Affine morphism

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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.


[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


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 .


[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
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Affine morphism. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. DanilovI.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article