On Invariants and Endomorphism Rings of Local Cohomology Modules

Abstract

Let $(R,\mathfrak{m})$ denote an $n$-dimensional Gorenstein ring. For an ideal $I\subset R$ with $\grade I = c$ we define new numerical invariants ${\tau }_{i,j}(I)$ as the socle dimensions of $H_{\mathfrak{m}}^i(H_I^{n-j}(R))$. In case of a regular local ring containing a field these numbers coincide with the Lyubeznik numbers ${\lambda }_{i,j}(R/I)$. We use ${\tau }_{d,d}(I),d=\dim R/I,$ to characterize the surjectivity of the natural homomorphism $f : \hat{R} \to \Hom_{\hat{R}}(H^c_{I\hat{R}}(\hat{R}),H^c_{I\hat{R}}(\hat{R}))$. As a technical tool we study several natural homomorphisms. Moreover we prove a few results on ${\tau }_{i,j}(I)$.

arXiv commentAccepted in Journal of Algebra