The Stacks project

Situation 75.13.1. Here we have an inverse system of rings $(A_ n)$ with surjective transition maps whose kernels are locally nilpotent. Set $A = \mathop{\mathrm{lim}}\nolimits A_ n$. We have an algebraic space $X$ proper, flat, and of finite presentation over $A$. We set $X_ n = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_ n)$ and we view it as a closed subspace of $X$. We assume further given a system $(K_ n, \varphi _ n)$ where $K_ n$ is a pseudo-coherent object of $D(\mathcal{O}_{X_ n})$ and

\[ \varphi _ n : K_ n \longrightarrow K_{n - 1} \]

is a map in $D(\mathcal{O}_{X_ n})$ which induces an isomorphism $K_ n \otimes _{\mathcal{O}_{X_ n}}^\mathbf {L} \mathcal{O}_{X_{n - 1}} \to K_{n - 1}$ in $D(\mathcal{O}_{X_{n - 1}})$.


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