Hilbert's syzygy theorem


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, which 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 that 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 concern the relations, or syzygies in Hilbert's terminology, between the generators of an ideal, or, more generally, a module. As the relations form a module, one may consider the relations between the relations; Hilbert's syzygy theorem asserts that, if one continues in this way, starting with a module over a polynomial ring in indeterminates over a field, one eventually finds a zero module of relations, after at most steps.
Hilbert's syzygy theorem is now considered to be an early result of homological algebra. It is the starting point of the use of homological methods in commutative algebra and algebraic geometry.

History

The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen". The paper is split into five parts: part I proves Hilbert's basis theorem over a field, while part II proves it over the integers. Part III contains the syzygy theorem, which is used in part IV to discuss the Hilbert polynomial. The last part, part V, proves finite generation of certain rings of invariants. Incidentally part III also contains a special case of the Hilbert–Burch theorem.

Syzygies (relations)

Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to modules over any ring.
Given a generating set of a module over a ring, a relation or first syzygy between the generators is a -tuple of elements of such that
Let be the free module with basis the relation may be identified with the element
and the relations form the kernel of the linear map defined by In other words, one has an exact sequence
This first syzygy module depends on the choice of a generating set, but, if is the module which is obtained with another generating set, there exist two free modules and such that
where denote the direct sum of modules.
The second syzygy module is the module of the relations between generators of the first syzygy module. By continuing in this way, one may define the th syzygy module for every positive integer.
If the th syzygy module is free for some, then by taking a basis as a generating set, the next syzygy module is the zero module. If one does not take a bases as generating sets, then all subsequent syzygy modules are free.
Let be the smallest integer, if any, such that the th syzygy module of a module is free or projective. The above property of invariance, up to the sum direct with free modules, implies that does not depend on the choice of generating sets. The projective dimension of is this integer, if it exists, or if not. This is equivalent with the existence of an exact sequence
where the modules are free and is projective. It can be shown that one may always choose the generating sets for being free, that is for the above exact sequence to be a free resolution.

Statement

Hilbert's syzygy theorem states that, if is a finitely generated module over a polynomial ring in indeterminates over a field, then the th syzygy module of is always a free module.
In modern language, this implies that the projective dimension of is at most, and thus that there exists a free resolution
of length.
This upper bound on the projective dimension is sharp, that is, there are modules of projective dimension exactly. The standard example is the field, which may be considered as a -module by setting for every and every. For this module, the th syzygy module is free, but not the th one.
The theorem is also true for modules that are not finitely generated. As the global dimension of a ring is the supremum of the projective dimensions of all modules, Hilbert's syzygy theorem may be restated as: the global dimension of is .

Low dimension

In the case of zero indeterminates, Hilbert's syzygy theorem is simply the fact that every 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.

Koszul complex

The Koszul complex, also called "complex of exterior algebra", allows, in some cases, an explicit description of all syzygy modules.
Let be a generating system of an ideal in a polynomial ring, and let be a free module of basis The exterior algebra of is the direct sum
where is the free module, which has, as a basis, the exterior products
such that In particular, one has , the two definitions of coincide, and for. For every positive, one may define a linear map by
where the hat means that the factor is omitted. A straightforward computation shows that the composition of two consecutive such maps is zero, and thus that one has a complex
This is the Koszul complex. In general the Koszul complex is not an exact sequence, but it is an exact sequence if one works with a polynomial ring and an ideal generated by a regular sequence of homogeneous polynomials.
In particular, the sequence is regular, and the Koszul complex is thus a projective resolution of In this case, the th syzygy module is free of dimension one ; the th syzygy module is thus the quotient of a free module of dimension by the submodule generated by This quotient may not be a projective module, as otherwise, there would exist polynomials such that which is impossible. This proves that the projective dimension of is exactly.
The same proof applies for proving that the projective dimension of is exactly if the form a regular sequence of homogeneous polynomials.

Computation

At Hilbert's time, there were 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. If the degree of these polynomials is bounded, the number of their monomials is also bounded. Expressing that one has a syzygy provides a system of linear equations whose unknowns are the coefficients of these monomials. Therefore, any algorithm for linear systems implies an algorithm for syzygies, as soon as a bound of the degrees is known.
The first bound for syzygies was given in 1926 by Grete Hermann: Let a submodule of a free module of dimension over if the coefficients over a basis of of a generating system of have a total degree at most, then there is a constant 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 of an element of.
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. At the present time, the best algorithms for computing syzygies are Gröbner basis algorithms. They allow the computation of the first syzygy module, and also, with almost no extra cost, all syzygies modules.

Syzygies and regularity

One might wonder which ring-theoretic property of causes the Hilbert syzygy theorem to hold. It turns out that
this is regularity, which is an algebraic formulation of the fact that affine -space is a variety without singularities. In fact the following generalization holds: Let be a Noetherian ring. Then has finite global dimension if and only if is regular and the Krull dimension of is finite; in that case the global dimension of is equal to the Krull dimension. This result may be proven using Serre's theorem on regular local rings.