Graded-Injective Modules and Bass Numbers of Veronese Submodules

TL;DR

This paper investigates how Bass numbers of graded modules and local cohomology modules behave under Veronese functors in standard graded algebras, providing explicit formulas and applications.

Contribution

It computes Bass numbers of Veronese modules over graded algebras and applies results to local cohomology, extending understanding of their invariants.

Findings

Bass numbers of $M^{(n)}$ are expressed in terms of those of $M$.

Bass numbers of local cohomology modules are determined under finite Bass number conditions.

Explicit formulas relate Bass numbers before and after applying Veronese functors.

Abstract

Let $R$ be a standard graded, finitely generated algebra over a field, and let $M$ be a graded module over $R$ with all Bass numbers finite. Set $(-)^{(n)}$ to be the $n$-th Veronese functor. We compute the Bass numbers of $M^{(n)}$ over the ring $R^{(n)}$ for all prime ideals of $R^{(n)}$ that are not the homogeneous maximal ideal in terms of the Bass numbers of $M$ over $R$. As an application to local cohomology modules, we determine the Bass numbers of $H_{I\cap R^{(n)}}^i(R^{(n)})$ over the ring $R^{(n)}$ in the case where $H_I^i(R)$ has finite Bass numbers over $R$ and $I$ is a graded ideal.