are elements of a (left) module M over a ring R (the case of a vector space over a field is a special case), a relation between
Hilbert's syzygy theorem states that, if
is a polynomial ring in n indeterminates over a field, then every nth syzygy module is free.
The case n = 0 is the fact that every finite dimensional vector space has a basis, and the case n = 1 is the fact that K[x] is a principal ideal domain and that every submodule of a finitely generated free K[x] module is also free.
The construction of higher order syzygy modules is generalized as the definition of free resolutions, which allows restating Hilbert's syzygy theorem as a polynomial ring in n indeterminates over a field has global homological dimension n. If a and b are two elements of the commutative ring R, then (b, –a) is a relation that is said trivial.
The module of trivial relations of an ideal is the submodule of the first syzygy module of the ideal that is generated by the trivial relations between the elements of a generating set of an ideal.
The concept of trivial relations can be generalized to higher order syzygy modules, and this leads to the concept of the Koszul complex of an ideal, which provides information on the non-trivial relations between the generators of an ideal.
because, although the syzygy module depends on the chosen generating set, most of its properties are independent; see § Stable properties, below.
Generally speaking, in the language of K-theory, a property is stable if it becomes true by making a direct sum with a sufficiently large free module.
A fundamental property of syzygies modules is that there are "stably independent" of choices of generating sets for involved modules.
The following result is the basis of these stable properties.
This proves that the first syzygy module is "stably unique".
For proving this, it suffices to apply twice the preceding proposition for getting two decompositions of the module of the relations between the union of the two generating sets.
It suffices to consider a generating set of M ⊕ L that consists of a generating set of M and a basis of L. For every relation between the elements of this generating set, the coefficients of the basis elements of L are all zero, and the syzygies of M ⊕ L are exactly the syzygies of M extended with zero coefficients.
Theorem — For every positive integer k, the kth syzygy module of a given module depends on choices of generating sets, but is unique up to the direct sum with a free module.
The kernel of this left arrow is a first syzygy module of M. One can repeat this construction with this kernel in place of M. Repeating again and again this construction, one gets a long exact sequence where all
By definition, such a long exact sequence is a free resolution of M. For every k ≥ 1, the kernel
is a kth syzygy module of M. It follows that the study of free resolutions is the same as the study of syzygy modules.
A free resolution is finite of length ≤ n if
(the zero module) for every k > n. This allows restating Hilbert's syzygy theorem: If
is a polynomial ring in n indeterminates over a field K, then every free resolution is finite of length at most n. The global dimension of a commutative Noetherian ring is either infinite, or the minimal n such that every free resolution is finite of length at most n. A commutative Noetherian ring is regular if its global dimension is finite.
So, Hilbert's syzygy theorem may be restated in a very short sentence that hides much mathematics: A polynomial ring over a field is a regular ring.
In a commutative ring R, one has always ab – ba = 0.
This implies trivially that (b, –a) is a linear relation between a and b.
of an ideal I, one calls trivial relation or trivial syzygy every element of the submodule the syzygy module that is generated by these trivial relations between two generating elements.
More precisely, the module of trivial syzygies is generated by the relations such that
The word syzygy came into mathematics with the work of Arthur Cayley.
[1] In that paper, Cayley used it in the theory of resultants and discriminants.
[2] As the word syzygy was used in astronomy to denote a linear relation between planets, Cayley used it to denote linear relations between minors of a matrix, such as, in the case of a 2×3 matrix: Then, the word syzygy was popularized (among mathematicians) by David Hilbert in his 1890 article, which contains three fundamental theorems on polynomials, Hilbert's syzygy theorem, Hilbert's basis theorem and Hilbert's Nullstellensatz.
In his article, Cayley makes use, in a special case, of what was later[3] called the Koszul complex, after a similar construction in differential geometry by the mathematician Jean-Louis Koszul.