Affine morphism
A morphism of schemes such that the pre-image of any open affine subscheme in
is an affine scheme. The scheme
is called an affine
-scheme.
Let be a scheme, let
be a quasi-coherent sheaf of
-algebras and let
be open affine subschemes in
which form a covering of
. Then the glueing of the affine schemes
determines an affine
-scheme, denoted by
. Conversely, any affine
-scheme definable by an affine morphism
is isomorphic (as a scheme over
) to the scheme
. The set of
-morphisms of an
-scheme
into the affine
-scheme
is in bijective correspondence with the homomorphisms of the sheaves of
-algebras
.
Closed imbeddings of schemes or arbitrary morphisms of affine schemes are affine morphisms; other examples of affine morphisms are entire morphisms and finite morphisms. Thus the morphism of normalization of a scheme is an affine morphism. Under composition and base change the property of a morphism to be an affine morphism is preserved.
References
[1] | A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , Proc. Internat. Math. Congress Edinburgh, 1958 , Cambridge Univ. Press (1960) pp. 103–118 MR0130879 Zbl 0119.36902 |
[2] | J. Dieudonné, A. Grothendieck, "Elements de géometrie algébrique" Publ. Math. IHES , 4 (1960) MR0217083 MR0163908 Zbl 0203.23301 Zbl 0136.15901 |
Comments
is a finite morphism if there exist a covering
of
by affine open subschemes such that
is affine for all
and such that the ring
of
is finitely generated as a module over the ring
of
. The morphism is entire if
is entire over
, i.e. if every
integral over
, which means that it is a root of a monic polynomial with coefficients in
, or, equivalently, if for each
the module
is a finitely-generated module over
.
References
[a1] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
Affine morphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_morphism&oldid=45046