The invertible sheaves in those theories are in effect the line bundles appropriately formulated. In fact, the abstract definition in scheme theory of invertible sheaf can be replaced by the condition of being locally free of rank 1. That is, the condition of a tensor inverse then implies, locally on X, that S is the sheaf form of a free rank-1 module over a commutative ring. Examples come from fractional ideals in algebraic number theory, so that the definition captures that theory. More generally, when X is an affine schemeSpec(R), the invertible sheaves come from projective modules over R, of rank 1.
Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product. This group generalises the ideal class group. In general it is written
invertible, sheaf, mathematics, invertible, sheaf, coherent, sheaf, ringed, space, which, there, inverse, with, respect, tensor, product, modules, equivalent, algebraic, geometry, topological, notion, line, bundle, their, interactions, with, cartier, divisors,. 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 OX modules It is the equivalent in algebraic geometry of the topological notion of a line bundle Due to their interactions with Cartier divisors they play a central role in the study of algebraic varieties Contents 1 Definition 2 The Picard group 3 See also 4 ReferencesDefinition EditAn invertible sheaf is a locally free sheaf S on a ringed space X for which there is an inverse T with respect to tensor product of OX modules that is we have S T displaystyle S otimes T isomorphic to OX which acts as identity element for the tensor product The most significant cases are those coming from algebraic geometry and complex geometry For spaces such as locally Noetherian schemes or complex manifolds one can actually replace locally free by coherent in the definition The invertible sheaves in those theories are in effect the line bundles appropriately formulated In fact the abstract definition in scheme theory of invertible sheaf can be replaced by the condition of being locally free of rank 1 That is the condition of a tensor inverse then implies locally on X that S is the sheaf form of a free rank 1 module over a commutative ring Examples come from fractional ideals in algebraic number theory so that the definition captures that theory More generally when X is an affine scheme Spec R the invertible sheaves come from projective modules over R of rank 1 The Picard group EditMain article Picard group Quite generally the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product This group generalises the ideal class group In general it is written P i c X displaystyle mathrm Pic X with Pic the Picard functor Since it also includes the theory of the Jacobian variety of an algebraic curve the study of this functor is a major issue in algebraic geometry The direct construction of invertible sheaves by means of data on X leads to the concept of Cartier divisor See also EditVector bundles in algebraic geometry Line bundle First Chern class Picard group Birkhoff Grothendieck theoremReferences EditSection 0 5 4 of Grothendieck Alexandre Dieudonne Jean 1960 Elements de geometrie algebrique I Le langage des schemas Publications Mathematiques de l IHES 4 doi 10 1007 bf02684778 MR 0217083 Retrieved from https en wikipedia org w index php title Invertible sheaf amp oldid 1110578833, wikipedia, wiki, book, books, library,
