Gregor Kemper
Universitaet Heidelberg

"The Depth of Invariant Rings and Cohomology" (with an appendix by Kay
Magaard),
Preprint 99-35, IWR, Heidelberg, 1999.

Abstract:

  Let $G$ be a finite group acting linearly on a vector space $V$ over
  a field $K$ of positive characteristic~$p$, and let $P \leq G$ be a
  Sylow $p$-subgroup. Ellingsrud and Skjelbred proved the lower bound
  \[
  \depth(K[V]^G) \geq \min\{\dim(V^P)+2,\dim(V)\}
  \]
  for the depth of the invariant ring, with equality if $G$ is a
  cyclic $p$-group.  Let us call the pair $(G,V)$ {\em flat\/} if
  equality holds in the above. In this paper we use cohomological
  methods to obtain information about the depth of invariant rings and
  in particular to study the question of flatness. For $G$ of order
  not divisible by~$p^2$ it ensues that $(G,V)$ is flat if and only if
  $\dim(V^P) \geq \dim(V) - 2$ or $H^1(G,K[V]) \neq 0$. We obtain a
  formula for the depth of the invariant ring in the case that $G$
  permutes a basis of $V$ and has order not divisible by~$p^2$. In
  this situation $(G,V)$ is usually not flat. Moreover, we introduce
  the notion of visible flatness of pairs $(G,V)$ and prove that this
  implies flatness. For example, the groups $\SL_2(q)$, $\SO_3(q)$,
  $\SU_3(q)$, $\Sz(q)$, and R$(q)$ with many interesting
  representations in defining characteristic are visibly flat. In
  particular, if $G = \SL_2(q)$ and $V$ is the space of binary forms
  of degree~$n$ or a direct sum of such spaces, then $(G,V)$ is flat
  for all $q = p^r$ with the exception of a finite number of
  primes~$p$.
  
  Along the way, we obtain results about the Buchsbaum property of
  invariant rings, and about the depth of the cohomology modules
  $H^i(G,K[V])$. We also determine the support of the positive
  cohomology $H^+(G,K[V])$ as a module over $K[V]^G$. In the appendix,
  the visibly flat pairs $(G,V)$ of a group with $BN$-pair and an
  irreducible representation are classified.