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...torsion abelian group]]. Similarly, for a [[prime number]] ''p'', we say a sheaf <math>\mathcal{F}</math> is '''''p''-torsion''' if every section over any o A torsion sheaf on an étale site is the union of its [[constructible sheaf|constructible subsheaves]].<ref>{{harvnb|Milne|2012|loc=Remark 17.6}}</ref> ...

...bundle]] modulo some singularity. The notion is important both in [[scheme theory]] and [[complex algebraic geometry]]. For the theory of reflexive sheaves, one works over an [[Integral domain|integral]] [[Noet ...

{{Short description|Sheaf theory concept}} ...of local information coming from its stalks. It is useful for computing [[sheaf cohomology]]. It was discovered by [[Roger Godement]]. ...

{{Short description|Sheaf theory}} ...en ''X'' is a [[stratified space]], a [[constructible sheaf]] is roughly a sheaf that is locally constant on each member of the stratification. ...

...ry of group schemes based on the notion of group functor instead of scheme theory. ...om <math>\mathsf{Sch}_S</math> to the category of groups that is a Zariski sheaf (i.e., satisfying the gluing axiom for the Zariski topology). ...

{{Short description|Theorem in complex analysis about the sheaf of holomorphic functions}} ...omorphic functions on a [[complex manifold]] <math>X</math>) is [[coherent sheaf|coherent]].<ref>{{harvtxt|Noguchi|2019}}</ref><ref>In {{harvtxt|Oka|1950}} ...

{{short description|Result in algebraic K-theory relating Chow groups to cohomology}} ...the cohomology of ''X'' with coefficients in the K-theory of the structure sheaf <math>\mathcal{O}_X</math>; that is, ...

...erential equation|linear partial differential equations]] by using [[sheaf theory]] and [[complex analysis]] to study properties and generalizations of [[Fun ...l definition|dimension]] ''n'', and let ''X'' be its complexification. The sheaf of '''microlocal functions''' on ''M'' is given as{{sfn|Kashiwara|Schapira| ...

...ules|sheaf of <math>\mathcal{O}_X</math>-modules]]. It is [[quasi-coherent sheaf|quasi-coherent]] if it is so as a module. ...]], just like a ring, one can take the [[global Spec]] of a quasi-coherent sheaf of algebras: this results in the contravariant [[functor]] <math>\operatorn ...

...in this case says that to give an equivariant sheaf on ''X'' is to give a sheaf on the quotient ''X''/''G''. ...df |title=Notes on Grothendieck topologies, fibered categories and descent theory |date=September 2, 2008}} ...

...proper scheme ''X'' of dimension ''n'' over a field ''k'' is a [[coherent sheaf]] <math>\omega_X</math> together with a linear functional for each coherent sheaf ''F'' on ''X'' (the superscript * refers to a [[dual vector space]]).<ref n ...

...ory and <math>\xi</math> in <math>\mathfrak{X}(S)</math>, a quasi-coherent sheaf <math>F_{\xi}</math> on ''S'' together with maps implementing the compatibi ...rello|Cornalba|Griffiths|2011|loc=Ch. XIII., § 2.}}</ref> A quasi-coherent sheaf on a [[Deligne–Mumford stack]] generalizes an [[orbibundle]] (in a sense). ...

...theory]] of [[algebraic curve]]s. Furthermore, it has applications to the theory of [[modular form]]s on [[reductive group|reductive]] [[algebraic groups]]< | mr=2409679 }}</ref> and [[string theory]].<ref>{{Citation ...

...plicial sheaf''' on a site is a [[simplicial object]] in the category of [[Sheaf (mathematics)|sheaves]] on the site.<ref>{{harvnb|Jardine|2007|loc=§1}}</re ...in the site, represents a simplicial presheaf (in fact, often a simplicial sheaf). ...

...e [[derived category]] of constructible sheaves, see a section in [[ℓ-adic sheaf]]. ...n étale cohomology states that the higher direct images of a constructible sheaf are constructible. ...

...An example of a torsion-free module that is not flat is the [[Ideal (ring theory)|ideal]] (''x'', ''y'') of the [[polynomial ring]] ''k''[''x'', ''y''] over ...f associated to a module|associated]] to some module ''M'' over ''R''. The sheaf ''F'' is said to be '''torsion-free''' if all those modules ''M'' are torsi ...

...is a [[duality (mathematics)|duality]] theorem for constructible abelian [[sheaf (mathematics)|sheaves]] over the [[spectrum of a ring]] of [[algebraic numb ...a [[constructible sheaf|constructible]] [[étale topology|étale]] [[abelian sheaf]] on ''X''. Then the [[Yoneda product|Yoneda pairing]] ...

...py theory is applied to [[algebraic geometry]], such as [[motivic homotopy theory]]. ...Algébrique du Bois Marie|SGA4]], Expose V, Sec. 7, Thm. 7.4.1, to compute sheaf cohomology in arbitrary Grothendieck topologies. For the étale site the def ...

...s reason, formal schemes frequently appear in topics such as [[deformation theory]]. But the concept is also used to prove a theorem such as the [[theorem on Formal schemes were motivated by and generalize Zariski's theory of [[formal holomorphic function]]s. ...

...constructible sheaf]] can be defined as a sheaf that is [[locally constant sheaf|locally constant]] on each stratum. *[[Perverse sheaf]] ...