Type of ringed space
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
-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
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
to
[1]
A
morphism of schemes
is called affine if
has an open affine cover
's such that
are affine.
[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.
[3]
Let
be an affine morphism between schemes and
a
locally ringed space together with a map
. Then the natural map between the sets:
![{\displaystyle \operatorname {Mor} _{Y}(E,X)\to \operatorname {Hom} _{{\mathcal {O}}_{Y}-{\text{alg}}}(f_{*}{\mathcal {O}}_{X},g_{*}{\mathcal {O}}_{E})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed470517150f7e48ca8051f705763a19495237ab)
is bijective.
[4]
- Let
be the normalization of an algebraic variety X. Then, since f is finite,
is quasi-coherent and
.
- Let
be a locally free sheaf of finite rank on a scheme X. Then
is a quasi-coherent
-algebra and
is the associated vector bundle over X (called the total space of
.)
- More generally, if F is a coherent sheaf on X, then one still has
, usually called the abelian hull of F; see
Cone (algebraic geometry)#Examples.
Given a ringed space S, there is the category
of pairs
consisting of a ringed space morphism
and an
-module
. 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
to the pair
.
Now assume S is a scheme and then let
be the subcategory consisting of pairs
such that
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
consisting of an
-algebra A and a quasi-coherent
-module
.
[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:
. Then, for each quasi-coherent A-module M, there is a corresponding quasi-coherent
-module
such that
called the sheaf associated to M. Put in another way,
determines an equivalence between the category of quasi-coherent
-modules and the quasi-coherent
-modules.