Depth and cohomological connectivity in modular invariant theory

Let G be a finite group acting linearly on a finite-dimensional vector

space V over a field K of characteristic p. Assume that p divides the

order of G so that V is a modular representation and let P be a Sylow

p-subgroup for G. De. ne the cohomological connectivity of the

symmetric algebra S( V *) to be the smallest positive integer m such

that H-m( G, S( V *)) not equal 0. We show that min {dim(K)(V-P) + m+

1, dim(K)( V)} is a lower bound for the depth of S( V *) G. We

characterize those representations for which the lower bound is sharp

and give several examples of representations satisfying the criterion.

In particular, we show that if G is p-nilpotent and P is cyclic, then,

for any modular representation, the depth of S( V *) G is min

{dim(K)(V-P) + 2, dim(K)(V)}.