In convex geometry, Gordan's lemma states that the semigroup of integral points in the dual cone of a rational convex polyhedral cone is finitely generated. In algebraic geometry, the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety; thus, the lemma says an affine toric variety is indeed an algebraic variety. The lemma is named after the German mathematician Paul Gordan (1837–1912).
There are topological and algebraic proofs.
Let
σ
be the cone as given in the lemma. Let
u
1
,
…
,
u
r
be the integral vectors so that
σ
=
{
x
∣
⟨
u
i
,
x
⟩
≥
0
,
1
≤
i
≤
r
}
.
Then the
u
i
's generate the dual cone
σ
∨
; indeed, writing C for the cone generated by
u
i
's, we have:
σ
⊂
C
∨
, which must be the equality. Now, if x is in the semigroup
S
σ
=
σ
∨
∩
Z
d
,
then it can be written as
x
=
∑
i
n
i
u
i
+
∑
i
r
i
u
i
where
n
i
are nonnegative integers and
0
≤
r
i
≤
1
. But since x and the first sum on the right-hand side are integral, the second sum is also integral and thus there can only be finitely many possibilities for the second sum (the topological reason). Hence,
S
σ
is finitely generated.
The proof is based on a fact that a semigroup S is finitely generated if and only if its semigroup algebra
C
[
S
]
is finitely generated algebra over
C
. To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the statement: for any unital subsemigroup S of
Z
d
,
If
S is finitely generated, then
S
+
=
S
∩
{
x
∣
⟨
x
,
v
⟩
≥
0
}
,
v an integral vector, is finitely generated.
Put
A
=
C
[
S
]
, which has a basis
χ
a
,
a
∈
S
. It has
Z
-grading given by
A
n
=
span
{
χ
a
∣
a
∈
S
,
⟨
a
,
v
⟩
=
n
}
.
By assumption, A is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that
C
[
S
+
]
=
⊕
0
∞
A
n
is a finitely generated algebra over
A
0
. Now, the semigroup
S
0
=
S
∩
{
x
∣
⟨
x
,
v
⟩
=
0
}
is the image of S under a linear projection, thus finitely generated and so
A
0
=
C
[
S
0
]
is finitely generated. Hence,
S
+
is finitely generated then.
Lemma: Let A be a
Z
-graded ring. If A is a Noetherian ring, then
A
+
=
⊕
0
∞
A
n
is a finitely generated
A
0
-algebra.
Proof: Let I be the ideal of A generated by all homogeneous elements of A of positive degree. Since A is Noetherian, I is actually generated by finitely many
f
i
′
s
, homogeneous of positive degree. If f is homogeneous of positive degree, then we can write
f
=
∑
i
g
i
f
i
with
g
i
homogeneous. If f has sufficieny large degree, then each
g
i
has degree positive and strictly less than that of f. Also, each degree piece
A
n
is a finitely generated
A
0
-module. (Proof: Let
N
i
be an increasing chain of finitely generated submodules of
A
n
with union
A
n
. Then the chain of the ideals
N
i
A
stabilizes in finite steps; so does the chain
N
i
=
N
i
A
∩
A
n
.
) Thus, by induction on degree, we see
A
+
is a finitely generated
A
0
-algebra.
