The Stacks project

37.65 Relatively perfect objects

This section is a continuation of the discussion in Derived Categories of Schemes, Section 36.35.

Lemma 37.65.1. Let $i : X \to X'$ be a finite order thickening of schemes. 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. To do this, we may reduce to the case of a first order thickening, see Section 37.2. 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, Lemma 20.51.4 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 Schemes, Lemmas 36.10.1, 36.3.9, and 36.4.1.

Assume $K'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$. The question is local on $X'$ hence we may assume $X'$ is affine. Say $X' = \mathop{\mathrm{Spec}}(A')$ and $X = \mathop{\mathrm{Spec}}(A)$ with $A = A'/I$ and $I$ nilpotent. Then $K'$ comes from an object $M' \in D(A')$, see Derived Categories of Schemes, Lemma 36.3.5. Thus $M = M' \otimes _{A'}^\mathbf {L} A$ is a pseudo-coherent object of $D(A)$ by Derived Categories of Schemes, Lemma 36.10.2 and our assumption on $K$. Hence we can represent $M$ by a bounded above complex of finite free $A$-modules $E^\bullet $, see More on Algebra, Lemma 15.63.5. By More on Algebra, Lemma 15.74.3 we conclude that $M'$ is pseudo-coherent as desired. $\square$

Lemma 37.65.2. Consider a cartesian diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & X' \ar[d]^{f'} \\ Y \ar[r]^ j & Y' } \]

of schemes. 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.

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 (Derived Categories of Schemes, Definition 36.35.1). By Lemma 37.65.1 we find that $E'$ is pseudo-coherent. In particular, $E'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$, see Derived Categories of Schemes, Lemma 36.10.1. To prove that $E'$ locally has finite tor dimension we may work locally on $X'$. Hence we may assume $X'$, $S'$, $X$, $S$ are affine, say given by rings $A'$, $R'$, $A$, $R$. Then we reduce to the commutative algebra version by Derived Categories of Schemes, Lemma 36.35.3. The commutative algebra version in More on Algebra, Lemma 15.82.8. $\square$

Lemma 37.65.3. 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 $f : X \to S$ be 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 finite affine open covering $X = U_1 \cup \ldots \cup U_ n$. For each $i$ we can choose a closed immersion $U_ i \to \mathbf{A}^{d_ i}_ S$. Set $U_{i, 0} = S_0 \times _ S U_ i$. For each $i$ the complex $E_0|_{U_{i, 0}}$ has tor amplitude in $[a_ i, b_ i]$ for some $a_ i, b_ i \in \mathbf{Z}$. Let $x \in X$ be a point. We will show that the tor amplitude of $E_ x$ over $R$ is in $[a_ i - d_ i, b_ i]$ for some $i$. This will finish the proof as the tor amplitude can be read off from the stalks by Cohomology, Lemma 20.45.5.

Since $f$ is proper $f(\overline{\{ x\} })$ is a closed subset of $S$. Since $I$ is contained in the Jacobson radical, we see that $f(\overline{\{ x\} })$ meeting the closed subset $S_0 \subset S$. Hence there is a specialization $x \leadsto x_0$ with $x_0 \in X_0$. Pick an $i$ with $x_0 \in U_ i$, so $x_0 \in U_{i, 0}$. We will fix $i$ for the rest of the proof. Write $U_ i = \mathop{\mathrm{Spec}}(A)$. Then $A$ is a flat, finitely presented $R$-algebra which is a quotient of a polynomial $R$-algebra in $d_ i$-variables. The restriction $E|_{U_ i}$ corresponds (by 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 $x \leadsto x_0$. Then $E_ x = K_{\mathfrak q}$ and $K_{\mathfrak q}$ is a localization of $K_{\mathfrak q_0}$. Hence it suffices to show that $K_{\mathfrak q_0}$ has tor amplitude in $[a_ i - d_ i, b_ i]$ 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 $a_ i, b_ i$ this complex has cohomology only in degrees in the interval $[a_ i, b_ i]$. 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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