Some numerical invariants of local rings

Abstract

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