The Stacks project

61.15 Compact generation

In this section we prove that various derived categories associated to our pro-étale sites are compactly generated as defined in Derived Categories, Definition 13.37.5.

Lemma 61.15.1. Let $S$ be a scheme. Let $\Lambda $ be a ring.

  1. $D(S_{pro\text{-}\acute{e}tale})$ is compactly generated,

  2. $D(S_{pro\text{-}\acute{e}tale}, \Lambda )$ is compactly generated,

  3. $D(S_{pro\text{-}\acute{e}tale}, \mathcal{A})$ is compactly generated for any sheaf of rings $\mathcal{A}$ on $S_{pro\text{-}\acute{e}tale}$,

  4. $D((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$ is compactly generated,

  5. $D((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}, \Lambda )$ is compactly generated, and

  6. $D((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}, \mathcal{A})$ is compactly generated for any sheaf of rings $\mathcal{A}$ on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$,

Proof. Proof of (3). Let $U$ be an affine object of $S_{pro\text{-}\acute{e}tale}$ which is weakly contractible. Then $j_{U!}\mathcal{A}_ U$ is a compact object of the derived category $D(S_{pro\text{-}\acute{e}tale}, \mathcal{A})$, see Cohomology on Sites, Lemma 21.52.6. Choose a set $I$ and for each $i \in I$ an affine weakly contractible object $U_ i$ of $S_{pro\text{-}\acute{e}tale}$ such that every affine weakly contractible object of $S_{pro\text{-}\acute{e}tale}$ is isomorphic to one of the $U_ i$. This is possible because $\mathop{\mathrm{Ob}}\nolimits (S_{pro\text{-}\acute{e}tale})$ is a set. To finish the proof of (3) it suffices to show that $\bigoplus j_{U_ i, !}\mathcal{A}_{U_ i}$ is a generator of $D(S_{pro\text{-}\acute{e}tale}, \mathcal{A})$, see Derived Categories, Definition 13.36.3. To see this, let $K$ be a nonzero object of $D(S_{pro\text{-}\acute{e}tale}, \mathcal{A})$. Then there exists an object $T$ of our site $S_{pro\text{-}\acute{e}tale}$ and a nonzero element $\xi $ of $H^ n(K)(T)$. In other words, $\xi $ is a nonzero section of the $n$th cohomology sheaf of $K$. We may assume $K$ is represented by a complex $\mathcal{K}^\bullet $ of sheaves of $\mathcal{A}$-modules and $\xi $ is the class of a section $s \in \mathcal{K}^ n(T)$ with $\text{d}(s) = 0$. Namely, $\xi $ is locally represented as the class of a section (so you get the result after replacing $T$ by a member of a covering of $T$). Next, we choose a covering $\{ T_ j \to T\} _{j \in J}$ as in Lemma 61.13.3. Since $H^ n(K)$ is a sheaf, we see that for some $j$ the restriction $\xi |_{T_ j}$ remains nonzero. Thus $s|_{T_ j}$ defines a nonzero map $j_{T_ j, !}\mathcal{A}_{T_ j} \to K$ in $D(S_{pro\text{-}\acute{e}tale}, \mathcal{A})$. Since $T_ j \cong U_ i$ for some $i \in I$ we conclude.

The exact same argument works for the big pro-étale site of $S$. $\square$


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 0994. Beware of the difference between the letter 'O' and the digit '0'.