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A morphism of schemes  $  f:  X \rightarrow S $
such that the pre-image of any open affine subscheme in  $  S $
is an affine [[Scheme|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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> A. Grothendieck, "The cohomology theory of abstract algebraic varieties" , ''Proc. Internat. Math. Congress Edinburgh, 1958'' , Cambridge Univ. Press (1960) pp. 103–118 {{MR|0130879}} {{ZBL|0119.36902}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> J. Dieudonné, A. Grothendieck, "Elements de géométrie algébrique" ''Publ. Math. IHES'' , '''4''' (1960) {{MR|0217083}} {{MR|0163908}} {{ZBL|0203.23301}} {{ZBL|0136.15901}} </TD></TR>
</table>

====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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>