The Stacks project

Situation 76.12.1. 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 $X$ separated 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 $(\mathcal{F}_ n, \varphi _ n)$ where $\mathcal{F}_ n$ is a finitely presented $\mathcal{O}_{X_ n}$-module, flat over $A_ n$, with support proper over $A_ n$, and

\[ \varphi _ n : \mathcal{F}_ n \otimes _{\mathcal{O}_{X_ n}} \mathcal{O}_{X_{n - 1}} \longrightarrow \mathcal{F}_{n - 1} \]

is an isomorphism (notation using the equivalence of Morphisms of Spaces, Lemma 66.14.1).


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