In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
discipline of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, Serre's theorem on affineness (also called Serre's cohomological characterization of affineness or Serre's criterion on affineness) is a theorem due to
Jean-Pierre Serre
Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ina ...
which gives sufficient conditions for a
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
to be
affine. The theorem was first published by Serre in 1957.
Statement
Let be a scheme with
structure sheaf
In mathematics, a ringed space is a family of ( commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
If:
:(1) is quasi-compact, and
:(2) for every quasi-coherent
ideal sheaf In algebraic geometry and other areas of mathematics, an ideal sheaf (or sheaf of ideals) is the global analogue of an ideal in a ring. The ideal sheaves on a geometric object are closely connected to its subspaces.
Definition
Let ''X'' be a to ...
of -modules, ,
then is
affine.
Related results
* A special case of this theorem arises when is an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
, in which case the conditions of the theorem imply that is an
affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. ...
.
* A similar result has stricter conditions on but looser conditions on the cohomology: if is a quasi-separated, quasi-compact scheme, and if for any quasi-coherent sheaf of ideals of finite type, then is affine.
[, Lemma 29.3.2.]
Notes
References
Bibliography
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Theorems in algebraic geometry
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