Remark 76.45.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of representable algebraic spaces over $S$ which is locally of finite type. Let $f_0 : X_0 \to Y_0$ be a morphism of schemes representing $f$ (awkward but temporary notation). Then $f_0$ is locally of finite type. If $E$ is an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$, then $E$ is the pullback of a unique object $E_0$ in $D_\mathit{QCoh}(\mathcal{O}_{X_0})$, see Derived Categories of Spaces, Lemma 75.4.2. In this situation the phrase “$E$ is $m$-pseudo-coherent relative to $Y$” will be taken to mean “$E_0$ is $m$-pseudo-coherent relative to $Y_0$” as defined in More on Morphisms, Section 37.59.
76.45 Relative pseudo-coherence
This section is the analogue of More on Morphisms, Section 37.59. However, in the treatment of this material for algebraic spaces we have decided to work exclusively with objects in the derived category whose cohomology sheaves are quasi-coherent. There are two reasons for this: (1) it greatly simplifies the exposition and (2) we currently have no use for the more general notion.
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:
for every commutative diagram
where $U$, $V$ are schemes and the vertical arrows are étale, the complex $E|_ U$ is $m$-pseudo-coherent relative to $V$,
for some commutative diagram as in (1) with $U \to X$ surjective, the complex $E|_ U$ is $m$-pseudo-coherent relative to $V$,
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 $\mathcal{O}_ Y(V)$.
\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]
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. $\square$
Definition 76.45.3. 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 $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Fix $m \in \mathbf{Z}$.
We say $E$ is $m$-pseudo-coherent relative to $Y$ if the equivalent conditions of Lemma 76.45.2 are satisfied.
We say $E$ is pseudo-coherent relative to $Y$ if $E$ is $m$-pseudo-coherent relative to $Y$ for all $m \in \mathbf{Z}$.
We say $\mathcal{F}$ is $m$-pseudo-coherent relative to $Y$ if $\mathcal{F}$ viewed as an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is $m$-pseudo-coherent relative to $Y$.
We say $\mathcal{F}$ is pseudo-coherent relative to $Y$ if $\mathcal{F}$ viewed as an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is pseudo-coherent relative to $Y$.
Most of the properties of pseudo-coherent complexes relative to a base will follow immediately from the corresponding properties in the case of schemes. We will add the relevant lemmas here as needed.
Lemma 76.45.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$. If $f$ is flat and locally of finite presentation, then the following are equivalent
$E$ is pseudo-coherent relative to $Y$, and
$E$ is pseudo-coherent on $X$.
Proof. By étale localization and the definitions we may assume $X$ and $Y$ are schemes. For the case of schemes this follows from More on Morphisms, Lemma 37.59.18. $\square$
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