Linear relation

Template:Short description 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 $e_1,\dots ,e_n$ are elements of a (left) module Template:Mvar over a ring Template:Mvar (the case of a vector space over a field is a special case), a relation between $e_1,\dots ,e_n$ is a sequence $(f_1,\dots ,f_n)$ of elements of Template:Mvar such that

$f_1{e}_1+\dots +f_n{e}_n=0.$

The relations between $e_1,\dots ,e_n$ form a module. One is generally interested in the case where $e_1,\dots ,e_n$ is a generating set of a finitely generated module Template:Mvar, in which case the module of the relations is often called a syzygy module of Template:Mvar. 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 $S_1$ and $S_2$ are syzygy modules corresponding to two generating sets of the same module, then they are stably isomorphic, which means that there exist two free modules $L_1$ and $L_2$ such that $S_1\oplus L_1$ and $S_2\oplus L_2$ are isomorphic.

Higher order syzygy modules are defined recursively: a first syzygy module of a module Template:Mvar is simply its syzygy module. For Template:Math, a Template:Mvarth syzygy module of Template:Mvar is a syzygy module of a Template:Math-th syzygy module. Hilbert's syzygy theorem states that, if $R=K[x_1,\dots ,x_n]$ is a polynomial ring in Template:Mvar indeterminates over a field, then every Template:Mvarth syzygy module is free. The case Template:Math is the fact that every finite dimensional vector space has a basis, and the case Template:Math is the fact that Template:Math is a principal ideal domain and that every submodule of a finitely generated free Template:Math 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 Template:Mvar indeterminates over a field has global homological dimension Template:Mvar.

If Template:Mvar and Template:Mvar are two elements of the commutative ring Template:Mvar, then Template:Math 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.

Basic definitions

Let Template:Mvar be a ring, and Template:Mvar be a left Template:Mvar-module. A linear relation, or simply a relation between Template:Mvar elements $x_1,\dots ,x_k$ of Template:Mvar is a sequence $(a_1,\dots ,a_k)$ of elements of Template:Mvar such that

$a_1{x}_1+\dots +a_k{x}_k=0.$

If $x_1,\dots ,x_k$ is a generating set of Template:Mvar, the relation is often called a syzygy of Template:Mvar. It makes sense to call it a syzygy of $M$ without regard to $x_1,..,x_k$ because, although the syzygy module depends on the chosen generating set, most of its properties are independent; see Template:Slink, below.

If the ring Template:Mvar is Noetherian, or, at least coherent, and if Template:Mvar is finitely generated, then the syzygy module is also finitely generated. A syzygy module of this syzygy module is a second syzygy module of Template:Mvar. Continuing this way one can define a Template:Mvarth syzygy module for every positive integer Template:Mvar.

Hilbert's syzygy theorem asserts that, if Template:Mvar is a finitely generated module over a polynomial ring $K[x_1,\dots ,x_n]$ over a field, then any Template:Mvarth 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" of choices of generating sets for involved modules. The following result is the basis of these stable properties.

Proof. As $\{x_1,\dots ,x_m\}$ is a generating set, each $y_i$ can be written $y_i=\sum {\alpha }_{i,j}{x}_j.$ This provides a relation $r_i$ between $x_1,\dots ,x_m,y_1,\dots ,y_n.$ Now, if $r=(a_1,\dots ,a_m,b_1,\dots ,b_n)$ is any relation, then $r-\sum b_i{r}_i$ is a relation between the $x_1,\dots ,x_m$ only. In other words, every relation between $x_1,\dots ,x_m,y_1,\dots ,y_n$ is a sum of a relation between $x_1,\dots ,x_m,$ and a linear combination of the $r_i$s. It is straightforward to prove that this decomposition is unique, and this proves the result. $◼$

This proves that the first syzygy module is "stably unique". More precisely, given two generating sets $S_1$ and $S_2$ of a module Template:Mvar, if $S_1$ and $S_2$ are the corresponding modules of relations, then there exist two free modules $L_1$ and $L_2$ such that $S_1\oplus L_1$ and $S_2\oplus L_2$ 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 Template:Mvar is any module, and Template:Mvar is a free module, then Template:Mvar and Template:Math have isomorphic syzygy modules. It suffices to consider a generating set of Template:Math that consists of a generating set of Template:Mvar and a basis of Template:Mvar. For every relation between the elements of this generating set, the coefficients of the basis elements of Template:Mvar are all zero, and the syzygies of Template:Math are exactly the syzygies of Template:Mvar extended with zero coefficients. This completes the proof to the following theorem.

Relationship with free resolutions

Given a generating set $g_1,\dots ,g_n$ of an Template:Mvar-module, one can consider a free module of Template:Mvar of basis $G_1,\dots ,G_n,$ where $G_1,\dots ,G_n$ are new indeterminates. This defines an exact sequence

$L⟶M⟶0,$

where the left arrow is the linear map that maps each $G_i$ to the corresponding $g_i.$ The kernel of this left arrow is a first syzygy module of Template:Mvar.

One can repeat this construction with this kernel in place of Template:Mvar. Repeating again and again this construction, one gets a long exact sequence

$\cdots ⟶L_k⟶L_{k-1}⟶\cdots ⟶L_0⟶M⟶0,$

where all $L_i$ are free modules. By definition, such a long exact sequence is a free resolution of Template:Mvar.

For every Template:Math, the kernel $S_k$ of the arrow starting from $L_{k-1}$ is a Template:Mvarth syzygy module of Template:Mvar. It follows that the study of free resolutions is the same as the study of syzygy modules.

A free resolution is finite of length Template:Math if $S_n$ is free. In this case, one can take $L_n=S_n,$ and $L_k=0$ (the zero module) for every Template:Math.

This allows restating Hilbert's syzygy theorem: If $R=K[x_1,\dots ,x_n]$ is a polynomial ring in Template:Mvar indeterminates over a field Template:Mvar, then every free resolution is finite of length at most Template:Mvar.

The global dimension of a commutative Noetherian ring is either infinite, or the minimal Template:Mvar such that every free resolution is finite of length at most Template:Mvar. 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 Template:Mvar, one has always Template:Math. This implies trivially that Template:Math is a linear relation between Template:Mvar and Template:Mvar. Therefore, given a generating set $g_1,\dots ,g_k$ of an ideal Template:Mvar, 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

$r_{i,j}=(x_1,\dots ,x_r)$

such that $x_i=g_j,$ $x_j=-g_i,$ and $x_h=0$ otherwise.

History

The word syzygy came into mathematics with the work of Arthur Cayley. In that paper, Cayley used it 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:

$a\left|\begin{array}{cc}b& c\\ e& f\end{array}\right|-b\left|\begin{array}{cc}a& c\\ d& f\end{array}\right|+c\left|\begin{array}{cc}a& b\\ d& e\end{array}\right|=0.$

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.

References

• Template:Cite book
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• David Eisenbud, The Geometry of Syzygies, Graduate Texts in Mathematics, vol. 229, Springer, 2005.