The Stacks project

Lemma 75.24.2. In Situation 75.24.1. Let $E_0$ and $K_0$ be objects of $D(\mathcal{O}_{X_0})$. Set $E_ i = Lf_{i0}^*E_0$ and $K_ i = Lf_{i0}^*K_0$ for $i \geq 0$ and set $E = Lf_0^*E_0$ and $K = Lf_0^*K_0$. Then the map

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{X_ i})}(E_ i, K_ i) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(E, K) \]

is an isomorphism if either

  1. $E_0$ is perfect and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$, or

  2. $E_0$ is pseudo-coherent and $K_0 \in D_\mathit{QCoh}(\mathcal{O}_{X_0})$ has finite tor dimension.

Proof. For every quasi-compact and quasi-separated object $U_0$ of $(X_0)_{spaces, {\acute{e}tale}}$ consider the condition $P$ that the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _{i \geq 0} \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_{U_ i})}(E_ i|_{U_ i}, K_ i|_{U_ i}) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(E|_ U, K|_ U) \]

is an isomorphism, where $U = X \times _{X_0} U_0$ and $U_ i = X_ i \times _{X_0} U_0$. We will prove $P$ holds for each $U_0$ by the induction principle of Lemma 75.9.3. Condition (2) of this lemma follows immediately from Mayer-Vietoris for hom in the derived category, see Lemma 75.10.4. Thus it suffices to prove the lemma when $X_0$ is affine.

If $X_0$ is affine, then the result follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.29.2. To see this use the equivalence of Lemma 75.4.2 and use the translation of properties explained in Lemmas 75.13.2, 75.13.3, and 75.13.5. $\square$


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Your email address will not be published.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 09RI. Beware of the difference between the letter 'O' and the digit '0'.