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 ${\displaystyle f_{*}{\mathcal {O}}_{Y}.}$

A morphism of schemes ${\displaystyle f:X\to Y}$ is called affine if ${\displaystyle Y}$ has an open affine cover ${\displaystyle U_{i}}$'s such that ${\displaystyle f^{-1}(U_{i})}$ 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 ${\displaystyle f:X\to Y}$ be an affine morphism between schemes and ${\displaystyle E}$ a locally ringed space together with a map ${\displaystyle g:E\to Y}$. Then the natural map between the sets:

is bijective.

Given a ringed space S, there is the category ${\displaystyle C_{S}}$ of pairs ${\displaystyle (f,M)}$ consisting of a ringed space morphism ${\displaystyle f:X\to S}$ and an ${\displaystyle {\mathcal {O}}_{X}}$-module ${\displaystyle M}$. Then the formation of direct images determines the contravariant functor from ${\displaystyle C_{S}}$ to the category of pairs consisting of an ${\displaystyle {\mathcal {O}}_{S}}$-algebra A and an A-module M that sends each pair ${\displaystyle (f,M)}$ to the pair ${\displaystyle (f_{*}{\mathcal {O}},f_{*}M)}$.

Now assume S is a scheme and then let ${\displaystyle \operatorname {Aff} _{S}\subset C_{S}}$ be the subcategory consisting of pairs ${\displaystyle (f:X\to S,M)}$ such that ${\displaystyle f}$ is an affine morphism between schemes and ${\displaystyle M}$ a quasi-coherent sheaf on ${\displaystyle X}$. Then the above functor determines the equivalence between ${\displaystyle \operatorname {Aff} _{S}}$ and the category of pairs ${\displaystyle (A,M)}$ consisting of an ${\displaystyle {\mathcal {O}}_{S}}$-algebra A and a quasi-coherent ${\displaystyle A}$-module ${\displaystyle M}$.