Affine morphism

A morphism of schemes $ f: X \rightarrow S $ such that the pre-image of any open affine subscheme in $ S $ is an affine scheme. The scheme $ X $ is called an affine $ S $- scheme.

Let $ S $ be a scheme, let $ A $ be a quasi-coherent sheaf of $ {\mathcal O} _ {S} $-algebras and let $ U _ {i} $ be open affine subschemes in $ S $ which form a covering of $ S $. Then the glueing of the affine schemes $ { \mathop{\rm Spec} } \Gamma (U _ {i} , A) $ determines an affine $ S $-scheme, denoted by $ { \mathop{\rm Spec} } A $. Conversely, any affine $ S $-scheme definable by an affine morphism $ f: X \rightarrow S $ is isomorphic (as a scheme over $ S $) to the scheme $ { \mathop{\rm Spec} } f _ {*} ( {\mathcal O} _ {X} ) $. The set of $ S $-morphisms of an $ S $-scheme $ f: Z \rightarrow S $ into the affine $ S $-scheme $ { \mathop{\rm Spec} } A $ is in bijective correspondence with the homomorphisms of the sheaves of $ {\mathcal O} _ {S} $-algebras $ A \rightarrow f _ {*} ( {\mathcal O} _ {Z} ) $.

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

$ f : X \rightarrow S $ is a finite morphism if there exist a covering $ ( S _ \alpha ) $ of $ S $ by affine open subschemes such that $ f ^ {-1} ( S _ \alpha ) $ is affine for all $ \alpha $ and such that the ring $ B _ \alpha $ of $ f ^ {-1} ( S _ \alpha ) $ is finitely generated as a module over the ring $ A _ \alpha $ of $ S _ \alpha $. The morphism is entire if $ B _ \alpha $ is entire over $ A _ \alpha $, i.e. if every $ x \in B _ \alpha $ integral over $ A _ \alpha $, which means that it is a root of a monic polynomial with coefficients in $ A _ \alpha $, or, equivalently, if for each $ x \in B _ \alpha $ the module $ A _ \alpha [ x ] $ is a finitely-generated module over $ A _ \alpha $.

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001