The Stacks project

Remark 61.11.8. Let $\Lambda _1 \to \Lambda _2$ be a homomorphism of torsion rings. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. The diagram

\[ \xymatrix{ D(X_{\acute{e}tale}, \Lambda _2) \ar[r]_{res} & D(X_{\acute{e}tale}, \Lambda _1) \\ D(Y_{\acute{e}tale}, \Lambda _2) \ar[r]^{res} \ar[u]^{Rf^!} & D(Y_{\acute{e}tale}, \Lambda _1) \ar[u]_{Rf^!} } \]

commutes where $res$ is the “restriction” functor which turns a $\Lambda _2$-module into a $\Lambda _1$-module using the given ring map. This holds by uniquenss of adjoints, the second commutative diagram of Remark 61.10.8 and because we have

\[ \mathop{\mathrm{Hom}}\nolimits _{\Lambda _2}(K_1 \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2, K_2) = \mathop{\mathrm{Hom}}\nolimits _{\Lambda _1}(K_1, res(K_2)) \]

This equality either for objects living over $X_{\acute{e}tale}$ or on $Y_{\acute{e}tale}$ is a very special case of Cohomology on Sites, Lemma 21.19.1.


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GLF. Beware of the difference between the letter 'O' and the digit '0'.