## 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 21.50.6. 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 $S$. $\square$

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