The Stacks project

Lemma 59.52.4. Let $I$, $g_ i : X_ i \to S_ i$, $g : X \to S$, $f_ i$, $g_ i$, $h_ i$ be as in Lemma 59.51.8. Let $0 \in I$ and $K_0 \in D^+(X_{0, {\acute{e}tale}})$. For $i \geq 0$ denote $K_ i$ the pullback of $K_0$ to $X_ i$. Denote $K$ the pullback of $K$ to $X$. Then

\[ R^ pg_*K = \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}K_ i \]

for all $p \in \mathbf{Z}$.

Proof. Fix an integer $p_0 \in \mathbf{Z}$. Let $a$ be an integer such that $H^ j(K_0) = 0$ for $j < a$. We will prove the formula holds for all $p \leq p_0$ by descending induction on $a$. If $a > p_0$, then we see that the left and right hand side of the formula are zero for $p \leq p_0$ by trivial vanishing, see Derived Categories, Lemma 13.16.1. Assume $a \leq p_0$. Consider the distinguished triangle

\[ H^ a(K_0)[-a] \to K_0 \to \tau _{\geq a + 1}K_0 \]

Pulling back this distinguished triangle to $X_ i$ and $X$ gives compatible distinguished triangles for $K_ i$ and $K$. For $p \leq p_0$ we consider the commutative diagram

\[ \xymatrix{ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^{p - 1}g_{i, *}(\tau _{\geq a + 1}K_ i) \ar[r]_-\alpha \ar[d] & R^{p - 1}g_*(\tau _{\geq a + 1}K) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}(H^ a(K_ i)[-a]) \ar[r]_-\beta \ar[d] & R^ pg_*(H^ a(K)[-a]) \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} h_ i^{-1}R^ pg_{i, *}K_ i \ar[r]_-\gamma \ar[d] & R^ pg_*K \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} R^ pg_{i, *}\tau _{\geq a + 1}K_ i \ar[r]_-\delta \ar[d] & R^ pg_*\tau _{\geq a + 1}K \ar[d] \\ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} R^{p + 1}g_{i, *}(H^ a(K_ i)[-a]) \ar[r]^-\epsilon & R^{p + 1}g_*(H^ a(K)[-a]) } \]

with exact columns. The arrows $\beta $ and $\epsilon $ are isomorphisms by Lemma 59.51.8. The arrows $\alpha $ and $\delta $ are isomorphisms by induction hypothesis. Hence $\gamma $ is an isomorphism as desired. $\square$


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Your email address will not be published.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