I-adic topology ($(I^{n}M{)}_{n\ge 0}$)

The $I$-adic topology on a module $M$ has the powers $I^{n}M$ as a neighbourhood basis of zero. It is generated by the I-adic filtration; separatedness means ${\bigcap }_n{I}^{n}M=0$.

For an ideal $I$, the $I$-adic filtration of a module $M$ is $M\supseteq IM\supseteq I^{2}M\supseteq \cdots$.

Let $R$ be a Noetherian local ring, $I$ an ideal contained in its maximal ideal, and $M$ a finitely generated module. The Krull intersection theorem statesIn particular ${\bigcap }_n{I}^n=0$ by taking $M=R$. It says that the I-adic topology is separated on every finite module.

For a proof, let $N={\bigcap }_n{I}^{n}M$. It is finitely generated because $M$ is Noetherian. The Artin-Rees lemma gives, for some $k$, $I^{k+1}M\cap N=I(I^{k}M\cap N)$. Both intersections equal $N$, so $N=IN$. The Nakayama lemma then gives $N=0$. Finite generation is essential to this argument; it cannot be omitted merely because the scalar ring is Noetherian.

The $x$-adic filtration is the filtration by powers of the principal ideal $(x)$.

An $I$-filtration $(M_n)$ is stable when $I{M}_n=M_{n+1}$ for all sufficiently large $n$. Every stable $I$-filtration is equivalent to the $I$-adic filtration.

For an ideal $I\subseteq R$, the Rees ring is the graded subringIt packages all powers of $I$ into one algebra and turns stable filtrations into finitely generated graded modules.

The Rees module of an $R$-module $M$ with respect to an ideal $I$ is ${\mathcal{R}}_I(M)={⨁}_{j\ge 0}{I}^{j}M{t}^j$, a graded module over the Rees ring $\mathcal{R}(I)$. Generators of $M$ in degree zero generate this module. For $N\subseteq M$, the graded submodule ${⨁}_j(I^{j}M\cap N)t^j$ is finitely generated when $R$ is a Noetherian ring and $M$ is a finitely generated module; a bound on its generators' degrees yields the Artin-Rees lemma.

If $R$ is Noetherian, $I$ is an ideal, and $N\subseteq M$ are finitely generated modules, then some $k$ satisfiesfor every $n\ge k$.