Gordan's lemma is a lemma in convex geometry and algebraic geometry.
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.
's generate the dual cone
; indeed, writing C for the cone generated by
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).
The proof[3] 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
-grading given by By assumption, A is finitely generated and thus is Noetherian.
is the image of S under a linear projection, thus finitely generated and so
Proof: Let I be the ideal of A generated by all homogeneous elements of A of positive degree.
, homogeneous of positive degree.
If f is homogeneous of positive degree, then we can write
If f has sufficiently large degree, then each
be an increasing chain of finitely generated submodules of
stabilizes in finite steps; so does the chain
(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.
be the maximum degree of an indecomposable multi-hypergraph on n vertices.
Gordan's lemma implies that
[1] Proof: for each subset S of vertices, define a variable xS (a non-negative integer).
Define another variable d (a non-negative integer).
Every solution (x,d) denotes a regular multi-hypergraphs on
, where x defines the hyperedges and d is the degree.
(since by definition, it cannot be generated by other multi-hypergraph).
Hence, the set of non-decomposable multi-hypergraphs is finite.