= Kernel of localization away from one plus an ideal
Let $R$ be <Noetherian>, $I\subseteq R$ an ideal, $S=1+I$, and $M$ a finitely generated $R$-module. Then
$$
\ker(M\longrightarrow S^{-1}M)=\bigcap_{j\geq1}I^jM.
$$
If $(1+a)m=0$ with $a\in I$, then $m=(-a)^jm\in I^jM$ for every $j$. Conversely, apply the <Artin-Rees lemma> to $Rm\subseteq M$. For sufficiently large $c$ it gives
$$
m\in I^{c+1}M\cap Rm=I(I^cM\cap Rm)\subseteq I(Rm),
$$
so $m=am$ for some $a\in I$ and $(1-a)m=0$.
Back to article page
