Gordan's lemma


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.

Proof

There are topological and algebraic proofs.

Topological proof

Let be the cone as given in the lemma. Let be the integral vectors so that Then the 's generate the dual cone ; indeed, writing C for the cone generated by 's, we have:, which must be the equality. Now, if x is in the semigroup
then it can be written as
where are nonnegative integers and. 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. Hence, is finitely generated.

Algebraic proof

The proof is based on a fact that a semigroup S is finitely generated if and only if its semigroup algebra is finitely generated algebra over. To prove Gordan's lemma, by induction, it is enough to prove the statement: for any unital subsemigroup S of,
Put, which has a basis. It has -grading given by
By assumption, A is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that is a finitely generated algebra over. Now, the semigroup is the image of S under a linear projection, thus finitely generated and so is finitely generated. Hence, is finitely generated then.
Lemma: Let A be a -graded ring. If A is a Noetherian ring, then is a finitely generated -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, homogeneous of positive degree. If f is homogeneous of positive degree, then we can write with homogeneous. If f has sufficiently large degree, then each has degree positive and strictly less than that of f. Also, each degree piece is a finitely generated -module. Thus, by induction on degree, we see is a finitely generated -algebra.