Let (R,m) be a local Noetherian ring, let I ⊂ R be any ideal and let M be a finitely generated R-module. It has been long conjectured that the local cohomology modules H I(M) have finitely many associated primes for all i (see Conjecture 5.1 in [H] and [L].) If R is not required to be local these sets of associated primes may be infinite, as shown by Anurag Singh in [S], where he constructed an...

Introduction 2 1. A Guide to Duality 3 1.1. Local Duality 3 1.2. Dualizing Complexes and Some Vanishing Theorems 10 1.3. Cohomological Annihilators 17 2. A Few Applications of Local Cohomology 21 2.1. On Ideal Topologies 21 2.2. On Ideal Transforms 25 2.3. Asymptotic Prime Divisors 28 2.4. The Lichtenbaum-Hartshorne Vanishing Theorem 35 2.5. Connectedness results 37 3. Local Cohomology and Syzy...

A projective algebraic variety X over an algebraically closed field k of positive characteristic is called globally F -regular if the section ring S(L) = ⊕n≥0 H (X,Ln) of an ample line bundle L on X is strongly F -regular in the sense of Hochster and Huneke [9] (cf. Definition 1.1). This important notion was introduced by Karen Smith in [18]. In this paper we prove that Schubert varieties are g...