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Concept
Gordan's lemma
Résumé
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 (this follows from the fact that the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety).
The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma".
There are topological and algebraic proofs.
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 (the topological reason). Hence, is finitely generated.
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 (cf. the proof above), it is enough to prove the following statement: for any unital subsemigroup S of ,
If S is finitely generated, then , v an integral vector, is finitely generated.
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.
Source officielle
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