On endomorphism rings of local cohomology modules

Let R be a local complete ring. For an R-module M the canonical ring map R -> End(R)(M) is in general neither injective nor surjective; we show that it is bijective for every local cohomology module M := H-I(h) (R) if H-I(l) (R) = 0 for every l not equal h (= height(I)) (I an ideal of R); furthermore the same holds for the Matlis dual of such a module. As an application we prove new criteria for an ideal to be a set-theoretic complete intersection.