In mathematics, Hilbert's syzygy theorem is one of the three fundamental theorems about polynomial rings over fields, first proved by David Hilbert in 1890, that were introduced for solving important open questions in invariant theory, and are at the basis of modern algebraic geometry.
The two other theorems are Hilbert's basis theorem, which asserts that all ideals of polynomial rings over a field are finitely generated, and Hilbert's Nullstellensatz, which establishes a bijective correspondence between affine algebraic varieties and prime ideals of polynomial rings.
Hilbert's syzygy theorem concerns the relations, or syzygies in Hilbert's terminology, between the generators of an ideal, or, more generally, a module.
Hilbert's syzygy theorem is now considered to be an early result of homological algebra.
The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890).
Incidentally part III also contains a special case of the Hilbert–Burch theorem.
Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring.
of a module M over a ring R, a relation or first syzygy between the generators is a k-tuple
In other words, one has an exact sequence This first syzygy module
If one does not take a basis as a generating set, then all subsequent syzygy modules are free.
The above property of invariance, up to the sum direct with free modules, implies that n does not depend on the choice of generating sets.
Hilbert's syzygy theorem states that, if M is a finitely generated module over a polynomial ring
In modern language, this implies that the projective dimension of M is at most n, and thus that there exists a free resolution of length k ≤ n. This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly n. The standard example is the field k, which may be considered as a
The theorem is also true for modules that are not finitely generated.
is n. In the case of zero indeterminates, Hilbert's syzygy theorem is simply the fact that every finitely generated vector space has a basis.
In the case of a single indeterminate, Hilbert's syzygy theorem is an instance of the theorem asserting that over a principal ideal ring, every submodule of a free module is itself free.
is the free module, which has, as a basis, the exterior products such that
and an ideal generated by a regular sequence of homogeneous polynomials.
In this case, the nth syzygy module is free of dimension one (generated by the product of all
is exactly n. The same proof applies for proving that the projective dimension of
At Hilbert's time, there was no method available for computing syzygies.
It was only known that an algorithm may be deduced from any upper bound of the degree of the generators of the module of syzygies.
In fact, the coefficients of the syzygies are unknown polynomials.
Expressing that one has a syzygy provides a system of linear equations whose unknowns are the coefficients of these monomials.
The first bound for syzygies (as well as for the ideal membership problem) was given in 1926 by Grete Hermann:[3] Let M a submodule of a free module L of dimension t over
if the coefficients over a basis of L of a generating system of M have a total degree at most d, then there is a constant c such that the degrees occurring in a generating system of the first syzygy module is at most
The same bound applies for testing the membership to M of an element of L.[4] On the other hand, there are examples where a double exponential degree necessarily occurs.
However such examples are extremely rare, and this sets the question of an algorithm that is efficient when the output is not too large.
It turns out that this is regularity, which is an algebraic formulation of the fact that affine n-space is a variety without singularities.
This result may be proven using Serre's theorem on regular local rings.