Lemma 76.54.1. Let $S$ be a scheme. Let $\{ f_ i : X_ i \to X\} $ be an fpqc covering of algebraic spaces over $S$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $m \in \mathbf{Z}$. Then $E$ is $m$-pseudo-coherent if and only if each $Lf_ i^*E$ is $m$-pseudo-coherent.
76.54 Descent of finiteness properties of complexes
This section is the analogue of More on Morphisms, Section 37.70 and Derived Categories of Schemes, Section 36.12.
Proof. Pullback always preserves $m$-pseudo-coherence, see Cohomology on Sites, Lemma 21.45.3. Thus it suffices to assume $Lf_ i^*E$ is $m$-pseudo-coherent and to prove that $E$ is $m$-pseudo-coherent. Then first we may assume $X_ i$ is a scheme for all $i$, see Topologies on Spaces, Lemma 73.9.5. Next, choose a surjective étale morphism $U \to X$ where $U$ is a scheme. Then $U_ i = U \times _ X X_ i$ is a scheme and we obtain an fpqc covering $\{ U_ i \to U\} $ of schemes, see Topologies on Spaces, Lemma 73.9.4. We know the result is true for $\{ U_ i \to U\} _{i \in I}$ by the case for schemes, see Derived Categories of Schemes, Lemma 36.12.2. On the other hand, the restriction $E|_ U$ comes from an object of $D_\mathit{QCoh}(\mathcal{O}_ U)$ (defined using the Zariski topology and the “usual” structure sheaf of $U$), see Derived Categories of Spaces, Lemma 75.4.2. The lemma follows as the two notions of pseudo-coherent (étale and Zariski) agree by Derived Categories of Spaces, Lemma 75.13.2. $\square$
Lemma 76.54.2. Let $S$ be a scheme. Let $\{ g_ i : Y_ i \to Y\} $ be an fpqc covering of algebraic spaces over $S$. Let $f : X \to Y$ be a morphism of algebraic spaces and set $X_ i = Y_ i \times _ Y X$ with projections $f_ i : X_ i \to Y_ i$ and $g'_ i : X_ i \to X$. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $a, b \in \mathbf{Z}$. Then the following are equivalent
$E$ has tor amplitude in $[a, b]$ as an object of $D(f^{-1}\mathcal{O}_ Y)$, and
$L(g'_ i)^*E$ has tor amplitude in $[a, b]$ as a object of $D(f_ i^{-1}\mathcal{O}_{Y_ i})$ for all $i$.
Also true if “tor amplitude in $[a, b]$” is replaced by “locally finite tor dimension”.
Proof. Pullback preserves “tor amplitude in $[a, b]$” by Derived Categories of Spaces, Lemma 75.20.7 Observe that $Y_ i$ and $X$ are tor independent over $Y$ as $Y_ i \to Y$ is flat. Let us assume (2) and prove (1). We can compute tor dimension at stalks, see Cohomology on Sites, Lemma 21.46.10 and Properties of Spaces, Theorem 66.19.12. Let $\overline{x}$ be a geometric point of $X$. Choose an $i$ and a geometric point $\overline{x}_ i$ in $X_ i$ with image $\overline{x}$ in $X$. Then
\[ (L(g_ i')^*E)_{\overline{x}_ i} = E_{\overline{x}} \otimes _{\mathcal{O}_{X, \overline{x}}}^\mathbf {L} \mathcal{O}_{X_, \overline{x}_ i} \]
Let $\overline{y}_ i$ in $Y_ i$ and $\overline{y}$ in $Y$ be the image of $\overline{x}_ i$ and $\overline{x}$. Since $X$ and $Y_ i$ are tor independent over $Y$, we can apply More on Algebra, Lemma 15.61.2 to see that the right hand side of the displayed formula is equal to $E_{\overline{x}} \otimes _{\mathcal{O}_{Y, \overline{y}}}^\mathbf {L} \mathcal{O}_{Y_ i, \overline{y}_ i}$ in $D(\mathcal{O}_{Y_ i, \overline{y}_ i})$. Since we have assume the tor amplitude of this is in $[a, b]$, we conclude that the tor amplitude of $E_{\overline{x}}$ in $D(\mathcal{O}_{Y, \overline{y}})$ is in $[a, b]$ by More on Algebra, Lemma 15.66.17. Thus (1) follows.
Using some elementary topology the case “locally finite tor dimension” follows too. $\square$
The following lemmas do not really belong in this section.
Lemma 76.54.3. Let $S$ be a scheme. Let $i : X \to X'$ be a finite order thickening of algebraic spaces. Let $K' \in D(\mathcal{O}_{X'})$ be an object such that $K = Li^*K'$ is pseudo-coherent. Then $K'$ is pseudo-coherent.
Proof. We first prove $K'$ has quasi-coherent cohomology sheaves; we urge the reader to skip this part. To do this, we may reduce to the case of a first order thickening, see Section 76.9. Let $\mathcal{I} \subset \mathcal{O}_{X'}$ be the quasi-coherent sheaf of ideals cutting out $X$. Tensoring the short exact sequence
\[ 0 \to \mathcal{I} \to \mathcal{O}_{X'} \to i_*\mathcal{O}_ X \to 0 \]
with $K'$ we obtain a distinguished triangle
\[ K' \otimes _{\mathcal{O}_{X'}}^\mathbf {L} \mathcal{I} \to K' \to K' \otimes _{\mathcal{O}_{X'}}^\mathbf {L} i_*\mathcal{O}_ X \to (K' \otimes _{\mathcal{O}_{X'}}^\mathbf {L} \mathcal{I})[1] \]
Since $i_* = Ri_*$ and since we may view $\mathcal{I}$ as a quasi-coherent $\mathcal{O}_ X$-module (as we have a first order thickening) we may rewrite this as
\[ i_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I}) \to K' \to i_*K \to i_*(K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{I})[1] \]
Please use Cohomology of Spaces, Lemma 69.4.4 to identify the terms. Since $K$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$ we conclude that $K'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$; this uses Derived Categories of Spaces, Lemmas 75.13.6, 75.5.6, and 75.6.1.
Assume $K'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$. The question is étale local on $X'$ hence we may assume $X'$ is affine. In this case the result follows from the case of schemes (More on Morphisms, Lemma 37.71.1). The translation into the language of schemes uses Derived Categories of Spaces, Lemmas 75.4.2 and 75.13.2 and Remark 75.6.3. $\square$
Lemma 76.54.4. Let $S$ be a scheme. Consider a cartesian diagram of algebraic spaces over $S$. Assume $X' \to Y'$ is flat and locally of finite presentation and $Y \to Y'$ is a finite order thickening. Let $E' \in D(\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect.
\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & X' \ar[d]^{f'} \\ Y \ar[r]^ j & Y' } \]
Proof. Recall that being $Y$-perfect for $E$ means $E$ is pseudo-coherent and locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_ Y$-modules (Definition 76.52.1). By Lemma 76.54.3 we find that $E'$ is pseudo-coherent. In particular, $E'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$, see Derived Categories of Spaces, Lemma 75.13.6. By Lemma 76.52.3 this reduces us to the case of schemes. The case of schemes is More on Morphisms, Lemma 37.71.2. $\square$
Lemma 76.54.5. Let $(R, I)$ be a pair consisting of a ring and an ideal $I$ contained in the Jacobson radical. Set $S = \mathop{\mathrm{Spec}}(R)$ and $S_0 = \mathop{\mathrm{Spec}}(R/I)$. Let $X$ be an algebraic space over $R$ whose structure morphism $f : X \to S$ is proper, flat, and of finite presentation. Denote $X_0 = S_0 \times _ S X$. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent. If the derived restriction $E_0$ of $E$ to $X_0$ is $S_0$-perfect, then $E$ is $S$-perfect.
Proof. Choose a surjective étale morphism $U \to X$ with $U$ affine. Choose a closed immersion $U \to \mathbf{A}^ d_ S$. Set $U_0 = S_0 \times _ S U$. The complex $E_0|_{U_0}$ has tor amplitude in $[a, b]$ for some $a, b \in \mathbf{Z}$. Let $\overline{x}$ be a geometric point of $X$. We will show that the tor amplitude of $E_{\overline{x}}$ over $R$ is in $[a - d, b]$. This will finish the proof as the tor amplitude can be read off from the stalks by Cohomology on Sites, Lemma 21.46.10 and Properties of Spaces, Theorem 66.19.12.
Let $x \in |X|$ be the point determined by $\overline{x}$. Recall that $|X| \to |S|$ is closed (by definition of proper morphisms). Since $I$ is contained in the Jacobson radical, any nonempty closed subset of $S$ contains a point of the closed subscheme $S_0$. Hence we can find a specialization $x \leadsto x_0$ in $|X|$ with $x_0 \in |X_0|$. Choose $u_0 \in U_0$ mapping to $x_0$. By Decent Spaces, Lemma 68.7.4 (or by Decent Spaces, Lemma 68.7.3 which applies directly to étale morphisms) we find a specialization $u \leadsto u_0$ in $U$ such that $u$ maps to $x$. We may lift $\overline{x}$ to a geometric point $\overline{u}$ of $U$ lying over $u$. Then we have $E_{\overline{x}} = (E|_ U)_{\overline{u}}$.
Write $U = \mathop{\mathrm{Spec}}(A)$. Then $A$ is a flat, finitely presented $R$-algebra which is a quotient of a polynomial $R$-algebra in $d$-variables. The restriction $E|_ U$ corresponds (by Derived Categories of Spaces, Lemmas 75.13.6, 75.4.2, and 75.13.2 and Derived Categories of Schemes, Lemma 36.3.5 and 36.10.2) to a pseudo-coherent object $K$ of $D(A)$. Observe that $E_0$ corresponds to $K \otimes _ A^\mathbf {L} A/IA$. Let $\mathfrak q \subset \mathfrak q_0 \subset A$ be the prime ideals corresponding to $u \leadsto u_0$. Then
\[ E_{\overline{x}} = (E|_ U)_{\overline{u}} = E_ u \otimes _{\mathcal{O}_{U, u}}^\mathbf {L} \mathcal{O}_{U, \overline{u}} = K_{\mathfrak q} \otimes _{A_\mathfrak q}^\mathbf {L} A_{\mathfrak q}^{sh} \]
(some details omitted). Since $A_\mathfrak q \to A_\mathfrak q^{sh}$ is flat, the tor amplitude of this as an $R$-module is the same as the tor amplitude of $K_\mathfrak q$ as an $R$-module (More on Algebra, Lemma 15.66.18). Also, $K_{\mathfrak q}$ is a localization of $K_{\mathfrak q_0}$. Hence it suffices to show that $K_{\mathfrak q_0}$ has tor amplitude in $[a - d, b]$ as a complex of $R$-modules.
Let $I \subset \mathfrak p_0 \subset R$ be the prime ideal corresponding to $f(x_0)$. Then we have
\begin{align*} K \otimes _ R^\mathbf {L} \kappa (\mathfrak p_0) & = (K \otimes _ R^\mathbf {L} R/I) \otimes _{R/I}^\mathbf {L} \kappa (\mathfrak p_0) \\ & = (K \otimes _ A^\mathbf {L} A/IA) \otimes _{R/I}^\mathbf {L} \kappa (\mathfrak p_0) \end{align*}
the second equality because $R \to A$ is flat. By our choice of $a, b$ this complex has cohomology only in degrees in the interval $[a, b]$. Thus we may finally apply More on Algebra, Lemma 15.83.9 to $R \to A$, $\mathfrak q_0$, $\mathfrak p_0$ and $K$ to conclude. $\square$
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