COMPLEXITY AND KRULL DIMENSION

Let k be an algebraically closed field of characteristic p> 0. In this lecture we want to take a first glance at the geometric approach towards the modular representation theory of finite groups. The results presented here also hold in the wider context of finite group schemes, mainly because the fundamental result, Theorem 1, concerning the cohomology ring also holds in that generality. In the following, G denotes a finite group with group algebra kG. Let M ∈ mod kG be a finite dimensional G-module, P: = (Pi)i≥0 be a minimal projective resolution of M. Then cxG(M): = min{s ∈ N ∪ {∞} ; ∃ λ> 0 such that dimk Pn ≤ λn s−1 ∀ n ≥ 1} is called the complexity of M. This notion, first introduced by Alperin [1] and further developed in [2], has proven to be an effective tool in the modular representation theory of finite groups. Example. Consider the group algebra k(Z/(p) × Z/(p)) ∼ = k[X,Y]/(Xp, Y p) ∼ = k[X]/(Xp)⊗k k[Y]/(Y p). Let P = (Pi)i≥0 be a minimal projective resolution of the trivial k[X]/(X p)-module k, i.e., Pi = k[X]/(Xp) for every i ≥ 0. Setting Qi:= ∑i j=0 Pj ⊗k Pi−j, we obtain a minimal projective resolution Q: = (Qi)i≥0 of the k(Z/(p) × Z/(p))-module k ⊗k k ∼ = k. Since dimkQi = (i+ 1)p 2, we have cxZ/(p)×Z/(p)(k) = 2. One can iterate this process to see that c(Z/(p))r (k) = r. Let G be a finite group. If M is a G-module, we denote by Hn(G,M): = ExtnG(k,M) (n ≥ 0) the n-th cohomology group of G with coefficients in M. Note that these are just the Hochschild cohomology groups of the augmented algebra (kG, ε). Given three G-modules X,Y,Z, we recall the Yoneda produc