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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