Sheaf of algebras explained
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
Affine morphism
is called
affine if
has an open affine cover
's such that
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.[1]
Let
be an affine morphism between schemes and
a
locally ringed space together with a map
. Then the natural map between the sets:
\operatorname{Mor}Y(E,X)\to\operatorname{Hom}l{OY-alg
}(f_* \mathcal_X, g_* \mathcal_E)is bijective.
Examples
be the normalization of an algebraic variety
X. Then, since
f is finite,
} is quasi-coherent and
\operatorname{Spec}X(f*l{O}\widetilde{X
}) = \widetilde.
be a locally free sheaf of finite rank on a scheme
X. Then
is a quasi-coherent
-algebra and
| *)) |
| \operatorname{Spec} | |
| X(\operatorname{Sym}(E |
\toX
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
\operatorname{Spec}X(\operatorname{Sym}(F))\toX
, usually called the abelian hull of
F; see Cone (algebraic geometry)#Examples.
The formation of direct images
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
\operatorname{Aff}S\subsetCS
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
.
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:
f:X=\operatorname{Spec}S(A)\toS
. 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.
See also
External links
- https://ncatlab.org/nlab/show/affine+morphism
Notes and References
- .
