Let ρ:G → GL(n,F) be a representation of a finite group over the field F of characteristic p, and h_1,…, h_m ∈ F[V]~G invariant polynomials that form a regular sequence in F[V]. In this note we introduce a tool to study the problem of whether they form a regular sequence in F[V]~G. Examples show they need not. We define the cohomology of G with coefficients in the Koszul complex (K, (partial deriv)) = (F[V]direct X E(s~(-1)h_1,…, s~(-1)h_n), partial deriv(s~(-1)h_i) = h_i:i = 1,…, n), which we denote by H~*(G; (K, (partial deriv))), and use it to study the homological codimension of rings of invariants of permutation representations of the cyclic group of order p, for p ≠ 0, and to answer the above question in this case.
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