Sheaf of algebras

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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 ${\displaystyle {\mathcal {O}}_{X}}$-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 ${\displaystyle \operatorname {Spec} _{X}}$ from the category of quasi-coherent (sheaves of) ${\displaystyle {\mathcal {O}}_{X}}$-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 ${\displaystyle f:Y\to X}$ to ${\displaystyle f_{*}{\mathcal {O}}_{Y}.}$^[1]