In algebraic geometry, local cohomology is an algebraic analogue of relative cohomology. Alexander Grothendieck introduced it in seminars in Harvard in 1961 written up by , and in 1961-2 at IHES written up as SGA2 - , republished as . Given a function (more generally, a section of a quasicoherent sheaf) defined on an open subset of an algebraic variety (or scheme), local cohomology measures the obstruction to extending that function to a larger domain. The rational function , for example, is defined only on the complement of on the affine line over a field , and cannot be extended to a function on the entire space. The local cohomology module (where is the coordinate ring of ) detects this in the nonvanishing of a cohomology class . In a similar manner, is defined away from the and axes in the affine plane, but cannot be extended to either the complement of the -axis or the complement of the -axis alone (nor can it be expressed as a sum of such functions); this obstruction corresponds precisely to a nonzero class in the local cohomology module .
Outside of algebraic geometry, local cohomology has found applications in commutative algebra, combinatorics, and certain kinds of partial differential equations.
In the most general geometric form of the theory, sections are considered of a sheaf of abelian groups, on a topological space , with support in a closed subset , The derived functors of form local cohomology groups
In the theory's algebraic form, the space X is the spectrum Spec(R) of a commutative ring R (assumed to be Noetherian throughout this article) and the sheaf F is the quasicoherent sheaf associated to an R-module M, denoted by . The closed subscheme Y is defined by an ideal I. In this situation, the functor ΓY(F) corresponds to the I-torsion functor, a union of annihilators
i.e., the elements of M which are annihilated by some power of I. As a right derived functor, the ith local cohomology module with respect to I is the ith cohomology group of the chain complex obtained from taking the I-torsion part of an injective resolution of the module .
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In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent.
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.
In abstract algebra, a dualizing module, also called a canonical module, is a module over a commutative ring that is analogous to the canonical bundle of a smooth variety. It is used in Grothendieck local duality. A dualizing module for a Noetherian ring R is a finitely generated module M such that for any maximal ideal m, the R/m vector space Ext(R/m,M) vanishes if n ≠ height(m) and is 1-dimensional if n = height(m). A dualizing module need not be unique because the tensor product of any dualizing module with a rank 1 projective module is also a dualizing module.
ln this course we will develop algebraic and coalgebraic models for homotopy types.
Among other things we will learn about Quillen's and Sullivan's model of rationâl homotopy types and about Mandell's
Explore la théorie de l'homotopie des complexes de chaînes, en se concentrant sur les catégories de modèles, les équivalences faibles, et l'axiome de rétractation.
We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring g ...
In this thesis, we give a modern treatment of Dwyer's tame homotopy theory using the language of ∞-categories.We introduce the notion of tame spectra and show it has a concrete algebraic description.We then carry out a study of ∞-operads and ...
EPFL2022
We prove the vanishing of the bounded cohomology of lamplighter groups for a wide range of coefficients. This implies the same vanishing for a number of groups with self-similarity properties, such as Thompson's group F. In particular, these groups are bou ...