The Stacks project

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. $\square$

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