On certain rings of differentiable type and finiteness properties of local cohomology

TL;DR

This paper studies rings of differentiable type over characteristic zero fields, showing their associated graded rings are regular and Noetherian, and uses these properties to prove finiteness of associated primes of local cohomology modules.

Contribution

It establishes the structure of the associated graded ring of differentiable operators and applies this to prove finiteness of associated primes in local cohomology modules.

Findings

Associated graded ring of D(R,F) is commutative, Noetherian, regular, with pure graded dimension 2n.

D(R,F) has weak global dimension equal to n.

Finiteness of associated primes of local cohomology modules.

Abstract

Let $R$ be a commutative $F$-algebra, where $F$ is a field of characteristic 0, satisfying the following conditions: $R$ is equidimensional of dimension $n$, every residual field with respect to a maximal ideal is an algebraic extension of $F,$ and $\Der_F (R)$ is a finitely generated projective $R$-module of rank $n$ such that $R_m\otimes_R \Der_F (R)=\Der_F(R_m)$. We show that the associated graded ring of the ring of differentiable operators, $D(R,F)$, is a commutative Noetherian regular with unity and pure graded dimension equal to $2\dim(R)$. Moreover, we prove that $D(R,F)$ has weak global dimension equal to $\dim(R)$ and that its Bernstein class is closed under localization at one element. Using these properties of $D(R,F)$, we show that the set of associated primes of every local cohomology module, $H^i_I(R)$, is finite. If $(S,m,K)$ is a complete regular local ring of mixed characteristic $p>0$, we show that the localization of $S$ at $p$, $S_p$, is such a ring. As a consequence, the set of associated primes of $H^i_I (S)$ that does not contain $p$ is finite. Moreover, we prove this finiteness property for a larger class of functors.