Chevalley scheme explained
A Chevalley scheme in algebraic geometry was a precursor notion of scheme theory.
Let X be a separated integral noetherian scheme, R its function field. If we denote by
the set of subrings
of
R, where
x runs through
X (when
, we denote
by
),
verifies the following three properties
,
R is the field of fractions of
M.
- There is a finite set of noetherian subrings
of
R so that
and that, for each pair of indices
i,j, the subring
of
R generated by
is an
-algebra of finite type.
in
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
's were algebras of finite type over a field too (this simplifies the second condition above).
Bibliography
- Grothendieck . Alexandre . Alexandre Grothendieck . . 1960 . . I. Le langage des schémas . . I.8. Online
