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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 nth 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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