In this note we describe aspects of the cohomology of coherent sheaves on a complete toric variety X over a field k and, more generally, the local cohomology, with supports in a monomial ideal, of a finitely generated module over a polynomial ring S. This leads to an efficient way of computing such cohomology, for which we give explicit algorithms. The problem is finiteness. The i local cohomol...

This article is based on five lectures the author gave during the summer school, Interactions between Homotopy Theory and Algebra, from July 26–August 6, 2004, held at the University of Chicago, organized by Lucho Avramov, Dan Christensen, Bill Dwyer, Mike Mandell, and Brooke Shipley. These notes introduce basic concepts concerning local cohomology, and use them to build a proof of a theorem Gr...

Local cohomology was discovered in the 1960s as a tool to study sheaves and their cohomology in algebraic geometry, but have since seen wide use in commutative algebra. An example of their use is to answer the question: how many elements are necessary to generate a given ideal, up to radical? For example, consider two planes in 4-space meeting at a point. The vanishing ideal I = (x, y) ∩ (u, v)...

We prove a duality theorem for graded algebras over a field that implies several known duality results: graded local duality, versions of Serre duality for local cohomology and of Suzuki duality for generalized local cohomology, and Herzog-Rahimi bigraded duality.

We introduce a generalization of formal local cohomology module, which we call a formal local cohomology module with respect to a pair of ideals and study its various properties. We analyze their structure, the upper and lower vanishing and non-vanishing. There are various exact sequences concerning the formal cohomology modules. Among them a MayerVietoris sequence for two ideals with respect t...

let $a_1$, $a_2$ be unital banach algebras and $x$ be an $a_1$-$a_2$- module. applying the concept of module maps, (inner) modulegeneralized derivations and  generalized first cohomology groups, wepresent several results concerning the relations between modulegeneralized derivations from $a_i$ into the dual space $a^*_i$ (for$i=1,2$) and such derivations  from  the triangular banach algebraof t...

In this paper we introduce a new definition of the first non-abelian cohomology of topological groups.  We relate the cohomology of a normal subgroup $N$ of a topological group $G$ and the quotient $G/N$ to the cohomology of $G$. We get the inflation-restriction exact sequence. Also, we obtain a seven-term exact cohomology sequence up to dimension 2. We give an interpretation of the first non-a...

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...