Gordan's lemma

Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.

Let be a matrix of integers. Let be the set of non-negative integer solutions of. Then there exists a finite subset of vectors in, such that every element of is a linear combination of these vectors with non-negative integer coefficients.
The semigroup of integral points in a rational convex polyhedral cone is finitely generated.
An affine toric variety is an algebraic variety.

The lemma is named after the mathematician Paul Gordan. Some authors have misspelled it as "Gordon's lemma".

Proofs

There are topological and algebraic proofs.

Topological proof

Let be the dual cone of the given rational polyhedral cone. Let be 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 a lattice point in a bounded region, and so there are only 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 a finitely generated algebra over. To prove Gordan's lemma, by induction, it is enough to prove the following 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.

Applications

A multi-hypergraph over a certain set is a multiset of subsets of '. A multi-hypergraph is called regular if all vertices have the same degree. It is called decomposable if it has a proper nonempty subset that is regular too. For any integer n, let be the maximum degree of an indecomposable multi-hypergraph on n vertices. Gordan's lemma implies that is finite. Proof: for each subset S of vertices, define a variable x_S. Define another variable d. Consider the following set of n equations :Every solution denotes a regular multi-hypergraphs on, where x' defines the hyperedges and d'' is the degree. By Gordan's lemma, the set of solutions is generated by a finite set of solutions, i.e., there is a finite set of multi-hypergraphs, such that each regular multi-hypergraph is a linear combination of some elements of. Every non-decomposable multi-hypergraph must be in . Hence, the set of non-decomposable multi-hypergraphs is finite.