Filtration of a module

Summary

A filtration of a module $M$ is a decreasing sequence of submodules $M = M_0 \supseteq M_1 \supseteq M_2 \supseteq \cdots$. For an ideal $I$, an $I$-filtration is one with $I M_n \subseteq M_{n+1}$, and it is stable when equality holds from some point on. The basic example is the $I$-adic filtration $M_n = I^n M$; filtrations lead to the associated graded ring, the $I$-adic topology and the Artin–Rees lemma.