙+Y′ → M′′ ˙+Y′′ is a kHhomomorphism given by the matrix

f′ = (\begin{matrix}α′ & β′ \\ γ′ & δ′ \end{matrix}) ,

then modulo 𝔛-projective homomorphisms we have

Φ(f′f) = α′α + β′γ ≡ α′α = Φf′ ∘ Φf,

so Φ is indeed a functor. The following result is now straightforward.

Proposition 3.3. The functors Φ and Ψ are k-linear equivalences of categories such that Φ ∘ Ψ and Ψ ∘ Φ are naturally isomorphic to identity functors.

To prove a categorical version of the Green correspondence, we would like to know that Res_{G,H} : Mod(kG) → Mod(kH) induces a restriction map

Res_{G,H} : \frac{𝔇-Proj(kG)}{𝔛-Proj(kG)} → \frac{𝔇-Proj(kH) + 𝔜-Proj(kH)}{𝔜-Proj(kH)}.

The Mackey Decomposition Theorem shows that the restriction from G to H of a relatively 𝔇projective kG-module is a summand of a module in 𝔇-Proj(kH) + 𝔜-Proj(kH). Without a Krull– Schmidt Theorem, however, we cannot conclude that the restriction itself lies in the subcategory 𝔇-Proj(kH) + 𝔜-Proj(kH). Indeed, this point seems to be the chief difficulty in proving a general version of the Green correspondence. It can be resolved by using the result of Proposition 2.5 on the splitting of idempotent endomorphisms.

Proposition 3.4. Idempotent endomorphisms split in (𝔇-Proj(kH) + 𝔜-Proj(kH))/𝔜-Proj(kH).

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 11

Proof. Because the splitting of idempotent endomorphisms is preserved by additive equivalences, it suffices to prove that idempotent endomorphisms split in 𝔇-Proj(kH)/𝔛-Proj(kH). Let M be a 𝔇-projective kH-module, and let e be a kH-endomorphism of M such that e is idempotent in Mod^{𝔛}(kH). Then e splits in Mod^{𝔛}(kH) by Proposition 2.5, so there is an 𝔛-projective kH-module X such that M ⊕ X = M_{e} ˙+M_{1−e}, and modulo 𝔛-projectives the idempotent e has the form

(\begin{matrix}1_{M_{e}} & 0 \\ 0 & 0 \end{matrix})

with respect to the direct sum decomposition M_{e} ˙+M_{1−e}. Because M ⊕ X is 𝔇-projective, the modules M_{e} and M_{1−e} are also 𝔇-projective. Thus e splits in 𝔇-Proj(kH)/𝔛-Proj(kH), as desired.

Corollary 3.5. Let M be a summand of a module in 𝔇-Proj(kH) + 𝔜-Proj(kH). Then there is a 𝔜-projective kH-module Y such that M ⊕ Y is an object of 𝔇-Proj(kH) + 𝔜-Proj(kH).

Proof. Let M′ be an object of 𝔇-Proj(kH) + 𝔜-Proj(kH) such that M is a summand of M′, and let e ∈ End_{kH}(M′) be the idempotent endomorphism corresponding to projection onto M. Then the image of e in (𝔇-Proj(kH)+𝔜-Proj(kH))/𝔜-Proj(kH) splits, so there is a 𝔜-projective kH-module Y′ such that M′ ⊕ Y′ = M_{e} ˙+M_{1−e} for some modules M_{e} and M_{1−e} in 𝔇-Proj(kH) + 𝔜-Proj(kH) with M_{e} ≅ M in Mod^{𝔜}(kH). Let Y_{e} and Y be 𝔜-projective kH-modules such that M_{e}⊕Y_{e} ≅ M ⊕Y in Mod(kH). Then M ⊕ Y is in 𝔇-Proj(kH) + 𝔜-Proj(kH), as desired.

The reader may wonder why we have used the sum of categories 𝔇-Proj(kH)+𝔜-Proj(kH) instead of the category (𝔇 ∪ 𝔜)-Proj(kH), which is closed under the process of taking direct summands. After all, 𝔇-Proj(kH) + 𝔜-Proj(kH) is a subcategory of (𝔇 ∪ 𝔜)-Proj(kH); moreover, if M is an object of (𝔇 ∪ 𝔜)-Proj(kH), then M is a summand of a module in 𝔇-Proj(kH) + 𝔜-Proj(kH), so Corollary 3.5 implies that there is a 𝔜-projective kH-module Y such that M ⊕ Y is an object of 𝔇-Proj(kH) + 𝔜-Proj(kH). It follows that the inclusion map induces an equivalence of categories

\frac{𝔇-Proj(kH) + 𝔜-Proj(kH)}{𝔜-Proj(kH)} → \frac{(𝔇 ∪ 𝔜)-Proj(kH)}{𝔜-Proj(kH)}.

The advantage of using the sum of categories is that it is easy to define the equivalence Φ. If the sum is replaced by (𝔇∪𝔜)-Proj(kH) at the outset, however, then it becomes unclear how to obtain such a functor. The equivalence Φ plays an important part in the Green correspondence, which we now define.

Let M be a 𝔇-projective kG-module. Then M↓_{H} is a summand of a module in 𝔇-Proj(kH)+𝔜-Proj(kH), and Corollary 3.5 shows that there is a 𝔜-projective kH-module Y such that M↓_{H} ⊕Y is in 𝔇-Proj(kH) + 𝔜-Proj(kH). Fix such a kH-module Y for each 𝔇-projective kG-module M; take Y = 0 whenever M↓_{H} is itself in 𝔇-Proj(kH) + 𝔜-Proj(kH). Set R_{G,H}(M) = M↓_{H} ⊕Y. If α : M → M′ is a kG-homomorphism, define R_{G,H}(α) : R_{G,H}(M) → R_{G,H}(M′) by the matrix

(\begin{matrix}α & 0 \\ 0 & 0 \end{matrix}) .

Note that R_{G,H}(1_{M}) is not always equal to 1_{R_{G,H}(M)} so that R_{G,H} does not define a functor from 𝔇-Proj(kG) to 𝔇-Proj(kH) + 𝔜-Proj(kH). But because the restriction of an 𝔛-projective kG-module to H is 𝔜-projective, R_{G,H} does induce a functor

R_{G,H} : \frac{𝔇-Proj(kG)}{𝔛-Proj(kG)} → \frac{𝔇-Proj(kH) + 𝔜-Proj(kH)}{𝔜-Proj(kH)}.

Moreover, up to natural isomorphism this functor R_{G,H} is independent of the choices of the 𝔜projective modules used to define it. The induction functor Ind_{H,G} : Mod(kH) → Mod(kG) induces a functor

Ind_{H,G} : \frac{𝔇-Proj(kH)}{𝔛-Proj(kH)} → \frac{𝔇-Proj(kG)}{𝔛-Proj(kG)},

so there is a diagram

\xymatrix{ & \frac{𝔇-Proj(kG)}{𝔛-Proj(kG)} \ar@{-->}[dr]^{_{R_{G,H}}} \\ \frac{𝔇-Proj(kH)}{𝔛-Proj(kH)} \ar[rr]^{Ψ} \ar@{-->}[ur]^{_{Ind_{H,G}}} &  & \frac{𝔇-Proj(kH) + 𝔜-Proj(kH)}{𝔜-Proj(kH)}. \ar[ll]^{Φ}}

The following result implies that all of the functors appearing in this diagram are equivalences and that the diagram commutes up to natural isomorphism.

Theorem 3.6 (Green Correspondence). Set f = Φ ∘ R_{G,H} and g = Ind_{H,G}. Then the k-linear functors f and g are adjoint equivalences of categories.

Proof. It is easy to see that f and g are k-linear, and the Mackey Decomposition Theorem shows that the composition R_{G,H} ∘ Ind_{H,G} is naturally isomorphic to Ψ, so f ∘ g is naturally isomorphic to the identity functor. To prove that g ∘ f is also naturally isomorphic to the identity functor, suppose that M is a 𝔇-projective kG-module. Then there is a 𝔜-projective kH-module Y such that R_{G,H}(M) = M↓_{H} ⊕Y is in 𝔇-Proj(kH) + 𝔜-Proj(kH). The functor Φ is defined on R_{G,H}(M) by writing M↓_{H} ⊕Y as an internal direct sum N ˙+Y′ of a 𝔇-projective kH-module N and a 𝔜projective kH-module Y′. Because M↓_{H} ⊕Y = N ˙+Y′, we can write the identity map on M↓_{H} ⊕Y as a matrix

(\begin{matrix}α & β \\ γ & δ \end{matrix})

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 13

with α ∈ Hom_{kH}(M↓_{H}, N), β ∈ Hom_{kH}(Y, N), γ ∈ Hom_{kH}(M↓_{H}, Y′), and δ ∈ Hom_{kH}(Y, Y′). Then Φ(M↓_{H} ⊕Y) = N, so (g ∘ f)(M) = N↑^{G}. Using the isomorphism Fr : Hom_{kH}(M↓_{H}, N) → Hom_{kG}(M, N↑^{G}), we define η_{M} : M → (g ∘ f)(M) by setting

η_{M} = Fr α.

We claim that η : 1 → g ∘ f is a natural transformation. Let M̃ be another 𝔇-projective kGmodule, and suppose that λ : M → M̃ is a kG-homomorphism. Write M̃↓_{H} ⊕Ỹ = Ñ ˙+Ỹ′ as above, and suppose that the identity map of M̃↓_{H} ⊕Ỹ is given by the matrix

(\begin{matrix}α̃ & β̃ \\ γ̃ & δ̃ \end{matrix}) .

The identity map of M↓_{H} ⊕Y = N ˙+Y′ can also be written as

(\begin{matrix}α′ & β′ \\ γ′ & δ′ \end{matrix})

with α′ ∈ Hom_{kH}(N, M↓_{H}), β′ ∈ Hom_{kH}(Y′, M↓_{H}), γ′ ∈ Hom_{kH}(N, Y), and δ′ ∈ Hom_{kH}(Y′, Y). Then α′α + β′γ = 1_{M↓_{H}}. In order to prove that η is natural, we must show that the diagram

\xymatrix{M \ar[r]^{Fr α} \ar[d]_{λ} & N↑^{G} \ar[d]^{(α̃λα′)↑^{G}} \\ M̃ \ar[r]_{_{Fr α̃}} & Ñ↑^{G}}

commutes modulo 𝔛-projective maps. But

(Fr α̃) ∘ λ − (α̃λα′)↑^{G} ∘Fr α = Fr (α̃λ) − Fr (α̃λα′α) = Fr (α̃λβ′γ),

and α̃λβ′γ : M↓_{H}→ Ñ is 𝔜-projective. Because Ñ is 𝔇-projective, Lemma 3.1 shows that α̃λβ′γ is 𝔛-projective and hence so is Fr (α̃λβ′γ), as desired.

Finally, we must show that if M is a 𝔇-projective kG-module, then η_{M} : M → (g ∘ f)(M) is an isomorphism in Mod^{𝔛}(kG). Fix a 𝔇-projective kH-module U such that M is a summand of U↑^{G}. Let M′ be a kG-module for which there is an isomorphism μ : M ⊕ M′ → U↑^{G}, and let ι : M → M ⊕ M′ and ι′ : M′ → M ⊕ M′ be the inclusion maps. The naturality of η implies that there is a commutative diagram

[figure: M ⊕ M′; η_{M} ⊕η_{M′}; (g ∘ f)(M) ⊕ (g ∘ f)(M′); ≅; (g∘f)(ι)⊕(g∘f)(ι′); M ⊕ M′; η_{M⊕M′}; ≅; μ; (g ∘ f)(M ⊕ M′); ≅; (g∘f)(μ); U↑^{G}; η_{U}; ≅; (g ∘ f)(U↑^{G}),]

14 D. J. BENSON AND WAYNE W. WHEELER

and the map η_{U} in the bottom row is an isomorphism by the Mackey Decomposition Theorem. Thus η_{M} is an isomorphism for all M. It follows that g ∘ f is naturally isomorphic to the identity, and this completes the proof.

If V and W are subgroups of G such that there is an h ∈ H with h^{−1}V h ⊆ W, then we write V ⊆_{H} W; if 𝔙 and 𝔚 are collections of subgroups such that for every V ∈ 𝔙 there is a W ∈ 𝔚 with V ⊆_{H} W, then we write 𝔙 ⊆_{H} 𝔚.

Corollary 3.7. Let M be a 𝔇-projective kG-module with Green correspondent N = f(M), and let 𝔙 be a collection of subgroups of H with 𝔛 ⊆_{H} 𝔙 ⊆_{H} 𝔇. Then M is 𝔙-projective if and only if N is 𝔙-projective.

Proof. If N is 𝔙-projective, then M is also 𝔙-projective because M ≅ N↑^{G} in Mod^{𝔛}(kG). Conversely, suppose that M is 𝔙-projective. Then there exist a kG-module M′ and a 𝔙-projective kH-module V such that M ⊕ M′ ≅ V ↑^{G}. The module M′ is 𝔇-projective, so it has a Green correspondent N′ = f(M′). Using the additivity of the functor f, we conclude that

N ⊕ N′ = f(M) ⊕ f(M′) ≅ f(V ↑^{G}) = (f ∘ g)(V) ≅ V

in Mod^{𝔛}(kH). Thus N is 𝔙-projective, as desired.

4. The Burry–Carlson–Puig Theorem

In this section we give a surprisingly short proof of the Burry–Carlson–Puig Theorem [5] that holds for arbitrary modules and arbitrary commutative coefficient rings. We continue to use the notation of the previous section. It is interesting to note, however, that the results presented here do not depend upon the Green correspondence.

Lemma 4.1. Let M and N be kG-modules, and suppose that α : M↓_{H}→ N↓_{H} is a 𝔇-projective kH-homomorphism. Then α = γ↓_{H} +δ, where γ : M → N is a 𝔇-projective kG-homomorphism and δ : M↓_{H}→ N↓_{H} is a 𝔜-projective kH-homomorphism.

Proof. The proof, which is nearly the same as that of Lemma 12.1 of [1], is left to the reader. Although that proof ends with an appeal to the Krull–Schmidt Theorem, it is entirely unnecessary.

Proposition 4.2. Assume that V is a 𝔇-projective kH-module, and set U = V ↑^{G}. Let M be any kG-module. Then the following statements are equivalent:

(1) there is an 𝔛-projective kG-module X such that U | M ⊕ X;

(2) there is a 𝔜-projective kH-module Y such that V | M↓_{H} ⊕Y.

THE GREEN CORRESPONDENCE FOR INFINITELY GENERATED MODULES 15

Proof. Suppose that there is an 𝔛-projective kG-module X such that U | M ⊕ X. Then X↓_{H} is 𝔜-projective, and V | M↓_{H} ⊕X↓_{H} because V | U↓_{H}.

Conversely, suppose that V | M↓_{H} ⊕Y for some 𝔜-projective kH-module Y. Then there are kH-homomorphisms α : V → M↓_{H} and β : M↓_{H}→ V with βα = 1_{V} + ξ for some 𝔜-projective kH-homomorphism ξ : V → V. Because V is 𝔇-projective, ξ is 𝔛-projective by Lemma 3.1. Let π_{1} : U↓_{H}→ V be the kH-homomorphism given by

π_{1}( ∑_{g∈H\G} v_{g} ⊗ g) = v_{1}.

Then απ_{1} ∈ Hom_{kH}(U↓_{H}, M↓_{H}) is 𝔇-projective, so Lemma 4.1 implies that απ_{1} = γ↓_{H} +δ for some kG-homomorphism γ : U → M and some 𝔜-projective kH-homomorphism δ : U↓_{H}→ M↓_{H}. It follows that

(Fr β) ∘ γ = Fr (β ∘ γ↓_{H}) = Fr (βαπ_{1} − βδ) = Fr (π_{1} + ξπ_{1} − βδ) = 1_{U} + Fr (ξπ_{1} − βδ).

Because V is 𝔇-projective and δ is 𝔜-projective, the map βδ : U↓_{H}→ V is 𝔛-projective by Lemma 3.1. Thus ξπ_{1} − βδ is 𝔛-projective, and so is Fr (ξπ_{1} − βδ). Let X be an 𝔛-projective kG-module such that there are maps σ : U → X and τ : X → U with Fr (ξπ_{1} − βδ) = τσ. Then the composition of the maps

(γ, σ) : U → M ⊕ X
(Fr β, −τ) : M ⊕ X → U

is 1_{U}. Thus U | M ⊕ X, and this completes the proof.

Note that Proposition 4.2 is essentially a statement of the Burry–Carlson–Puig Theorem. Indeed, suppose that k is a field of characteristic p > 0 and 𝔇 = {D} for some p-subgroup D with N_{G}(D) ⊆ H. If V is a finitely generated indecomposable kH-module with vertex D and g(V) is its indecomposable Green correspondent, then Proposition 4.2 shows that g(V) | M if and only if V | M↓_{H}. When M is a finitely generated indecomposable module, this means that g(V) ≅ M if and only if V | M↓_{H}, and this statement is precisely the usual formulation of the Burry–Carlson– Puig Theorem.

References

[1] J. L. Alperin, Local Representation Theory, Cambridge Univ. Press, Cambridge, 1986.

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

[3] D. J. Benson and W. Wheeler, Direct sum decompositions of infinitely generated modules, Trans. Amer. Math. Soc. 351 (1999), 3843–3855.

[4] M. Bökstedt and A. Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209–234.

[5] D. Burry and J. F. Carlson, Restrictions of modules to local subgroups, Proc. Amer. Math. Soc. 84 (1982), 181–184.

[6] J. F. Carlson, C. Peng, and W. Wheeler, Transfer maps and virtual projectivity, J. Algebra 204 (1998), 286–311.

[7] J. F. Carlson and W. Wheeler, Homomorphisms in higher complexity quotient categories; in Group Representations: Cohomology, Group Actions, and Topology, A. Adem et. al., eds., Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 115–155.

[8] A. Dress, Vertices of integral representations, Math. Zeit. 114 (1970), 159–169.

[9] J. A. Green, A transfer theorem for modular representations, J. Algebra 1 (1964), 73–84.

[10] J. A. Green, Relative module categories for finite groups, J. Pure Appl. Algebra 2 (1972), 371–393.

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

[12] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 56 (1997), 149–170.

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

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

Center for Communications Research, 4320 Westerra Court, San Diego, CA 92117, USA

  E-mail address: wheeler@member.ams.org