Asymptotic behavior of Ext for pairs of modules of large complexity over graded complete intersections

TL;DR

This paper investigates the asymptotic behavior of Ext modules over graded complete intersections, revealing that the Hilbert polynomials for large even and odd degrees share the same degree and leading coefficient under certain conditions.

Contribution

It establishes a new relationship between the Hilbert polynomials of Ext modules for modules over graded complete intersections, especially regarding their degrees and leading coefficients.

Findings

Even and odd Hilbert polynomials have the same degree and leading coefficient.

Results depend on the highest degree of the Hilbert polynomials relative to module dimension.

Refinements are provided when the ring is regular in small codimensions.

Abstract

Let $M$ and $N$ be finitely generated graded modules over a graded complete intersection $R$ such that $\operatorname{Ext}_R^i(M,N)$ has finite length for all $i\gg 0$. We show that the even and odd Hilbert polynomials, which give the lengths of $\operatorname{Ext}^i_R(M,N)$ for all large even $i$ and all large odd $i$, have the same degree and leading coefficient whenever the highest degree of these polynomials is at least the dimension of $M$ or $N$. Refinements of this result are given when $R$ is regular in small codimensions.