A Chevalley scheme in

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 ...

was a precursor notion of

scheme theory In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...

. Let ''X'' be a separated integral

noetherian scheme In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets \operatorname A_i, A_i noetherian rings. More generally, a scheme is locally noetherian if it is covered by spectra of noetherian rings. Thus ...

, ''R'' its function field. If we denote by

X'

the set of subrings

\mathcal O_x

of ''R'', where ''x'' runs through ''X'' (when

X=\mathrm(A)

, we denote

X'

L(A)

X'

verifies the following three properties * For each

M\in X'

, ''R'' is the field of fractions of ''M''. * There is a finite set of noetherian subrings

A_i

of ''R'' so that

X'=\cup_i L(A_i)

and that, for each pair of indices ''i,j'', the subring

A_

of ''R'' generated by

A_i \cup A_j

is an

A_i

-algebra of finite type. * If

M\subseteq N

X'

are such that the maximal ideal of ''M'' is contained in that of ''N'', then ''M=N''. Originally, Chevalley also supposed that R was an extension of finite type of a field K and that the

A_i

's were algebras of finite type over a field too (this simplifies the second condition above).

Bibliography

Online
{{Webarchive, url=https://web.archive.org/web/20160306015028/http://numdam.org/numdam-bin/feuilleter?id=pmihes_1960__4_ , date=2016-03-06 Scheme theory