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

Abstract

Set $A:=Q/(\mathbf{z})$, where $Q$ is a polynomial ring over a field, and $\mathbf{z}=z_1,\dots ,z_c$ 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 ρ_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 ρ_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b' \mbox{ for all } i, n \ge 0 $, where $b$ and $b^{\prime }$ are some constants, $f:=\min \{\mathrm{deg}(z_j):1\le j\le c\}$, 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.

14 pages, updated version, to appear in Journal of Algebra

References in corpus (2)

• Asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules
• Asymptotic Prime Divisors Related to Ext, Regularity of Powers of Ideals, and Syzygy Modules

Cited by in corpus (1)

• The (ir)regularity of Tor and Ext