The Stacks project

Lemma 75.13.2. Let $X$ be a scheme. Let $E$ be an object of $D(\mathcal{O}_ X)$. The following are equivalent

  1. $E$ is $m$-pseudo-coherent, and

  2. $\epsilon ^*E$ is $m$-pseudo-coherent on the small ├ętale site of $X$.

Here $\epsilon $ is as in (

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

\[ \epsilon ^*H^ i(E) = H^ i(\epsilon ^*E). \]

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 ├ętale 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

\[ \mathcal{O}_ X^{\oplus t}[-n] \to E \to C \to \mathcal{O}_ X^{\oplus t}[-n + 1] \]

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. $\square$

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