[permalink] [id link]
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.
from
Wikipedia
Some Related Sentences
mathematics and sheaf
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
** Part of a sheaf ( mathematics )
# REDIRECT Sheaf ( mathematics )# The étalé space of a sheaf
# REDIRECT Sheaf ( mathematics )# The étalé space of a sheaf
# REDIRECT Sheaf ( mathematics )# The étale space of a sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of O < sub > X </ sub >- modules.
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X.
In mathematics, the Borel – Weil-Bott theorem is a basic result in the representation theory of Lie groups, showing how a family of representations can be obtained from holomorphic sections of certain complex vector bundles, and, more generally, from higher sheaf cohomology groups associated to such bundles.
In mathematics, a sheaf spanned by global sections is a sheaf F on a locally ringed space X, with structure sheaf O < sub > X </ sub > that is of a rather simple type.
In mathematics, the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy, given that it is a presheaf, which is by definition a contravariant functor
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology ( and other derived functors, such as sheaf Ext ).
In mathematics, more specifically complex analysis, a holomorphic sheaf ( often also called an analytic sheaf ) is a natural generalization of the sheaf of holomorphic functions on a complex manifold.
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.
mathematics and cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex.
From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century ; from the initial idea of homology as a topologically invariant relation on chains, the range of applications of homology and cohomology theories has spread out over geometry and abstract algebra.
In mathematics, de Rham cohomology ( after Georges de Rham ) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes.
In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X.
In mathematics, particularly algebraic topology and homology theory, the Mayer – Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups.
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures.
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of differential forms of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M.
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul ( see Lie algebra cohomology ).
In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.
In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups.
In mathematics, particularly in algebraic topology, Alexander – Spanier cohomology is a cohomology theory for topological spaces, introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of A. D. Wallace.
Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s.
In mathematics, more specifically in cohomology theory, a-cocycle in the cochain group is associated with a unique equivalence class known as the cocycle class or coclass of
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space.
* Brown – Peterson cohomology, a generalized cohomology theory in mathematics
In mathematics, the Chern – Weil homomorphism is a basic construction in the Chern – Weil theory, relating for a smooth manifold M the curvature of M to the de Rham cohomology groups of M, i. e., geometry to topology.
In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds.
mathematics and is
This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.
Next September, after receiving a degree from Yale's Master of Arts in Teaching Program, I will be teaching somewhere -- that much is guaranteed by the present shortage of mathematics teachers.
But because science is based on mathematics doesn't mean that a hot rodder must necessarily be a mathematician.
Like primitive numbers in mathematics, the entire axiological framework is taken to rest upon its operational worth.
In the new situation, philosophy is able to provide the social sciences with the same guidance that mathematics offers the physical sciences, a reservoir of logical relations that can be used in framing hypotheses having explanatory and predictive value.
So, too, is the mathematical competence of a college graduate who has majored in mathematics.
The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.
In mathematics and statistics, the arithmetic mean, or simply the mean or average when the context is clear, is the central tendency of a collection of numbers taken as the sum of the numbers divided by the size of the collection.
The term " arithmetic mean " is preferred in mathematics and statistics because it helps distinguish it from other means such as the geometric and harmonic mean.
In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.
The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation.
In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".
The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.
The axiom of choice has also been thoroughly studied in the context of constructive mathematics, where non-classical logic is employed.
:" A choice function exists in constructive mathematics, because a choice is implied by the very meaning of existence.
Although the axiom of countable choice in particular is commonly used in constructive mathematics, its use has also been questioned.
One of the most interesting aspects of the axiom of choice is the large number of places in mathematics that it shows up.
There is no prize awarded for mathematics, but see Abel Prize.
0.161 seconds.