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
X
′
the set of subrings
O
x
of R, where x runs through X (when
X
=
S
p
e
c
(
A
)
, we denote
X
′
by
L
(
A
)
),
X
′
verifies the following three properties
For each
M
∈
X
′
, R is the field of fractions of M.
There is a finite set of noetherian subrings
A
i
of R so that
X
′
=
∪
i
L
(
A
i
)
and that, for each pair of indices i,j, the subring
A
i
j
of R generated by
A
i
∪
A
j
is an
A
i
-algebra of finite type.
If
M
⊆
N
in
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).
