The Stacks project

Lemma 15.4.3. Let $A \to B$ be a flat map of Noetherian rings. Let $I \subset A$ be an ideal. Let $f : M \to N$ be a homomorphism of finite $A$-modules. Assume that $c$ works for $f$ in the Artin-Rees lemma. Then $c$ works for $f \otimes 1 : M \otimes _ A B \to N \otimes _ A B$ in the Artin-Rees lemma for the ideal $IB$.

Proof. Note that

\[ (f \otimes 1)(M) \cap I^ n N \otimes _ A B = (f \otimes 1)\left((f \otimes 1)^{-1}(I^ n N \otimes _ A B)\right) \]

On the other hand,

\begin{align*} (f \otimes 1)^{-1}(I^ n N \otimes _ A B) & = \mathop{\mathrm{Ker}}(M \otimes _ A B \to N \otimes _ A B/(I^ n N \otimes _ A B)) \\ & = \mathop{\mathrm{Ker}}(M \otimes _ A B \to (N/I^ nN) \otimes _ A B) \end{align*}

As $A \to B$ is flat taking kernels and cokernels commutes with tensoring with $B$, whence this is equal to $f^{-1}(I^ nN) \otimes _ A B$. By assumption $f^{-1}(I^ nN)$ is contained in $\mathop{\mathrm{Ker}}(f) + I^{n - c}M$. Thus the lemma holds. $\square$


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