The Stacks project

Proof. Special case of Derived Categories of Schemes, Example 36.21.5. $\square$

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, Section 20.44. Hence by Derived Categories of Schemes, Lemma 36.30.7 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, Lemma 20.50.5. $\square$

Proof. Combinging Lemma 38.28.3 and Derived Categories of Schemes, Lemma 36.34.3 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, Lemma 37.69.4. $\square$

Proof. Fix $n$. By Derived Categories of Schemes, Lemma 36.33.4 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 38.28.3 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 $U \subset X$ be an affine open neighbourhood of $x$. Say $U = \mathop{\mathrm{Spec}}(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma 38.28.4 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 38.28.5

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 $x$ is maximal. Hence in an open neighbourhood of $x$ 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 support of $\mathcal{F}$. This is a closed subset as $\mathcal{F}$ is of finite presentation. By part (3) of Lemma 38.28.6 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, Lemma 37.53.9 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. Set $S_ n = \mathop{\mathrm{Spec}}(A_ n)$. Let $T_0 \subset T$ be the common image of the factorizations $S_ n \to T$. Then $T_0$ is quasi-compact. Let $T' \subset T$ be a quasi-compact open containing $T_0$. Then $S_ n \to T$ factors through $T'$. If we can show that $T' = S$, then $T' = T = S$. Hence we may assume $T$ is quasi-compact.

Assume $T$ is quasi-compact. In this case $T \to S$ is separated and quasi-finite (Morphisms, Lemma 29.20.15). Using Zariski's Main Theorem (in the form of More on Morphisms, Lemma 37.43.3) we choose a factorization $T \to W \to S$ with $W \to S$ finite and $T \to W$ an open immersion. Write $W = \mathop{\mathrm{Spec}}(B)$. The (unique) factorizations $S_ n \to T$ may be viewed as morphisms into $W$ and we obtain

Consider the morphism $h : S = \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(B) = W$ coming from the arrow on the right. Then

is an open subscheme of $S$ containing the image of $S_ n \to S$ for all $n$. To finish the proof it suffices to show that any open $U \subset S$ containing the image of $S_ n \to S$ for some $n \geq 1$ is equal to $S$. This is true because $(A, \mathop{\mathrm{Ker}}(A \to A_ n))$ is a henselian pair (More on Algebra, Lemma 15.11.3) and hence every closed point of $S$ is contained in the image of $S_ n \to S$. $\square$

Proof. Apply Lemmas 38.28.2, 38.28.3, 38.28.4, 38.28.5, 38.28.6, and 38.28.7 to get an open subscheme $W \subset X$ 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 38.17.1 we see that $\mathcal{F}$ is universally pure relative to $\mathop{\mathrm{Spec}}(A)$. By Theorem 38.27.5 (for explanation, see Lemma 38.27.6) 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. 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 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. $\square$