Linear relation

In linear algebra, a linear relation, or simply relation, between elements of a vector space or a module is a linear equation that has these elements as a solution. 
More precisely, if formula_1 are elements of a (left) module over a ring (the case of a vector space over a field is a special case), a relation between formula_1 is a sequence formula_3 of elements of such that 
The relations between formula_1 form a module. One is generally interested to the case where formula_1 is a generating set of a finitely generated module , in which case the module of the relations is often called a syzygy module of . The syzygy module depends on the choice of a generating set, but it is unique up to the direct sum with a free module. That is, if formula_7 and formula_8 are syzygy modules corresponding to two generating sets of the same module, then there are stably isomorphic, which means that there exist two free modules formula_9 and formula_10 such that formula_11 and formula_12 are isomorphic.
Higher order syzygy modules are defined recursively: a first syzygy module of a module is simply its syzygy module. For , a th syzygy module of is a syzygy module of a -th syzygy module. Hilbert's syzygy theorem states that, if formula_13 is a polynomial ring in indeterminates over a field, then every th syzygy module is free. The case is the fact that every finite dimensional vector space has a basis, and the case is the fact that is a principal ideal domain and that every submodule of a finitely generated free 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 indeterminates over a field has global homological dimension .
If and are two elements of the commutative ring , then 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 the 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.
Basic definitions.
Let be a ring, and be a left -module. A " linear relation", or simply a "relation" between elements formula_14 of is a sequence formula_15 of elements of such that
If formula_14 is a generating set of , the relation is often called a "syzygy" of . This terminology makes sense, since, although the syzygy module depends on the chosen generating set, most of its properties are independent; see , below.
If the ring is Noetherian, or, at least coherent, and if is finitely generated, then the syzygy module is also finitely generated. A syzygy module of this syzygy module is a "second syzygy module" of . Continuing this way one can define a th syzygy module for every positive integer .
Hilbert's syzygy theorem asserts that, if is a finitely generated module over a polynomial ring formula_18 over a field, then any th syzygy module is a free module.
Stable properties.
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" on choices of generating sets for involved modules. The following result is the basis of these stable properties.
"Proof." As formula_19 is a generating set, each formula_20 can be written
formula_21
This provides a relation formula_22 between formula_23 Now, if formula_24 is any relation, then 
formula_25 
is a relation between the formula_26 only. In other words, every relation between formula_27 is a sum of a relation between formula_28 and a linear combination of the formula_22s. It is straightforward to prove that this decomposition is unique, and this proves the result. formula_30
This prove that the first syzygy module is "stably unique". More precisely, given two generating sets formula_7 and formula_8 of a module , if formula_7 and formula_8 are the corresponding modules of relations, then there exist two free modules formula_9 and formula_10 such that formula_11 and formula_12 are isomorphic. 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.
For obtaining a similar result for higher syzygy modules, it remains to prove that, if is any module, and is a free module, then and have isomorphic syzygy modules. It suffices to consider a generating set of that consists of a generating set of and a basis of . For every relation between the elements of this generating set, the coefficients of the basis elements of are all zero, and the syzygies of are exactly the syzygies of extended with zero coefficients. This completes the proof to the following theorem.
Relationship with free resolutions.
Given a generating set formula_39 of an -module, one can consider a free module of of basis formula_40 where formula_41 are new indeterminates. This defines an exact sequence
where the left arrow is the linear map that maps each formula_43 to the corresponding formula_44 The kernel of this left arrow is a first syzygy module of .
One can repeat this construction with this kernel in place of . Repeating again and again this construction, one gets a long exact sequence
where all formula_46 are free modules. By definition, such a long exact sequence is a free resolution of .
For every , the kernel formula_47 of the arrow starting from formula_48 is a th syzygy module of . It follows that the study of free resolutions is the same as the study of syzygy modules. 
A free resolution is "finite" of length if formula_49 is free. In this case, one can take formula_50 and formula_51 (the zero module) for every .
This allows restating Hilbert's syzygy theorem: If formula_52 is a polynomial ring in indeterminates over a field , then every free resolution is finite of length at most .
The global dimension of a commutative Noetherian ring is either infinite, or the minimal such that every free resolution is finite of length at most . A commutative Noetherian ring is regular if its global dimension is finite. In this case, the global dimension equals its Krull dimension. 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."
Trivial relations.
In a commutative ring , one has always . This implies "trivially" that is a linear relation between and . Therefore, given a generating set formula_53 of an ideal , 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 formula_55 formula_56 and formula_57 otherwise.
History.
The word "syzygy" came into mathematics with the work of Arthur Cayley. In that paper, Cayley used it for in the theory of resultants and discriminants.
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 called the Koszul complex, after a similar construction in differential geometry by the mathematician Jean-Louis Koszul.