Ext functors, support varieties and Hilbert polynomials over complete intersection rings

Let (A,m) be a complete intersection of dimension d ≥ 1 and codimension c ≥ 1. Let I be an m-primary ideal and let M be a finitely generated A-module. For i ≥ 1 let ψᵢI(M) be the degree of the polynomial type function n → (ExtⁱA(M, A/Iⁿ)). We show that for $j = 0, 1$ and for all i 0 we have ψ2i +jI(M) is a constant and let r₀I(M) and r₁I(M) denote these constant values. Set rI(M) = max\ r₀I(M), r₁I(M) \. We show that rI(M) is an invariant of $I, A$ and the support variety of M. We set the degree of the zero polynomial to be -∞. If rI(M) ≤ 0 then we show that reg \ GI(Ωⁱ(M)) for i ≥ 0 is bounded. We give an application of this result to syzgetic Artin-Rees property of M. We also give several examples which illustrate our results.