The Stacks project

21.21 Localization and cohomology

Let $\mathcal{C}$ be a site. Let $f : X \to Y$ be a morphism of $\mathcal{C}$. Then we obtain a morphism of topoi

\[ j_{X/Y} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/Y) \]

See Sites, Sections 7.25 and 7.27. Some questions about cohomology are easier for this type of morphisms of topoi. Here is an example where we get a trivial type of base change theorem.

Lemma 21.21.1. Let $\mathcal{C}$ be a site. Let

\[ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } \]

be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^{-1} \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^{-1}$ as functors $D(\mathcal{C}/X) \to D(\mathcal{C}/Y')$.

Proof. Let $E \in D(\mathcal{C}/X)$. Choose a K-injective complex $\mathcal{I}^\bullet $ of abelian sheaves on $\mathcal{C}/X$ representing $E$. By Lemma 21.20.1 we see that $j_{X'/X}^{-1}\mathcal{I}^\bullet $ is K-injective too. Hence we may compute $Rj_{X'/Y'}(j_{X'/X}^{-1}E)$ by $j_{X'/Y', *}j_{X'/X}^{-1}\mathcal{I}^\bullet $. Thus we see that the equality holds by Sites, Lemma 7.27.5. $\square$

If we have a ringed site $(\mathcal{C}, \mathcal{O})$ and a morphism $f : X \to Y$ of $\mathcal{C}$, then $j_{X/Y}$ becomes a morphism of ringed topoi

\[ j_{X/Y} : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/Y), \mathcal{O}_ Y) \]

See Modules on Sites, Lemma 18.19.5.

Lemma 21.21.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let

\[ \xymatrix{ X' \ar[d] \ar[r] & X \ar[d] \\ Y' \ar[r] & Y } \]

be a cartesian diagram of $\mathcal{C}$. Then we have $j_{Y'/Y}^* \circ Rj_{X/Y, *} = Rj_{X'/Y', *} \circ j_{X'/X}^*$ as functors $D(\mathcal{O}_ X) \to D(\mathcal{O}_{Y'})$.

Proof. Since $j_{Y'/Y}^{-1}\mathcal{O}_ Y = \mathcal{O}_{Y'}$ we have $j_{Y'/Y}^* = Lj_{Y'/Y}^* = j_{Y'/Y}^{-1}$. Similarly we have $j_{X'/X}^* = Lj_{X'/X}^* = j_{X'/X}^{-1}$. Thus by Lemma 21.20.7 it suffices to prove the result on derived categories of abelian sheaves which we did in Lemma 21.21.1. $\square$

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Your email address will not be published.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