The Stacks project

Example 48.9.8. If $i : Z \to X$ is closed immersion of Noetherian schemes, then the diagram

\[ \xymatrix{ i_*a(K) \ar[rr]_-{\text{Tr}_{i, K}} \ar@{=}[d] & & K \ar@{=}[d] \\ i_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Z, K) \ar@{=}[r] & R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(i_*\mathcal{O}_ Z, K) \ar[r] & K } \]

is commutative for $K \in D_\mathit{QCoh}^+(\mathcal{O}_ X)$. Here the horizontal equality sign is Lemma 48.9.3 and the lower horizontal arrow is induced by the map $\mathcal{O}_ X \to i_*\mathcal{O}_ Z$. The commutativity of the diagram is a consequence of Lemma 48.9.7.

Comments (0)

There are also:

  • 2 comment(s) on Section 48.9: Right adjoint of pushforward for closed immersions

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Your email address will not be published.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 0B6M. Beware of the difference between the letter 'O' and the digit '0'.