Some numerical invariants of local rings

Abstract

Let $R$ be a formal power series ring over a field of characteristic zero and $I\subseteq R$ be any ideal. The aim of this work is to introduce some numerical invariants of the local rings $R/I$ by using theory of algebraic $\mathcal D$-modules. More precisely, we will prove that the multiplicities of the characteristic cycle of the local cohomology modules $H_I^{n-i}(R)$ and $H_{\mathfrak{p}}^p(H_I^{n-i}(R))$, where $\mathfrak{p} \subseteq R$ is any prime ideal that contains $I$, are invariants of $R/I$.