Gordan's lemma is a lemma in
convex geometry
In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational geometry, convex analysis, discrete geometry, functional analysis, geometry of num ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
. 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
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively (just notation, not necessarily th ...
of integral points in a rational convex
polyhedral cone
In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C is a cone if x\in C implies sx\in C for e ...
is finitely generated.
* An
affine toric variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...
is an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the solution set, set of solutions of a system of polynomial equations over the real number, ...
(this follows from the fact that the
prime spectrum
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
of the
semigroup algebra of such a semigroup is, by definition, an
affine toric variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be n ...
).
The lemma is named after the mathematician
Paul Gordan
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a German mathematician known for work in invariant theory and for the Clebsch–Gordan coefficients and Gordan's lemma. He was called "the king of invariant theory". His most famous ...
(1837–1912). Some authors have misspelled it as "Gordon's lemma".
Proofs
There are topological and algebraic proofs.
Topological proof
Let
be the
dual cone
Dual cone and polar cone are closely related concepts in convex analysis, a branch of mathematics.
Dual cone
In a vector space
The dual cone ''C'' of a subset ''C'' in a linear space ''X'' over the real numbers, reals, e.g. Euclidean spac ...
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.
Algebraic proof
The proof
[ , Lemma 4.12.] is based on a fact that a semigroup ''S'' is finitely generated if and only if its semigroup algebra