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

On the other hand,

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: