On connectedness and indecomposibility of local cohomology modules

Abstract

Let $I$ denote an ideal of a local Gorenstein ring $(R,\mathfrak{m})$. Then we show that the local cohomology module $H^c_I(R), c = \height I,$ is indecomposable if and only if $V(I_d)$ is connected in codimension one. Here $I_d$ denotes the intersection of the highest dimensional primary components of $I.$ This is a partial extension of a result shown by Hochster and Huneke in the case $I$ the maximal ideal. Moreover there is an analysis of connectedness properties in relation to various aspects of local cohomology. Among others we show that the endomorphism ring of $H_I^c(R)$ is a local Noetherian ring if $\dim R/I=1.$

References in corpus (2)

• On Lyubeznik's invariants and endomorphisms of local cohomology modules
• On some local cohomology invariants of local rings