The Stacks project

Proof. Let $(R_ n)$ be an inverse system of $S$-algebras with surjective transition maps whose kernels are locally nilpotent. Set $R = \mathop{\mathrm{lim}}\nolimits R_ n$. Let $(\xi _ n)$ be a system of objects of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ lying over $(\mathop{\mathrm{Spec}}(R_ n))$. We have to show $(\xi _ n)$ is effective, i.e., there exists an object $\xi $ of $\mathcal{C}\! \mathit{omplexes}_{X/B}$ lying over $\mathop{\mathrm{Spec}}(R)$.

Write $X_ R = \mathop{\mathrm{Spec}}(R) \times _ S X$ and $X_ n = \mathop{\mathrm{Spec}}(R_ n) \times _ S X$. Of course $X_ n$ is the base change of $X_ R$ by $R \to R_ n$. Since $S = B$, we see that $\xi _ n$ corresponds simply to an $R_ n$-perfect object $E_ n \in D(\mathcal{O}_{X_ n})$ satisfying condition (2) of Lemma 99.16.2. In particular $E_ n$ is pseudo-coherent. The isomorphisms $\xi _{n + 1}|_{\mathop{\mathrm{Spec}}(R_ n)} \cong \xi _ n$ correspond to isomorphisms $L(X_ n \to X_{n + 1})^*E_{n + 1} \to E_ n$. Therefore by Flatness on Spaces, Theorem 77.13.6 we find a pseudo-coherent object $E$ of $D(\mathcal{O}_{X_ R})$ with $E_ n$ equal to the derived pullback of $E$ for all $n$ compatible with the transition isomorphisms.

Observe that $(R, \mathop{\mathrm{Ker}}(R \to R_1))$ is a henselian pair, see More on Algebra, Lemma 15.11.3. In particular, $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical of $R$. Then we may apply More on Morphisms of Spaces, Lemma 76.54.5 to see that $E$ is $R$-perfect.

Finally, we have to check condition (2) of Lemma 99.16.2. By Lemma 99.16.1 the set of points $t$ of $\mathop{\mathrm{Spec}}(R)$ where the negative self-exts of $E_ t$ vanish is an open. Since this condition is true in $V(\mathop{\mathrm{Ker}}(R \to R_1))$ and since $\mathop{\mathrm{Ker}}(R \to R_1)$ is contained in the Jacobson radical of $R$ we conclude it holds for all points. $\square$