Leray cover

Last updated April 17, 2019

In mathematics, a Leray cover(ing) is a cover of a topological space which allows for easy calculation of its cohomology. Such covers are named after Jean Leray.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

In mathematics, a cover of a set $is a collection of sets whose union contains as a subset. Formally, if$

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

Sheaf cohomology measures the extent to which a locally exact sequence on a fixed topological space, for instance the de Rham sequence, fails to be globally exact. Its definition, using derived functors, is reasonably natural, if technical. Moreover, important properties, such as the existence of a long exact sequence in cohomology corresponding to any short exact sequence of sheaves, follow directly from the definition. However, it is virtually impossible to calculate from the definition. On the other hand, Čech cohomology with respect to an open cover is well-suited to calculation, but of limited usefulness because it depends on the open cover chosen, not only on the sheaves and the space. By taking a direct limit of Čech cohomology over arbitrarily fine covers, we obtain a Čech cohomology theory that does not depend on the open cover chosen. In reasonable circumstances (for instance, if the topological space is paracompact), the derived-functor cohomology agrees with this Čech cohomology obtained by direct limits. However, like the derived functor cohomology, this cover-independent Čech cohomology is virtually impossible to calculate from the definition. The Leray condition on an open cover ensures that the cover in question is already "fine enough." The derived functor cohomology agrees with the Čech cohomology with respect to any Leray cover.

In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. The central figure of this study is Alexander Grothendieck and his 1957 Tohuku paper.

De Rham cohomology cohomology with real coefficients computed using differential forms

In mathematics, de Rham cohomology 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. It is a cohomology theory based on the existence of differential forms with prescribed properties.

In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.

Let ${\mathfrak {U}}=\{U_{i}\}$ be an open cover of the topological space $X$ , and ${\mathcal {F}}$ a sheaf on X. We say that ${\mathfrak {U}}$ is a Leray cover with respect to ${\mathcal {F}}$ if, for every nonempty finite set $\{i_{1},\ldots ,i_{n}\}$ of indices, and for all $k>0$ , we have that $H^{k}(U_{i_{1}}\cap \cdots \cap U_{i_{n}},{\mathcal {F}})=0$ , in the derived functor cohomology.^[1] For example, if $X$ is a separated scheme, and ${\mathcal {F}}$ is quasicoherent, then any cover of $X$ by open affine subschemes is a Leray cover.^[2]

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In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

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. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with reference to a sheaf of rings that codifies this geometric information.

In mathematics, the derived categoryD(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology.

In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced in Tôhoku paper, is a spectral sequence that computes the derived functors of the composition of two functors $, from knowledge of the derived functors of F and G .$

In algebraic topology and algebraic geometry, Leray's theorem relates abstract sheaf cohomology with Čech cohomology.

In mathematics, in the field of sheaf theory and especially in algebraic geometry, the direct image functor generalizes the notion of a section of a sheaf to the relative case.

In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.

In mathematics, Verdier duality is a duality in sheaf theory that generalizes Poincaré duality for manifolds. Verdier duality was introduced by Verdier as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck. It is commonly encountered when studying constructible or perverse sheaves.

In mathematics, especially in sheaf theory, a domain applied in areas such as topology, logic and algebraic geometry, there are four image functors for sheaves which belong together in various senses.

In algebraic topology, a branch of mathematics, the Čech-to-derived functor spectral sequence is a spectral sequence that relates Čech cohomology of a sheaf and sheaf cohomology.

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

In mathematics, an open cover of a topological space $is a family of open subsets such that is the union of all of the open sets. In algebraic topology, an open cover is called a good cover if all open sets in the cover and all intersections of finitely many open sets,, are contractible.$

In mathematics, a sheaf of O-modules or simply an O-module over a ringed space is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) →F(V) are compatible with the restriction maps O(U) →O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).

References

↑ Taylor, Joseph L. Several complex variables with connections to algebraic geometry and Lie groups. Graduate Studies in Mathematics v. 46. American Mathematical Society, Providence, RI. 2002.
↑ Macdonald, Ian G. Algebraic geometry. Introduction to schemes. W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp.

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[1] Taylor, Joseph L. Several complex variables with connections to algebraic geometry and Lie groups. Graduate Studies in Mathematics v. 46. American Mathematical Society, Providence, RI. 2002.

[2] Macdonald, Ian G. Algebraic geometry. Introduction to schemes. W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp.