Editing Scheme (mathematics) (section)

==Coherent sheaves==
{{Main|Coherent sheaf}}
A central part of scheme theory is the notion of [[coherent sheaf|coherent sheaves]], generalizing the notion of (algebraic) [[vector bundle]]s. For a scheme ''X'', one starts by considering the [[abelian category]] of '''[[sheaf of modules|''O''<sub>''X''</sub>-modules]]''', which are sheaves of abelian groups on ''X'' that form a [[module (mathematics)|module]] over the sheaf of regular functions ''O''<sub>''X''</sub>. In particular, a module ''M'' over a commutative ring ''R'' determines an [[sheaf associated to a module|associated]] ''O''<sub>''X''</sub>-module {{overset|~|''M''}} on ''X'' = Spec(''R''). A '''[[quasi-coherent sheaf]]''' on a scheme ''X'' means an ''O''<sub>''X''</sub>-module that is the sheaf associated to a module on each affine open subset of ''X''. Finally, a '''coherent sheaf''' (on a Noetherian scheme ''X'', say) is an ''O''<sub>''X''</sub>-module that is the sheaf associated to a [[finitely generated module]] on each affine open subset of ''X''.

Coherent sheaves include the important class of '''vector bundles''', which are the sheaves that locally come from finitely generated [[free module]]s. An example is the [[tangent bundle]] of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme ''Y'' of ''X'' can be viewed as a coherent sheaf on ''X'' that is zero outside ''Y'' (by the [[direct image]] construction). In this way, coherent sheaves on a scheme ''X'' include information about all closed subschemes of ''X''. Moreover, [[sheaf cohomology]] has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of [[coherent sheaf cohomology]] is perhaps the main technical tool in algebraic geometry.{{sfn|Dieudonné|1985|loc=sections VIII.2 and VIII.3}}{{sfn|Hartshorne|1997|loc=Chapter III}}