Cartier isomorphism information

In algebraic geometry, the Cartier isomorphism is a certain isomorphism between the cohomology sheaves of the de Rham complex of a smooth algebraic variety...

with positive characteristic or more arithmetic structure, additional isomorphism types exist. For example, if 2 is invertible over the base, all group...

interactions with Cartier divisors, they play a central role in the study of algebraic varieties. Let (X, OX) be a ringed space. Isomorphism classes of sheaves...

geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor in a scheme X over...

Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group...

A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is...

abelian group with 1 its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, R {\displaystyle...

bundle. The restriction from the divisor class group Cl(X) to Cl(U) is an isomorphism, and (since U is smooth) Cl(U) can be identified with the Picard group...

from the group of Cartier divisors to the Picard group Pic ⁡ ( X ) {\displaystyle \operatorname {Pic} (X)} of X, the group of isomorphism classes of line...

the first Chern class is the isomorphism from the Picard group of line bundles on a variety X to the group of Cartier divisors modulo linear equivalence...

the pair ( Q , ℓ ) {\displaystyle (Q,\ell )} to Q. This morphism is an isomorphism on the open subset of all points ( Q , ℓ ) {\displaystyle (Q,\ell )}...

as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor K {\displaystyle K} on V {\displaystyle V} giving rise to the canonical...

observation gives an isomorphism between Σ n {\displaystyle \Sigma _{n}} and Σ − n {\displaystyle \Sigma _{-n}} since there is the isomorphism vector bundles...

{\text{End}}({\widehat {G}})^{\text{op}}} . More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories...

\mathrm {Pic} \ \mathbf {P} _{\mathbf {k} }^{n}=\mathbb {Z} } , and the isomorphism is given by the degree of divisors. The invertible sheaves, or line bundles...

dual morphisms fv: Bv → Av in a compatible way, and there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle)....

This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle) and...

classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of)...

dimension. Let D ↪ X {\displaystyle D\hookrightarrow X} be an effective Cartier divisor. Then the normal bundle to it (or equivalently the normal cone...

{F}}(U)\to {\mathcal {F}}(V),\,f\otimes s\mapsto f\cdot s|_{V}} is an isomorphism. For each open affine subscheme U = Spec ⁡ A {\displaystyle U=\operatorname...

between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from...

to) local class field theory. More precisely the Artin map gives an isomorphism from the group GL1(K)= K* to the abelianization of the Weil group. In...