The Stacks project

Lemma 74.54.5. 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 ├ętale 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 21.44.10 and Properties of Spaces, Theorem 64.19.12.

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 66.7.3 (or by Decent Spaces, Lemma 66.7.2 which applies directly to ├ętale 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 73.13.6, 73.4.2, and 73.13.2 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 15.65.18). 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 15.82.9 to $R \to A$, $\mathfrak q_0$, $\mathfrak p_0$ and $K$ to conclude. $\square$

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