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On an Endomorphism Ring of Local Cohomology

Let I be an ideal of a local ring (R, 𝔪) with d = dim R. For the local cohomology module it is a well-known fact that it vanishes for i > d and is an Artinian R-module for i = d. In the case that the Hartshorne–Lichtenbaum Vanishing Theorem fails, that is we explore its fine structure. In particular, we investigate its endomorphism ring and related connectedness properties. In the case R is complete we prove—as a technical tool—that for a certain ideal J ⊂ R. Thus, properties of and its Matlis dual might be described in terms of the local cohomology supported in the maximal ideal.