# 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

$f : X \rightarrow S$ 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 $A _ \alpha$.