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- Affine_variety abstract "In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine n-space of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subspace of an affine variety is called a quasi-affine variety.If X is an affine variety defined by a prime ideal I, then the quotient ringis called the coordinate ring of X. This ring is precisely the set of all regular functions on X; in other words, it is the space of global sections of the structure sheaf of X. A theorem of Serre gives a cohomological characterization of an affine variety: that is, an algebraic variety is affine if and only iffor any and any quasi-coherent sheaf F on X. (cf. Cartan's theorem B.) This makes the cohomological study of an affine variety non-existent, in a sharp contrast to the projective case in which cohomology groups of line bundles are of central interest.An affine variety plays a role of a local chart for algebraic varieties; that is to say, general algebraic varieties such as projective varieties are obtained by gluing affine varieties. Linear structures that are attached to varieties are also (trivially) affine varieties; e.g., tangent spaces.An affine variety is, up to an equivalence of categories a special case of an affine scheme, which is precisely the spectrum of a ring. In complex geometry, an affine variety is an analog of a Stein manifold.".
- Affine_variety thumbnail Cubic_with_double_point.svg?width=300.
- Affine_variety wikiPageExternalLink ag.html.
- Affine_variety wikiPageExternalLink lec.html.
- Affine_variety wikiPageID "320468".
- Affine_variety wikiPageRevisionID "602216456".
- Affine_variety hasPhotoCollection Affine_variety.
- Affine_variety subject Category:Algebraic_geometry.
- Affine_variety comment "In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine n-space of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set.".
- Affine_variety label "Affine variety".
- Affine_variety label "Variété algébrique affine".
- Affine_variety sameAs Variété_algébrique_affine.
- Affine_variety sameAs m.0rfdmts.
- Affine_variety sameAs Q3554813.
- Affine_variety sameAs Q3554813.
- Affine_variety wasDerivedFrom Affine_variety?oldid=602216456.
- Affine_variety depiction Cubic_with_double_point.svg.
- Affine_variety isPrimaryTopicOf Affine_variety.