Ext functors, support varieties and Hilbert polynomials over complete intersection rings
Tony J. Puthenpurakal
TL;DR
The work develops a deep link between Ext-module growth over complete intersections and the geometry of support varieties. It proves stabilization of polynomial-type degrees ψ_{2i+j}^I(M) for i≫0, defines invariants r_j^I(M) and r^I(M) tied to V_A(M), and shows boundedness of reg G_I(Ω^i(M)) in the r^I(M) ≤ 0 case, yielding uniform Artin–Rees-type results for syzygies. Through MCM-approximations, base-change techniques, and complexity-reduction methods, it provides a robust framework connecting algebraic invariants to geometric data, with a suite of examples illustrating the phenomena. The results have implications for the uniform behavior of free resolutions and Artin–Rees type properties in families of modules over complete intersections.
Abstract
Let $(A,\mathfrak{m})$ be a complete intersection of dimension $d \geq 1$ and codimension $c \geq 1$. Let $I$ be an $\mathfrak{m}$-primary ideal and let $M$ be a finitely generated $A$-module. For $i \geq 1$ let $ψ_i^I(M)$ be the degree of the polynomial type function $n \rightarrow \ell(Ext^i_A(M, A/I^n))$. We show that for $j = 0, 1$ and for all $i \gg 0$ we have $ψ_{2i +j}^I(M)$ is a constant and let $r_0^I(M)$ and $r_1^I(M)$ denote these constant values. Set $r^I(M) = \max\{ r_0^I(M), r_1^I(M) \}$. We show that $r^I(M)$ is an invariant of $I, A$ and the support variety of $M$. We set the degree of the zero polynomial to be $-\infty$. If $r^I(M) \leq 0$ then we show that $reg \ G_I(Ω^i(M))$ for $i \geq 0$ 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.