ON A THEOREM BY BENSON AND GNACADJA

AMNON NEEMAN

Abstract. I have a simplification of part of the argument in a recent paper by Benson and Gnacadja [2]. In this note I remind the reader of Benson and Gnacadja’s work, and explain the simplification.

1. A review of the results in Benson and Gnacadja [2]

We begin by reminding the reader of some notation. Let G be a finite group, k a field. Let StMod(kG) be the stable module category. The objects are modules, the morphisms are equivalence classes of homomorphisms. Two homomorphisms of modules M ⟶ N are identified in StMod(kG) if the difference factors through a projective module. A morphism M ⟶ N in StMod(kG) is called phantom if, whenever K is a finitely generated kG-module, every composite K ⟶ M ⟶ N factors through a projective.

The first result of [2] is to produce a five-term exact sequence involving the phantom maps Ph_{kG}(M, N). From this exact sequence the authors deduce

Theorem 1.3. Let x be non-zero element in Pext_{kG}^{2}(M, N). There are two possibilities:

(1) If the image of x ∈Pext_{kG}^{2}(M, N) under the map

Pext_{kG}^{2}(M, N) ⟶ Ext_{kG}^{2}(M, N)

is non-zero, then there is a non-vanishing composite of two phantom maps M ⟶ K ⟶ N[2].

(2) If the image of x ∈Pext_{kG}^{2}(M, N) under the map

Pext_{kG}^{2}(M, N) ⟶ Ext_{kG}^{2}(M, N)

vanishes, then there exists a filtered system of objects in StMod(kG) which does not lift to a filtered system of kG-modules.

The authors then compute the differential in the spectral sequence carefully enough to show that the situation (2) can actually occur. In part III of this series, which has just appeared (see [1]), Benson shows that (1) can also happen.

2. The simplification

We simplify the argument of sections 3 and 4 of [2]. Benson and Gnacadja want to produce an example of two kG-modules M and N, and an element in Pext_{kG}^{2}(M, N) which maps to zero in Ext_{kG}^{2}(M, N).

Suppose we are given a non-vanishing x ∈Pext_{kG}^{2}(M, K_{2}). The map Pext_{kG}^{2}(M, K_{2}) ⟶ Ext_{kG}^{2}(M, K_{2}) takes it to an element of Ext_{kG}^{2}(M, K_{2}), that is to a map in StMod(kG) of the form M[−2] ⟶ K_{2}. The fact that this map comes from a pure extension says that it is a phantom map in StMod(kG).

We may complete to a distinguished triangle

M[−2] ⟶ K_{2} ⟶ Y ⟶ M[−1];

in fact, we may realise this triangle as a short exact sequence

0 ⟶ K_{2} ⟶ Y ⟶ M[−1] ⟶ 0.

The fact that M[−2] ⟶ K_{2} is phantom means that this short exact sequence is pure exact. But now consider the commutative square

Pext_{kG}^{2}(M, K_{2}) ⟶ Pext_{kG}^{2}(M, Y)
↓  ↓
Ext_{kG}^{2}(M, K_{2}) ⟶ Ext_{kG}^{2}(M, Y)

The image of x ∈Pext_{kG}^{2}(M, K_{2}) by the composite

Pext_{kG}^{2}(M, K_{2})
↓
Ext_{kG}^{2}(M, K_{2}) ⟶ Ext_{kG}^{2}(M, Y)

clearly vanishes, since it is the composite

M[−2] ⟶ K_{2} ⟶ Y

of two morphisms in a triangle. This means that the image of x in Pext_{kG}^{2}(M, Y) is an element of the kernel of Pext_{kG}^{2}(M, Y) ⟶ Ext_{kG}^{2}(M, Y). But as in the end of the proof of Theorem 4.1 (bottom of page 401 of [2]), we have an exact sequence

Pext_{kG}^{1}(M, M[−1]) ⟶ Pext_{kG}^{2}(M, K_{2}) ⟶ Pext_{kG}^{2}(M, Y).

Assume Pext_{kG}^{1}(M, M[−1]) = 0. The map Pext_{kG}^{2}(M, K_{2}) ⟶ Pext_{kG}^{2}(M, Y) must then be injective. The image of x under the map

Pext_{kG}^{2}(M, K_{2}) ⟶ Pext_{kG}^{2}(M, Y)

is therefore a non-zero element of the kernel of

Pext_{kG}^{2}(M, Y) ⟶ Ext_{kG}^{2}(M, Y).

References

1. David J. Benson, Phantom maps and purity in modular representation theory III, J. Algebra. 248 (2002), 747–754.

2. David J. Benson and Gilles Ph. Gnacadja, Phantom maps and purity in modular representation theory II, Algebras and Rep. Thy. 4 (2001), 395–404.

Center for Mathematics and its Applications, School of Mathematical Sciences, John Dedman Building, The Australian National University, Canberra, ACT 0200, AUSTRALIA

  E-mail address: Amnon.Neeman@anu.edu.au