Smooth Algebra and Finiteness of the Set of Associated Primes of Local Cohomology Modules

Abstract

In this article, we study the behaviour of smooth algebra $R$ over local Noetherian local ring $A$. At first, we observe that for every $f\in R$, $R_f$ has finite length in the category of $D(R,A)$-module if dimension of $A$ is zero. This extends the result of Theorem 2 of \cite{Ly3}. We use this fact to generalize the result of Theorem 4.1 of \cite{BBLSZ}, from the finiteness of the set of associated primes of local cohomology module to that of Lyubeznik functor. Finally, we introduce the definition of $\Sigma$-finite $D$-modulue for smooth algebra and we extend the result of Theorem 1.3 of \cite{Nu3} from polynomial and power series algebra to smooth algebra. Theorem 1.3 of \cite{Nu3} comes out as a partial answer to a question raised by Melvin Hochster. Thus, we extend the partial answer to the above question from polynomial and power series algebra to smooth algebra over an arbitrary Noetherian local ring.

arXiv commentarXiv admin note: text overlap with arXiv:1207.1896 by other authors