Lemma 10.51.3. Suppose that $0 \to K \to M \xrightarrow {f} N$ is an exact sequence of finitely generated modules over a Noetherian ring $R$. Let $I \subset R$ be an ideal. Then there exists a $c$ such that

$f^{-1}(I^ nN) = K + I^{n-c}f^{-1}(I^ cN) \quad \text{and}\quad f(M) \cap I^ nN \subset f(I^{n - c}M)$

for all $n \geq c$.

Proof. Apply Lemma 10.51.2 to $\mathop{\mathrm{Im}}(f) \subset N$ and note that $f : I^{n-c}M \to I^{n-c}f(M)$ is surjective. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).