Gordan's lemma
Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways. The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma". Proofs. There are topological and algebraic proofs. Topological proof. Let formula_6 be the dual cone of the given rational polyhedral cone. Let formula_7 be integral vectors so that formula_8 Then the formula_9's generate the dual cone formula_10; indeed, writing "C" for the cone generated by formula_9's, we have: formula_12, which must be the equality. Now, if "x  is in the semigroup then it can be written as where formula_15 are nonnegative integers and formula_16. 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, formula_17 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 formula_18 is a finitely generated algebra over formula_19. 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 formula_20, Put formula_22, which has a basis formula_23. It has formula_24-grading given by By assumption, "A" is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that formula_26 is a finitely generated algebra over formula_27. Now, the semigroup formula_28 is the image of "S" under a linear projection, thus finitely generated and so formula_29 is finitely generated. Hence, formula_30 is finitely generated then. Lemma: Let "A" be a formula_24-graded ring. If "A" is a Noetherian ring , then formula_32 is a finitely generated formula_27-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 formula_34, homogeneous of positive degree. If "f" is homogeneous of positive degree, then we can write formula_35 with formula_36 homogeneous. If "f" has sufficiently large degree, then each formula_36 has degree positive and strictly less than that of "f". Also, each degree piece formula_38 is a finitely generated formula_27-module. (Proof: Let formula_40 be an increasing chain of finitely generated submodules of formula_38  with union formula_38. Then the chain of the ideals formula_43 stabilizes in finite steps; so does the chain formula_44) Thus, by induction on degree, we see formula_45 is a finitely generated formula_27-algebra. Applications. A "multi-hypergraph" over a certain set formula_47 is a multiset of subsets of "formula_47" (it is called "multi-hypergraph" since each hyperedge may appear more than once). 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 formula_49 be the maximum degree  of an indecomposable multi-hypergraph on "n" vertices. Gordan's lemma implies that formula_49 is finite. "Proof": for each subset "S" of vertices, define a variable "x" (a non-negative integer). Define another variable "d" (a non-negative integer). Consider the following set of "n" equations (one equation per vertex):formula_51Every solution (x,"d") denotes a regular multi-hypergraphs on formula_47, 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 formula_2 of multi-hypergraphs, such that each regular multi -hypergraph is a linear combination of some elements of formula_2. Every non-decomposable multi-hypergraph must be in formula_2 (since by definition, it cannot be generated by other multi-hypergraph). Hence, the set of non-decomposable multi-hypergraphs is finite.