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 (1837–1912).
There are topological and algebraic proofs.
Let σ be the cone as given in the lemma. Let u 1 , … , u r be the integral vectors so that σ = { x ∣ ⟨ u i , x ⟩ ≥ 0 , 1 ≤ i ≤ r } . Then the u i 's generate the dual cone σ ∨ ; indeed, writing C for the cone generated by u i 's, we have: σ ⊂ C ∨ , which must be the equality. Now, if x is in the semigroup
S σ = σ ∨ ∩ Z d , then it can be written as
x = ∑ i n i u i + ∑ i r i u i where n i are nonnegative integers and 0 ≤ r i ≤ 1 . 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 (the topological reason). Hence, S σ is finitely generated.
The proof is based on a fact that a semigroup S is finitely generated if and only if its semigroup algebra C [ S ] is finitely generated algebra over C . To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the statement: for any unital subsemigroup S of Z d ,
If
S is finitely generated, then
S + = S ∩ { x ∣ ⟨ x , v ⟩ ≥ 0 } ,
v an integral vector, is finitely generated.
Put A = C [ S ] , which has a basis χ a , a ∈ S . It has Z -grading given by
A n = span { χ a ∣ a ∈ S , ⟨ a , v ⟩ = n } .
By assumption, A is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that C [ S + ] = ⊕ 0 ∞ A n is a finitely generated algebra over A 0 . Now, the semigroup S 0 = S ∩ { x ∣ ⟨ x , v ⟩ = 0 } is the image of S under a linear projection, thus finitely generated and so A 0 = C [ S 0 ] is finitely generated. Hence, S + is finitely generated then.
Lemma: Let A be a Z -graded ring. If A is a Noetherian ring, then A + = ⊕ 0 ∞ A n is a finitely generated A 0 -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 f i ′ s , homogeneous of positive degree. If f is homogeneous of positive degree, then we can write f = ∑ i g i f i with g i homogeneous. If f has sufficieny large degree, then each g i has degree positive and strictly less than that of f. Also, each degree piece A n is a finitely generated A 0 -module. (Proof: Let N i be an increasing chain of finitely generated submodules of A n with union A n . Then the chain of the ideals N i A stabilizes in finite steps; so does the chain N i = N i A ∩ A n . ) Thus, by induction on degree, we see A + is a finitely generated A 0 -algebra.
