ON ENDOMORPHISM RINGS AND DIMENSIONS OF LOCAL COHOMOLOGY MODULES

Let (R, m) denote an n-dimensional complete local Gorenstein ring. For an ideal I of R let H-I(i)(R), i is an element of Z, denote the local cohomology modules of R with respect to I. If H-I(i)(R) = 0 for all i not equal c = height I, then the endomorphism ring of H-I(c) (R) is isomorphic to R. Here we prove that this is true if and only if H-I(i) (R) = 0 for i = n, n - 1, provided c >= 2 and R/I has an isolated singularity, resp. if I is set-theoretically a complete intersection in codimension at most one. Moreover, there is a vanishing result of H-I(i)(R) for all i > m, m a given integer, and an estimate of the dimension of H-I(i)(R).