Local cohomology modules of polynomial or power series rings over rings of small dimension

TL;DR

This paper proves finiteness of Bass numbers and associated primes of local cohomology modules over polynomial or power series rings when the base ring has small dimension, extending known results without requiring a field.

Contribution

It establishes finiteness properties of local cohomology modules over rings of small dimension without the need for the base ring to contain a field, and extends results on injective dimension in mixed characteristic.

Findings

Finite Bass numbers and associated primes when base ring has dimension zero.

Finiteness extends to certain prime ideals when base ring has dimension one.

Extended results on injective dimension of local cohomology in mixed characteristic.

Abstract

Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$. When $A$ has dimension zero, we show that the Bass numbers and the associated primes of the local cohomology modules over $R$ are finite. Moreover, if $A$ has dimension one and $\pi$ is an nonzero divisor, then the same properties hold for prime ideals that contain $\pi.$ These results do not require that $A$ contains a field. As a consequence, we give a different proof for the finiteness properties of local cohomology over unramified regular local rings. In addition, we extend previous results on the injective dimension of local cohomology modules over certain regular rings of mixed characteristic.