Friday, 14 June 2013

What is a Syzygy?

Linear algebra over rings is lots more fun than over fiels. Tha main reason is that most modules over a ring do not have bases-that is, their generators usually satisfy some nontrivial relations or ``syzygies''. Given a finitely generated $R$-modules $M$ (where $R$ is a commutative ring) and a set $z_1,\dots,z_n$ of generators, a syzygy of $M$ is an element $(a_1,\dots,a_n) \in R^n$ for which $a_1z_1+\cdots+a_nz_n=0$. The set of all syzygies (relative to the given generating set) is a submodule of $R^n$, called the module of syzygies. Thus the module of syzygies of $M$ is the kernal of the map $R^n \longrightarrow M $ that takes the standard basis elements of $R^n$ to the given set of generators.