Lemma 15.96.6. Let $A \to B$ be a ring map. Let $f \in A$ be a nonzerodivisor. Let $M^\bullet $ be a bounded complex of finite free $A$-modules. Assume $f$ maps to a nonzerodivisor $g$ in $B$ and $I_ i(M^\bullet , f)$ is a principal ideal for all $i \in \mathbf{Z}$. Then there is a canonical isomorphism $\eta _ fM^\bullet \otimes _ A B = \eta _ g(M^\bullet \otimes _ A B)$.

**Proof.**
Set $N^ i = M^ i \otimes _ A B$. Observe that $f^ iM^ i \otimes _ A B = g^ iN^ i$ as submodules of $(N^ i)_ g$. The maps

\[ (\eta _ fM)^ i \otimes _ A B \to g^ iN^ i \otimes g^{i + 1}N^{i + 1} \quad \text{and}\quad (\eta _ gN)^ i \to g^ iN^ i \otimes g^{i + 1}N^{i + 1} \]

are inclusions of direct summands by Lemma 15.96.5. Since their images agree after localizing at $g$ we conclude. $\square$

## 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Your email address will not be published.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)