An ideal I of a local Gorenstein ring (R, m) is called cohomologically complete intersection whenever H i I (R) = 0 for all i = height I. Here H i I (R), i ∈ Z, denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of H c I (R), c = height I. As a main result it is shown that the vanishing H i I (R) = 0 for all i = c is completely encoded in homological properties of H c I (R), in particular in its Bass numbers.

Cohomological patterns of coherent sheaves over projective schemes Brodmann, M; Hellus, M Brodmann, M; Hellus, M (2002). Cohomological patterns of coherent sheaves over projective schemes. Journal of Pure and Applied Algebra, 172(2-3):165-182. AbstractWe study the sets P(X, ℱ) = (i,n) ∈ ℕ0 × ℤ Hi(X, ℱ(n)) ≠0}, where X is a projective scheme over a noetherian ring R0 and where ℱ is a coherent sheaf of OX-modules. In particular we show that P(X, ℱ) is a so called tame combinatorial pattern if the base ring R0 is semilocal and of dimension ≤ 1. If X = ℙR0d is a projective space over such a base ring R0, the possible sets P(X, ℱ) are shown to be precisely all tame combinatorial patterns of width ≤ d. We also discuss the "tameness problem" for arbitrary noetherian base rings R0 and prove some stability results for the R0-associated primes of the R0-modules Hi(X, ℱ (n)).

Abstract. 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 h I (R) if H l I (R) = 0 for every l = 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.

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Abstract. We compare the Castelnuovo-Mumford regularity and the reduction number of some classes of monomial projective curves with at most one singular point. Furthermore, for smooth monomial curves we prove an upper bound on the regularity which is stronger than the one given by L'vovsky.