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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 polynomial rings are Noetherian, 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 n indeterminates over a field, one eventually finds a zero module of relations, after at most n 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.

Contents

HistoryEdit

The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890).[1] 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 (Theorem III), 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)Edit

Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring.

Given a generating set   of a module M over a ring R, a relation or first syzygy between the generators is a k-tuple   of elements of R such that[2]

 

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 kth syzygy module for every positive integer k.

If, for some k, the kth syzygy module is free, then, by taking a basis as a generating set, the next syzygy module (and every subsequent one) is the zero module. If one does not take a bases as generating sets, then all subsequent syzygy modules are free.

Let n be the smallest integer, if any, such that the nth syzygy module of a module M is free or projective. 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. The projective dimension of M 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.

StatementEdit

Hilbert's syzygy theorem states that, if M is a finitely generated module over a polynomial ring   in n indeterminates over a field k, then the nth syzygy module of M is always a free module.

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 kn.

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  -module by setting   for every i and every ck. For this module, the nth syzygy module is free, but not the (n − 1)th one (for a proof, see § Koszul complex, below).

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 n.

Low dimensionEdit

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 complexEdit

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 I 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   (because of the definition of the empty product), the two definitions of   coincide, and   for t > k. For every positive t, 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 nth syzygy module is free of dimension one (generated by the product of all  ); the (n − 1)th syzygy module is thus the quotient of a free module of dimension n by the submodule generated by   This quotient may not be a projective module, as otherwise, there would exist polynomials   such that   which is impossible (substituting the   by 0 in the latter equality provides 1 = 0). This proves that the projective dimension of   is exactly n.

The same proof applies for proving that the projective dimension of   is exactly t if the   form a regular sequence of homogeneous polynomials.

ComputationEdit

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 (as well as for 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. 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 regularityEdit

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 n-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.

See alsoEdit

ReferencesEdit

  1. ^ D. Hilbert, Über die Theorie der algebraischen Formen, Mathematische Annalen 36, 473–530.
  2. ^ The theory is presented for finitely generated modules, but extends easily to arbitrary modules.
  3. ^ Grete Hermann: Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Unter Benutzung nachgelassener Sätze von K. Hentzelt, Mathematische Annalen, Volume 95, Number 1, 736-788, doi:10.1007/BF01206635 (abstract in German language) — The question of finitely many steps in polynomial ideal theory (review and English-language translation)
  4. ^ G. Hermann claimed c = 1, but did not prove this.