The Stacks project

Proof. The functor $DQ_ X$ exists because $X$ is quasi-compact and quasi-separated, see Derived Categories of Spaces, Lemma 75.19.1. Since $DQ_ X$ is a right adjoint it commutes with products and therefore with derived limits. Hence the equality in the statement of the lemma.

By Derived Categories of Spaces, Lemma 75.19.4 the functor $DQ_ X$ has bounded cohomological dimension. Hence it suffices to show that $R\mathop{\mathrm{lim}}\nolimits K_ n \in D^-(\mathcal{O}_ X)$. To see this, let $U \to X$ be étale with $U$ affine. Then there is a canonical exact sequence

by Cohomology on Sites, Lemma 21.23.2. Since $U$ is affine and $K_ n$ is pseudo-coherent (and hence has quasi-coherent cohomology sheaves by Derived Categories of Spaces, Lemma 75.13.6) we see that $H^ m(U, K_ n) = H^ m(K_ n)(U)$ by Derived Categories of Schemes, Lemma 36.3.5. Thus we conclude that it suffices to show that $K_ n$ is bounded above independent of $n$.

Since $K_ n$ is pseudo-coherent we have $K_ n \in D^-(\mathcal{O}_{X_ n})$. Suppose that $a_ n$ is maximal such that $H^{a_ n}(K_ n)$ is nonzero. Of course $a_1 \leq a_2 \leq a_3 \leq \ldots $. Note that $H^{a_ n}(K_ n)$ is an $\mathcal{O}_{X_ n}$-module of finite presentation (Cohomology on Sites, Lemma 21.45.7). We have $H^{a_ n}(K_{n - 1}) = H^{a_ n}(K_ n) \otimes _{\mathcal{O}_{X_ n}} \mathcal{O}_{X_{n - 1}}$. Since $X_{n - 1} \to X_ n$ is a thickening, it follows from Nakayama's lemma (Algebra, Lemma 10.20.1) that if $H^{a_ n}(K_ n) \otimes _{\mathcal{O}_{X_ n}} \mathcal{O}_{X_{n - 1}}$ is zero, then $H^{a_ n}(K_ n)$ is zero too (argue by checking on stalks for example; small detail omitted). Thus $a_{n - 1} = a_ n$ for all $n$ and we conclude. $\square$

Proof. Write $E_ n = E|_{X_ n}$ and $M_ n = R\Gamma (X_ n, K_ n \otimes ^\mathbf {L} E|_{X_ n})$. By Derived Categories of Spaces, Lemma 75.25.5 we see that $M_ n$ is a pseudo-coherent object of $D(A_ n)$ whose formation commutes with base change. Thus the maps $M_ n \otimes _{A_ n}^\mathbf {L} A_{n - 1} \to M_{n - 1}$ coming from $\varphi _ n$ are isomorphisms. By More on Algebra, Lemma 15.97.1 we find that $R\mathop{\mathrm{lim}}\nolimits M_ n$ is pseudo-coherent and that its base change back to $A_ n$ recovers $M_ n$. On the other hand, the exact functor $R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence

as desired. $\square$

Proof. Combinging Lemma 77.13.3 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 it follows from More on Morphisms of Spaces, Lemma 76.51.4 that $K$ is pseudo-coherent relative to $A$. Since $X$ is of flat and of finite presentation over $A$, this is the same as being pseudo-coherent on $X$, see More on Morphisms of Spaces, Lemma 76.45.4. $\square$

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 $K_ n$ we obtain

Using Lemma 77.13.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

Proof. Apply Lemmas 77.13.2, 77.13.3, 77.13.4 to get a pseudo-coherent object $K$ of $D(\mathcal{O}_ X)$. Choosing affine $U$ in Lemma 77.13.5 it follows immediately that $K$ restricts to $K_ n$ over $X_ n$. $\square$