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.
Contents
- History
- Syzygies relations
- Statement
- Low dimension
- Koszul complex
- Computation
- Syzygies and regularity
- References
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.
History
The syzygy theorem first appeared in Hilbert's seminal paper "Über die Theorie der algebraischen Formen" (1890). 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)
Originally, Hilbert defined syzygies for ideals in polynomial rings, but the concept generalizes trivially to (left) modules over any ring.
Given a generating set
Let
and the relations form the kernel
This first syzygy module depends on the choice of a generating set, but, if
where
The second syzygy module is the module of the relations between generators of the first syzygies 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 lower 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 means the existence of an exact sequence
where the modules
Statement
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. 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
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
where
such that
where the hat means that the factor is ommitted. 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
In particular, the sequence
The same proof applies for proving that the projective dimension of
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 (as well as for ideal membership problem were given in 1926 by Grete Hermann: Let M a submodule of a free module L of dimension t over
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