The Stacks project

Proof. The implication (1) $\Rightarrow $ (2) is a general fact, see Modules on Sites, Lemma 18.23.4. Assume (2). By assumption there exists an étale covering $\{ f_ i : X_ i \to X\} $ such that $\epsilon ^*\mathcal{F}|_{(X_ i)_{\acute{e}tale}}$ is generated by finitely many sections. Let $x \in X$. We will show that $\mathcal{F}$ is generated by finitely many sections in a neighbourhood of $x$. Say $x$ is in the image of $X_ i \to X$ and denote $X' = X_ i$. Let $s_1, \ldots , s_ n \in \Gamma (X', \epsilon ^*\mathcal{F}|_{X'_{\acute{e}tale}})$ be generating sections. As $\epsilon ^*\mathcal{F} = \epsilon ^{-1}\mathcal{F} \otimes _{\epsilon ^{-1}\mathcal{O}_ X} \mathcal{O}_{\acute{e}tale}$ we can find an étale morphism $X'' \to X'$ such that $x$ is in the image of $X' \to X$ and such that $s_ i|_{X''} = \sum s_{ij} \otimes a_{ij}$ for some sections $s_{ij} \in \epsilon ^{-1}\mathcal{F}(X'')$ and $a_{ij} \in \mathcal{O}_{\acute{e}tale}(X'')$. Denote $U \subset X$ the image of $X'' \to X$. This is an open subscheme as $f'' : X'' \to X$ is étale (Morphisms, Lemma 29.36.13). After possibly shrinking $X''$ more we may assume $s_{ij}$ come from elements $t_{ij} \in \mathcal{F}(U)$ as follows from the construction of the inverse image functor $\epsilon ^{-1}$. Now we claim that $t_{ij}$ generate $\mathcal{F}|_ U$ which finishes the proof of the lemma. Namely, the corresponding map $\mathcal{O}_ U^{\oplus N} \to \mathcal{F}|_ U$ has the property that its pullback by $f''$ to $X''$ is surjective. Since $f'' : X'' \to U$ is a surjective flat morphism of schemes, this implies that $\mathcal{O}_ U^{\oplus N} \to \mathcal{F}|_ U$ is surjective by looking at stalks and using that $\mathcal{O}_{U, f''(z)} \to \mathcal{O}_{X'', z}$ is faithfully flat for all $z \in X''$. $\square$

Proof. The implication (1) $\Rightarrow $ (2) is a general fact, see Cohomology on Sites, Lemma 21.45.3. Assume $\epsilon ^*E$ is $m$-pseudo-coherent. We will use without further mention that $\epsilon ^*$ is an exact functor and that therefore

To show that $E$ is $m$-pseudo-coherent we may work locally on $X$, hence we may assume that $X$ is quasi-compact (for example affine). Since $X$ is quasi-compact every étale covering $\{ U_ i \to X\} $ has a finite refinement. Thus we see that $\epsilon ^*E$ is an object of $D^{-}(\mathcal{O}_{\acute{e}tale})$, see comments following Cohomology on Sites, Definition 21.45.1. By Lemma 75.4.1 it follows that $E$ is an object of $D^-(\mathcal{O}_ X)$.

Let $n \in \mathbf{Z}$ be the largest integer such that $H^ n(E)$ is nonzero; then $n$ is also the largest integer such that $H^ n(\epsilon ^*E)$ is nonzero. We will prove the lemma by induction on $n - m$. If $n < m$, then the lemma is clearly true. If $n \geq m$, then $H^ n(\epsilon ^*E)$ is a finite $\mathcal{O}_{\acute{e}tale}$-module, see Cohomology on Sites, Lemma 21.45.7. Hence $H^ n(E)$ is a finite $\mathcal{O}_ X$-module, see Lemma 75.13.1. After replacing $X$ by the members of an open covering, we may assume there exists a surjection $\mathcal{O}_ X^{\oplus t} \to H^ n(E)$. We may locally on $X$ lift this to a map of complexes $\alpha : \mathcal{O}_ X^{\oplus t}[-n] \to E$ (details omitted). Choose a distinguished triangle

Then $C$ has vanishing cohomology in degrees $\geq n$. On the other hand, the complex $\epsilon ^*C$ is $m$-pseudo-coherent, see Cohomology on Sites, Lemma 21.45.4. Hence by induction we see that $C$ is $m$-pseudo-coherent. Applying Cohomology on Sites, Lemma 21.45.4 once more we conclude. $\square$

Proof. The easy implication follows from Cohomology on Sites, Lemma 21.46.5. For the converse, assume that $\epsilon ^*E$ has tor amplitude in $[a, b]$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. As $\epsilon $ is a flat morphism of ringed sites (Lemma 75.4.1) we have

Thus the (assumed) vanishing of cohomology sheaves on the right hand side implies the desired vanishing of the cohomology sheaves of $E \otimes ^\mathbf {L}_{\mathcal{O}_ X} \mathcal{F}$ via Lemma 75.4.1. $\square$

Proof. The easy direction in (1) follows from Cohomology on Sites, Lemma 21.46.5. Let $x \in X$ be a point and let $\overline{x}$ be a geometric point lying over $x$. Let $y = f(x)$ and denote $\overline{y}$ the geometric point of $Y$ coming from $\overline{x}$. Then $(f^{-1}\mathcal{O}_ Y)_ x = \mathcal{O}_{Y, y}$ (Sheaves, Lemma 6.21.5) and

is the strict henselization (by Étale Cohomology, Lemmas 59.36.2 and 59.33.1). Since the stalk of $\mathcal{O}_{X_{\acute{e}tale}}$ at $X$ is $\mathcal{O}_{X, x}^{sh}$ we obtain

by transitivity of pullbacks. If $\epsilon ^*E$ has tor amplitude in $[a, b]$ as a complex of $f_{small}^{-1}\mathcal{O}_{Y_{\acute{e}tale}}$-modules, then $(\epsilon ^*E)_{\overline{x}}$ has tor amplitude in $[a, b]$ as a complex of $\mathcal{O}_{Y, y}^{sh}$-modules (because taking stalks is a pullback and lemma cited above). By More on Flatness, Lemma 38.2.6 we find the tor amplitude of $(\epsilon ^*E)_{\overline{x}}$ as a complex of $\mathcal{O}_{Y, y}$-modules is in $[a, b]$. Since $\mathcal{O}_{X, x} \to \mathcal{O}_{X, x}^{sh}$ is faithfully flat (More on Algebra, Lemma 15.45.1) and since $(\epsilon ^*E)_{\overline{x}} = E_ x \otimes _{\mathcal{O}_{X, x}}^\mathbf {L} \mathcal{O}_{X, x}^{sh}$ we may apply More on Algebra, Lemma 15.66.18 to conclude the tor amplitude of $E_ x$ as a complex of $\mathcal{O}_{Y, y}$-modules is in $[a, b]$. By Cohomology, Lemma 20.48.5 we conclude that $E$ as an object of $D(f^{-1}\mathcal{O}_ Y)$ has tor amplitude in $[a, b]$. This gives the reverse implication in (1). Part (2) follows formally from (1). $\square$

Proof. The easy implication follows from the general result contained in Cohomology on Sites, Lemma 21.47.5. For the converse, we can use the equivalence of Cohomology on Sites, Lemma 21.47.4 and the corresponding results for pseudo-coherent and complexes of finite tor dimension, namely Lemmas 75.13.2 and 75.13.3. Some details omitted. $\square$

Proof. Locally $H^ i(E)$ is isomorphic to $H^ i(\mathcal{E}^\bullet )$ with $\mathcal{E}^\bullet $ strictly perfect. The sheaves $\mathcal{E}^ i$ are direct summands of finite free modules, hence quasi-coherent. The lemma follows. $\square$

Proof. As $X$ is quasi-compact we can find an affine scheme $U$ and a surjective étale morphism $U \to X$ (Properties of Spaces, Lemma 66.6.3). Observe that $U$ is Noetherian. Note that $E$ is $m$-pseudo-coherent if and only if $E|_ U$ is $m$-pseudo-coherent (follows from the definition or from Cohomology on Sites, Lemma 21.45.2). Similarly, $H^ i(E)$ is coherent if and only if $H^ i(E)|_ U = H^ i(E|_ U)$ is coherent (see Cohomology of Spaces, Lemma 69.12.2). Thus we may assume that $X$ is representable.

If $X$ is representable by a scheme $X_0$ then (Lemma 75.4.2) we can write $E = \epsilon ^*E_0$ where $E_0$ is an object of $D_\mathit{QCoh}(\mathcal{O}_{X_0})$ and $\epsilon : X_{\acute{e}tale}\to (X_0)_{Zar}$ is as in (75.4.0.1). In this case $E$ is $m$-pseudo-coherent if and only if $E_0$ is by Lemma 75.13.2. Similarly, $H^ i(E_0)$ is of finite type (i.e., coherent) if and only if $H^ i(E)$ is by Lemma 75.13.1. Finally, $H^ i(E_0) = 0$ if and only if $H^ i(E) = 0$ by Lemma 75.4.1. Thus we reduce to the case of schemes which is Derived Categories of Schemes, Lemma 36.10.3. $\square$

Proof. It is clear that (1) implies (2). Assume (2). Let $j : U \to X$ be an étale morphism with $U$ affine. As $X$ is quasi-separated $j : U \to X$ is quasi-compact and separated, hence $j_*$ transforms quasi-coherent modules into quasi-coherent modules (Morphisms of Spaces, Lemma 67.11.2). Thus the functor $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U)$ is essentially surjective. It follows that condition (2) implies the vanishing of $H^ i(E|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \mathcal{G})$ for $i \not\in [a, b]$ for all quasi-coherent $\mathcal{O}_ U$-modules $\mathcal{G}$. Since it suffices to prove that $E|_ U$ has tor amplitude in $[a, b]$ we reduce to the case where $X$ is representable.

If $X$ is representable by a scheme $X_0$ then (Lemma 75.4.2) we can write $E = \epsilon ^*E_0$ where $E_0$ is an object of $D_\mathit{QCoh}(\mathcal{O}_{X_0})$ and $\epsilon : X_{\acute{e}tale}\to (X_0)_{Zar}$ is as in (75.4.0.1). For every quasi-coherent module $\mathcal{F}_0$ on $X_0$ the module $\epsilon ^*\mathcal{F}_0$ is quasi-coherent on $X$ and

as $\epsilon $ is flat (Lemma 75.4.1). Moreover, the vanishing of these sheaves for $i \not\in [a, b]$ implies the same thing for $H^ i(E_0 \otimes _{\mathcal{O}_{X_0}}^\mathbf {L} \mathcal{F}_0)$ by the same lemma. Thus we've reduced the problem to the case of schemes which is treated in Derived Categories of Schemes, Lemma 36.10.6. $\square$

Proof. Recall that $\epsilon $ is flat (Lemma 75.4.1) and hence $\epsilon ^* = L\epsilon ^*$. There is a canonical map from left to right by Cohomology on Sites, Remark 21.35.11. To see this is an isomorphism we can work locally, i.e., we may assume $X$ is an affine scheme.

In case (1) we can represent $E$ by a bounded above complex $\mathcal{E}^\bullet $ of finite free $\mathcal{O}_ X$-modules, see Derived Categories of Schemes, Lemma 36.13.3. We may also represent $F$ by a bounded below complex $\mathcal{F}^\bullet $ of $\mathcal{O}_ X$-modules. Applying Cohomology, Lemma 20.46.11 we see that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, F)$ is represented by the complex with terms

Applying Cohomology on Sites, Lemma 21.44.10 we see that $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\epsilon ^*E, \epsilon ^*F)$ is represented by the complex with terms

Thus the statement of the lemma boils down to the true fact that the canonical map

is an isomorphism for any $\mathcal{O}_ X$-module $\mathcal{F}$ and finite free $\mathcal{O}_ X$-module $\mathcal{E}$.

In case (2) we can represent $E$ by a strictly perfect complex $\mathcal{E}^\bullet $ of $\mathcal{O}_ X$-modules, use Derived Categories of Schemes, Lemmas 36.3.5 and 36.10.7 and the fact that a perfect complex of modules is represented by a finite complex of finite projective modules. Thus we can do the exact same proof as above, replacing the reference to Cohomology, Lemma 20.46.11 by a reference to Cohomology, Lemma 20.46.9. $\square$

Proof. This follows from the analogue for schemes (Derived Categories of Schemes, Lemma 36.10.8) via the criterion of Lemma 75.5.2, the criterion of Lemmas 75.13.2 and 75.13.5, and the result of Lemma 75.13.9. $\square$

Proof. Checking whether or not the map is an isomorphism can be done étale locally hence we may assume $X$ is an affine scheme. Then we can write $K, L, M$ as $\epsilon ^*K_0, \epsilon ^*L_0, \epsilon ^*M_0$ for some $K_0, L_0, M_0$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 75.4.2. Then we see that Lemma 75.13.9 reduces cases (1) and (3) to the case of schemes which is Derived Categories of Schemes, Lemma 36.10.9. If $K$ is perfect but no other assumptions are made, then we do not know that either side of the arrow is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ but the result is still true because $K$ will be represented (after localizing further) by a finite complex of finite free modules in which case it is clear. $\square$