The Stacks project

Lemma 76.45.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $m \in \mathbf{Z}$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. With notation as explained in Remark 76.45.1 the following are equivalent:

  1. for every commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    where $U$, $V$ are schemes and the vertical arrows are étale, the complex $E|_ U$ is $m$-pseudo-coherent relative to $V$,

  2. for some commutative diagram as in (1) with $U \to X$ surjective, the complex $E|_ U$ is $m$-pseudo-coherent relative to $V$,

  3. for every commutative diagram as in (1) with $U$ and $V$ affine the complex $R\Gamma (U, E)$ of $\mathcal{O}_ X(U)$-modules is $m$-pseudo-coherent relative to $\mathcal{O}_ Y(V)$.

Proof. Part (1) implies (3) by More on Morphisms, Lemma 37.59.7.

Assume (3). Pick any commutative diagram as in (1) with $U \to X$ surjective. Choose an affine open covering $V = \bigcup V_ j$ and affine open coverings $(U \to V)^{-1}(V_ j) = \bigcup U_{ij}$. By (3) and More on Morphisms, Lemma 37.59.7 we see that $E|_ U$ is $m$-pseudo-coherent relative to $V$. Thus (3) implies (2).

Assume (2). Choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

where $U$, $V$ are schemes, the vertical arrows are étale, the morphism $U \to X$ is surjective, and $E|_ U$ is $m$-pseudo-coherent relative to $V$. Next, suppose given a second commutative diagram

\[ \xymatrix{ U' \ar[d] \ar[r] & V' \ar[d] \\ X \ar[r] & Y } \]

with étale vertical arrows and $U', V'$ schemes. We want to show that $E|_{U'}$ is $m$-pseudo-coherent relative to $V'$. The morphism $U'' = U \times _ X U' \to U'$ is surjective étale and $U'' \to V'$ factors through $V'' = V' \times _ Y V$ which is étale over $V'$. Hence it suffices to show that $E|_{U''}$ is $m$-pseudo-coherent relative to $V''$, see More on Morphisms, Lemmas 37.70.1 and 37.70.2. Using the second lemma once more it suffices to show that $E|_{U''}$ is $m$-pseudo-coherent relative to $V$. This is true by More on Morphisms, Lemma 37.59.16 and the fact that an étale morphism of schemes is pseudo-coherent by More on Morphisms, Lemma 37.60.6. $\square$

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