356 research outputs found

### Asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules

Let $A$ be a Noetherian standard $\mathbb{N}$-graded algebra over an Artinian
local ring $A_0$. Let $I_1,\ldots,I_t$ be homogeneous ideals of $A$ and $M$ a
finitely generated $\mathbb{N}$-graded $A$-module. We prove that there exist
two integers $k$ and $k'$ such that \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t}
M) \leq (n_1 + \cdots + n_t) k + k'
\quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. Comment: 9 page

### A short proof of a result of Katz and West

We give a short proof of a result due to Katz and West: Let $R$ be a
Noetherian ring and $I_1,\ldots,I_t$ ideals of $R$. Let $M$ and $N$ be finitely
generated $R$-modules and $N' \subseteq N$ a submodule. For every fixed $i \ge
0$, the sets $\mathrm{Ass}_R\left( \mathrm{Ext}_R^i(M, N/I_1^{n_1}\cdots
I_t^{n_t} N') \right)$ and $\mathrm{Ass}_R\left( \mathrm{Tor}_i^R(M,
N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ are independent of $(n_1,\ldots,n_t)$
for all sufficiently large $n_1,\ldots,n_t$.Comment: 3 pages, revised versio

### The (ir)regularity of Tor and Ext

We investigate the asymptotic behaviour of Castelnuovo-Mumford regularity of
Ext and Tor, with respect to the homological degree, over complete intersection
rings. We derive from a theorem of Gulliksen a linearity result for the
regularity of Ext modules in high homological degrees. We show a similar result
for Tor, under the additional hypothesis that high enough Tor modules are
supported in dimension at most one; we then provide examples showing that the
behaviour could be pretty hectic when the latter condition is not satisfied.Comment: 24 pages, Comments and suggestions are welcom

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