Linear relation

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.

Proposition — Let ${\displaystyle \{x_1,\dots ,x_m\}}$ be a generating set of an R-module M, and ${\displaystyle y_1,\dots ,y_n}$ be other elements of M. The module of the relations between ${\displaystyle x_1,\dots ,x_m,y_1,\dots ,y_n}$ is the direct sum of the module of the relations between ${\displaystyle x_1,\dots ,x_m,}$ and a free module of rank n.

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

This proves that the first syzygy module is "stably unique". More precisely, given two generating sets ${\displaystyle S_1}$ and ${\displaystyle S_2}$ of a module M, if ${\displaystyle S_1}$ and ${\displaystyle S_2}$ are the corresponding modules of relations, then there exist two free modules ${\displaystyle L_1}$ and ${\displaystyle L_2}$ such that ${\displaystyle S_1\oplus L_1}$ and ${\displaystyle S_2\oplus L_2}$ 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 M is any module, and L is a free module, then M and M ⊕ L have isomorphic syzygy modules. 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. This completes the proof to the following theorem.

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. More precisely, if ${\displaystyle S_1}$ and ${\displaystyle S_2}$ are kth syzygy modules that are obtained by different choices of generating sets, then there are free modules ${\displaystyle L_1}$ and ${\displaystyle L_2}$ such that ${\displaystyle S_1\oplus L_1}$ and ${\displaystyle S_2\oplus L_2}$ are isomorphic.

Relationship with free resolutions

Given a generating set ${\displaystyle g_1,\dots ,g_n}$ of an R-module, one can consider a free module of L of basis ${\displaystyle G_1,\dots ,G_n,}$ where ${\displaystyle G_1,\dots ,G_n}$ are new indeterminates. This defines an exact sequence

${\displaystyle L⟶M⟶0,}$

where the left arrow is the linear map that maps each ${\displaystyle G_i}$ to the corresponding ${\displaystyle g_i.}$ 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

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

where all ${\displaystyle L_i}$ are free modules. By definition, such a long exact sequence is a free resolution of M.

For every k ≥ 1, the kernel ${\displaystyle S_k}$ of the arrow starting from ${\displaystyle L_{k-1}}$ 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 ${\displaystyle S_n}$ is free. In this case, one can take ${\displaystyle L_n=S_n,}$ and ${\displaystyle L_k=0}$ (the zero module) for every k > n.

This allows restating Hilbert's syzygy theorem: If ${\displaystyle R=K[x_1,\dots ,x_n]}$ 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. 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 R, one has always ab – ba = 0. This implies trivially that (b, –a) is a linear relation between a and b. Therefore, given a generating set ${\displaystyle g_1,\dots ,g_k}$ 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

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

such that ${\displaystyle x_i=g_j,}$ ${\displaystyle x_j=-g_i,}$ and ${\displaystyle x_h=0}$ 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:

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

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.