On Lyubeznik’s Invariants and the Vanishing of Local Cohomology

This paper contributes to the study of local cohomology and topological, in particular connectedness properties. Let I denote an ideal of a regular local ring (R,m) containing a field. There is a description of the Lyubeznik number λd,d(R/I), d = dimR/I, in terms of the topology of V (IR̂). In particular, λd,d(R/I) = 1 if and only if V (IR̂) is connected in codimension one. This provides an alternative proof of results shown by Lyubeznik in case of prime characteristic and by Zhang. Further results on the Lyubeznik numbers are shown. They provide vanishing results on the local cohomology modules H I(R), in particular for i = dimR− 1.