The Stacks project

Proof. Special case of Derived Categories of Spaces, Example 75.19.5. $\square$

Proof. Proof of (2). Write $E_ n = E|_{X_ n}$ and $N_ n = R\mathop{\mathrm{Hom}}\nolimits _{X_ n}(E_ n, \mathcal{F}_ n)$. Recall that $R\mathop{\mathrm{Hom}}\nolimits _{X_ n}(-, -)$ is equal to $R\Gamma (X_ n, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, -))$, see Cohomology on Sites, Section 21.36. Hence by Derived Categories of Spaces, Lemma 75.25.8 we see that $N_ n$ is a perfect object of $D(A_ n)$ whose formation commutes with base change. Thus the maps $N_ n \otimes _{A_ n}^\mathbf {L} A_{n - 1} \to N_{n - 1}$ coming from $\varphi _ n$ are isomorphisms. By More on Algebra, Lemma 15.97.3 we find that $R\mathop{\mathrm{lim}}\nolimits N_ n$ is perfect and that its base change back to $A_ n$ recovers $N_ n$. On the other hand, the exact functor $R\mathop{\mathrm{Hom}}\nolimits _ X(E, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence

This proves (2). To see that (1) holds, translate it into (2) using Cohomology on Sites, Lemma 21.48.4. $\square$

Proof. Combinging Lemma 77.12.3 and Derived Categories of Spaces, Lemma 75.25.7 we see that $R\Gamma (X, K \otimes ^\mathbf {L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Thus the lemma follows from More on Morphisms of Spaces, Lemma 76.51.4. $\square$

Proof. Fix $n$. By Derived Categories of Spaces, Lemma 75.27.3 there exists a system of perfect complexes $E_ m$ on $X$ such that $R\Gamma (U, K) = \text{hocolim} R\Gamma (X, K \otimes ^\mathbf {L} E_ m)$. In fact, this formula holds not just for $K$ but for every object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Applying this to $\mathcal{F}_ n$ we obtain

Using Lemma 77.12.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

Proof. Fix $n \geq 1$. By construction there is a canonical map $K \to \mathcal{F}_ n$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and hence a canonical map $H^0(K) \to \mathcal{F}_ n$ of quasi-coherent sheaves. This explains the meaning of part (3).

Let $x \in X_0$ be a point. We will find an open neighbourhood $W$ of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact this will prove the lemma. Let $U \to X$ be an étale morphism with $U$ affine and $u \in U$ a point mapping to $x$. Since $|U| \to |X|$ is open it suffices to find an open neighbourhood of $u$ in $U$ where (1), (2), and (3) are true. Say $U = \mathop{\mathrm{Spec}}(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma 77.12.4 and the definition of relative pseudo-coherence there exists a bounded above complex $F^\bullet $ of finite free $P$-modules representing $Ri_*K$ where $i : U \to \mathop{\mathrm{Spec}}(P)$ is the closed immersion induced by the presentation. Let $M_ n$ be the $B$-module corresponding to $\mathcal{F}_ n|_ U$. By Lemma 77.12.5

Let $i$ be the maximal index such that $F^ i$ is nonzero. If $i \leq 0$, then (1), (2), and (3) are true. If not, then $i > 0$ and we see that the rank of the map

in the point $u$ is maximal. Hence in an open neighbourhood of $u$ inside $\mathop{\mathrm{Spec}}(P)$ the rank is maximal. Thus after replacing $P$ by a principal localization we may assume that the displayed map is surjective. Since $F^ i$ is finite free we may choose a splitting $F^{i - 1} = F' \oplus F^ i$. Then we may replace $F^\bullet $ by the complex

and we win by induction on $i$. $\square$

Proof. Fix $n \geq 1$. Let $I_ n = \mathop{\mathrm{Ker}}(A \to A_ n)$. By More on Algebra, Lemma 15.11.3 the pair $(A, I_ n)$ is henselian. Let $Z \subset W$ be the scheme theoretic support of $\mathcal{F}$. This is a closed subspace as $\mathcal{F}$ is of finite presentation. By part (3) of Lemma 77.12.6 we see that $Z \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_ n)$ is equal to the support of $\mathcal{F}_ n$ and hence proper over $\mathop{\mathrm{Spec}}(A/I)$. By More on Morphisms of Spaces, Lemma 76.36.10 we can write $Z = Z_1 \amalg Z_2$ with $Z_1, Z_2$ open and closed in $Z$, with $Z_1$ proper over $A$, and with $Z_1 \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I_ n)$ equal to the support of $\mathcal{F}_ n$. In other words, $|Z_2|$ does not meet $X_0$. Hence after replacing $W$ by $W \setminus Z_2$ we obtain the lemma. $\square$

Proof. Apply Lemmas 77.12.2, 77.12.3, 77.12.4, 77.12.5, 77.12.6, and 77.12.7 to get an open subspace $W \subset X$ containing all points lying over $\mathop{\mathrm{Spec}}(A_ n)$ and a finitely presented $\mathcal{O}_ W$-module $\mathcal{F}$ whose support is proper over $A$ with $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ W} \mathcal{O}_{X_ n}$ for all $n \geq 1$. (This makes sense as $X_ n \subset W$.) By Lemma 77.3.6 we see that $\mathcal{F}$ is universally pure relative to $\mathop{\mathrm{Spec}}(A)$. By Theorem 77.11.7 (for explanation, see Lemma 77.11.8) there exists a universal flattening $S' \to \mathop{\mathrm{Spec}}(A)$ of $\mathcal{F}$ and moreover the morphism $S' \to \mathop{\mathrm{Spec}}(A)$ is a monomorphism of finite presentation. In particular $S'$ is a scheme (this follows from the proof of the theorem but it also follows a postoriori by Morphisms of Spaces, Proposition 67.50.2). Since the base change of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(A_ n)$ is $\mathcal{F}_ n$ we find that $\mathop{\mathrm{Spec}}(A_ n) \to \mathop{\mathrm{Spec}}(A)$ factors (uniquely) through $S'$ for each $n$. By More on Flatness, Lemma 38.28.8 we see that $S' = \mathop{\mathrm{Spec}}(A)$. This means that $\mathcal{F}$ is flat over $A$. Finally, since the scheme theoretic support $Z$ of $\mathcal{F}$ is proper over $\mathop{\mathrm{Spec}}(A)$, the morphism $Z \to X$ is closed. Hence the pushforward $(W \to X)_*\mathcal{F}$ is supported on $W$ and has all the desired properties. $\square$