In algebraic geometry, a sheaf of algebras on a ringed space X is a sheaf of commutative rings on X that is also a sheaf of O X {\displaystyle {\mathcal {O}}_{X}} -modules. It is quasi-coherent if it is so as a module. When X is a scheme, just like a ring, one can take the global Spec of a quasi-coherent sheaf of algebras: this results in the contravariant functor Spec X {\displaystyle \operatorname {Spec} _{X}} from the category of quasi-coherent (sheaves of) O X {\displaystyle {\mathcal {O}}_{X}} -algebras on X to the category of schemes that are affine over X (defined below). Moreover, it is an equivalence: the quasi-inverse is given by sending an affine morphism f : Y → X {\displaystyle f:Y\to X} to f ∗ O Y . {\displaystyle f_{*}{\mathcal {O}}_{Y}.}
Affine morphism A morphism of schemes f : X → Y {\displaystyle f:X\to Y} is called affine if Y {\displaystyle Y} has an open affine cover U i {\displaystyle U_{i}} 's such that f − 1 ( U i ) {\displaystyle f^{-1}(U_{i})} are affine. For example, a finite morphism is affine. An affine morphism is quasi-compact and separated; in particular, the direct image of a quasi-coherent sheaf along an affine morphism is quasi-coherent. The base change of an affine morphism is affine. Let f : X → Y {\displaystyle f:X\to Y} be an affine morphism between schemes and E {\displaystyle E} a locally ringed space together with a map g : E → Y {\displaystyle g:E\to Y} . Then the natural map between the sets:
Mor Y ( E , X ) → Hom O Y -alg ( f ∗ O X , g ∗ O E ) {\displaystyle \operatorname {Mor} _{Y}(E,X)\to \operatorname {Hom} _{{\mathcal {O}}_{Y}{\text{-alg}}}(f_{*}{\mathcal {O}}_{X},g_{*}{\mathcal {O}}_{E})}
is bijective.
Examples Let f : X ~ → X {\displaystyle f:{\widetilde {X}}\to X} be the normalization of an algebraic variety X. Then, since f is finite, f ∗ O X ~ {\displaystyle f_{*}{\mathcal {O}}_{\widetilde {X}}} is quasi-coherent and Spec X ( f ∗ O X ~ ) = X ~ {\displaystyle \operatorname {Spec} _{X}(f_{*}{\mathcal {O}}_{\widetilde {X}})={\widetilde {X}}} . Let E {\displaystyle E} be a locally free sheaf of finite rank on a scheme X. Then Sym ( E ∗ ) {\displaystyle \operatorname {Sym} (E^{*})} is a quasi-coherent O X {\displaystyle {\mathcal {O}}_{X}} -algebra and Spec X ( Sym ( E ∗ ) ) → X {\displaystyle \operatorname {Spec} _{X}(\operatorname {Sym} (E^{*}))\to X} is the associated vector bundle over X (called the total space of E {\displaystyle E} .) More generally, if F is a coherent sheaf on X, then one still has Spec X ( Sym ( F ) ) → X {\displaystyle \operatorname {Spec} _{X}(\operatorname {Sym} (F))\to X} , usually called the abelian hull of F; see Cone (algebraic geometry)#Examples.
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