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
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
from the category of quasi-coherent (sheaves of)
-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
has an open affine cover
[2] 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.
be an affine morphism between schemes and
a locally ringed space together with a map
Then the natural map between the sets: is bijective.
[4] Given a ringed space S, there is the category
consisting of a ringed space morphism
Then the formation of direct images determines the contravariant functor from
to the category of pairs consisting of an
-algebra A and an A-module M that sends each pair
Now assume S is a scheme and then let
be the subcategory consisting of pairs
is an affine morphism between schemes and
a quasi-coherent sheaf on
Then the above functor determines the equivalence between
and the category of pairs
[5] The above equivalence can be used (among other things) to do the following construction.
As before, given a scheme S, let A be a quasi-coherent
-algebra and then take its global Spec:
called the sheaf associated to M. Put in another way,
determines an equivalence between the category of quasi-coherent