Infinite dimensional modules for finite groups.

Three lectures by Dave Benson, Bielefeld, September 1998.

This series of lectures is about the recent developments in modular representation theory of finite groups, centered around the properties of certain infinite dimensional modules and their influence on the theory of finite dimensional modules.

Let G be a finite group and k a field of characteristic p dividing the order of G. Then kG is a finite dimensional symmetric algebra, and we are interested in the category mod(kG) of finite dimensional kG-modules and the category Mod(kG) of all kG-modules. The corresponding stable categories stmod(kG) and StMod(kG) are triangulated categories, and homomorphisms of degree n in this category correspond to elements of Tate Ext in degree n.

Some of the problems I wish to discuss are as follows:

1. How do you tell if a kG-module is projective?

2. What are the thick subcategories of stmod(kG)?

3. Given a simple module S in the principal block of kG, is it true that there exists n > 0 such that H^{n}(G, S) ≠ 0?

4. For what groups G is it true that for all nonprojective finite dimensional modules M in the principal block of kG, there exists n > 0 such that H^{n}(G, M) ≠ 0?

5. Given a finite dimensional indecomposable periodic module, can we form a direct sum with an appropriate projective module in such a way that the result is induced from a suitable p-local subgroup? What about nonperiodic modules?

The study of these questions leads naturally to the study of infinitely generated modules, and in particular to some infinitely generated modules introduced by Rickard. Associated to any thick subcategory C of the stable module category stmod(kG) and any module M, we have a short exact sequence

0 → E_{C}(M) → M ⊕ (projective) → F_{C}(M) → 0.

The functors E_{C} and F_{C} are idempotent in the sense that repeated application is equivalent to just one application. If C is closed under tensor products with arbitrary finitely generated modules then the functors E_{C} and F_{C} are given by tensor product with modules E_{C} and F_{C}. These modules are orthogonal idempotents in the sense that

(1) E_{C} ⊗ E_{C} ≅ E_{C} ⊕ (projective)

(2) F_{C} ⊗ F_{C} ≅ F_{C} ⊕ (projective)

(3) E_{C} ⊗ F_{C} ≅ (projective).

My goal in these lectures is to introduce these idempotent modules and explain their relevance to the questions described above. Some time will be spent developing the general theory of infinitely generated modules, and the last lecture will be devoted to the theory of phantom maps and their relationship with purity. A homomorphism between infinitely generated modules M → N is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. I shall describe some work of my student Gilles Gnacadja and some of our joint work on these phantom maps.

References

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

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

3. , Complexity and varieties for infinitely generated modules, II, Math. Proc. Camb. Phil. Soc. 120 (1996), 597–615.

4. , Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59–80.

5. D. J. Benson, J. F. Carlson, and G. R. Robinson, On the vanishing of group cohomology, J. Algebra 131 (1990), 40–73.

6. D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, I, preprint, 1997.

7. G. Ph. Gnacadja, Phantom maps in the stable module category, to appear in Journal of Algebra.

8. J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149–170.