Let A be a regular ring containing field of characteristic zero and let $$R = A[X_1,\ldots , X_m]$$ . Consider R as standard graded with $$\deg 0$$ X_i 1$$ for all i. In this paper we present comprehensive study components local cohomology s $$H^i_I(R)$$ where I is an arbitrary homogeneous ideal in R. Our seems to the first regard.

The aim of this paper is to construct and examine three candidates for local-to-global spectral sequences for the cohomology of diagrams of algebras with directed indexing. In each case, the E-terms can be viewed as a type of local cohomology relative to a map or an object in the diagram.

This paper primarily concerns Galois cohomology groups associated to Galois representations over a complete local ring R. The underlying Galois module and the corresponding cohomology groups which we consider are discrete R-modules. Under certain hypotheses, we prove that the first cohomology group is an almost divisible Rmodule. We also consider the subgroup of locally trivial elements in the ...

Introduction. 1 1. Derivations and differentials 6 2. Log Calabi-Yau spaces: local structure and deformation theory 16 2.1. Local structure 16 2.2. Deformation theory 25 3. Cohomology of log Calabi-Yau spaces 38 3.1. Local calculations 38 3.2. Global calculations 46 3.3. The Hodge decomposition 59 4. Basechange and the cohomology of smoothings 67 5. Monodromy and the logarithmic Gauss-Manin con...

We reinterpret algebraic de Rham cohomology for a possibly singular complex variety X as sheaf cohomology in the site of smooth schemes over X with Voevodsky’s h-topology. Our results extend to the algebraic de Rham complex as well. Our main technique is to extend Čech cohomology of hypercovers to arbitrary local acyclic fibrations of simplicial presheaves.

This paper generalizes the notion of local coefficients and fundamental groups of spaces to simplicial coalgebras. We define a Hopf algebra π1(C) from a simplicial coalgebra C as a generalization of a fundamental group, and show that a module overπ1(C) corresponds to a local coefficient of C. As a consequence, the Hoschild cohomology of a Hopf algebraH with a coefficient M coincides with the co...