Lemma 76.52.3 (0DKQ)—The Stacks project

Lemma 76.52.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. The following are equivalent:

$E$ is $Y$-perfect,
for every commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_ g & V \ar[d] \\ X \ar[r]^ f & Y } \]

where $U$, $V$ are schemes and the vertical arrows are étale, the complex $E|_ U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition 36.35.1,
for some commutative diagram as in (2) with $U \to X$ surjective, the complex $E|_ U$ is $V$-perfect in the sense of Derived Categories of Schemes, Definition 36.35.1,
for every commutative diagram as in (2) with $U$ and $V$ affine the complex $R\Gamma (U, E)$ is $\mathcal{O}_ Y(V)$-perfect.

Proof. To make sense of parts (2), (3), (4) of the lemma, observe that the object $E|_ U$ of $D_\mathit{QCoh}(\mathcal{O}_ U)$ corresponds to an object $E_0$ of $D_\mathit{QCoh}(\mathcal{O}_{U_0})$ where $U_0$ denotes the scheme underlying $U$, see Derived Categories of Spaces, Lemma 75.4.2. Moreover, in this case $E_0$ is pseudo-coherent if and only if $E|_ U$ is pseudo-coherent, see Derived Categories of Spaces, Lemma 75.13.2. Also, $E|_ U$ locally has finite tor dimension over $f^{-1}\mathcal{O}_ Y|_ U = g^{-1}\mathcal{O}_ V$ if and only if $E_0$ locally has finite tor dimension over $g_0^{-1}\mathcal{O}_{V_0}$ by Derived Categories of Spaces, Lemma 75.13.4. Here $g_0 : U_0 \to V_0$ is the morphism of schemes representing $g : U \to V$ (notation as in Derived Categories of Spaces, Remark 75.6.3). Finally, observe that “being pseudo-coherent” is étale local and of course “having locally finite tor dimension” is étale local. Thus we see that it suffices to check $Y$-perfectness étale locally and by the above discussion we see that (1) implies (2) and (3) implies (1). Since part (4) is equivalent to (2) and (3) by Derived Categories of Schemes, Lemma 36.35.3 the proof is complete. $\square$

The Stacks project

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