An asymptotic bound for Castelnuovo-Mumford regularity of certain Ext modules over graded complete intersection rings

TL;DR

This paper establishes asymptotic bounds on the Castelnuovo-Mumford regularity of certain Ext modules over graded complete intersection rings, providing explicit inequalities and demonstrating their sharpness.

Contribution

It introduces new bounds on the regularity of Ext modules over graded complete intersection rings, extending understanding of their asymptotic behavior.

Findings

Derived explicit bounds for regularity of Ext modules.

Proved inequalities are sharp with concrete examples.

Connected regularity bounds to invariants of reduction ideals.

Abstract

Set $ A := Q/({\bf z}) $, where $ Q $ is a polynomial ring over a field, and $ {\bf z} = z_1,\ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be a homogeneous ideal of $ A $. We show that (1) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M, I^nN) \right) \le \rho_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b \mbox{ for all } i, n \ge 0 $, (2) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M,N/I^nN) \right) \le \rho_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b' \mbox{ for all } i, n \ge 0 $, where $ b $ and $ b' $ are some constants, $ f := \mathrm{min}\{ \mathrm{deg}(z_j) : 1 \le j \le c \} $, and $ \rho_N(I) $ is an invariant defined in terms of reduction ideals of $ I $ with respect to $ N $. There are explicit examples which show that these inequalities are sharp.