The Stacks project

Lemma 15.90.1. Let $A = \mathop{\mathrm{lim}}\nolimits A_ n$ be a limit of an inverse system $(A_ n)$ of rings. Suppose given $K_ n \in D(A_ n)$ and maps $K_{n + 1} \to K_ n$ in $D(A_{n + 1})$. Assume

  1. the transition maps $A_{n + 1} \to A_ n$ are surjective with locally nilpotent kernels,

  2. $K_1$ is pseudo-coherent, and

  3. the maps induce isomorphisms $K_{n + 1} \otimes _{A_{n + 1}}^\mathbf {L} A_ n \to K_ n$.

Then $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is a pseudo-coherent object of $D(A)$ and $K \otimes _ A^\mathbf {L} A_ n \to K_ n$ is an isomorphism for all $n$.

Proof. By assumption we can find a bounded above complex of finite free $A_1$-modules $P_1^\bullet $ representing $K_1$, see Definition 15.62.1. By Lemma 15.71.4 we can, by induction on $n > 1$, find complexes $P_ n^\bullet $ of finite free $A_ n$-modules representing $K_ n$ and maps $P_ n^\bullet \to P_{n - 1}^\bullet $ representing the maps $K_ n \to K_{n - 1}$ inducing isomorphisms (!) of complexes $P_ n^\bullet \otimes _{A_ n} A_{n - 1} \to P_{n - 1}^\bullet $. Thus $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ is represented by $P^\bullet = \mathop{\mathrm{lim}}\nolimits P_ n^\bullet $, see Lemma 15.81.1 and Remark 15.81.6. Since $P_ n^ i$ is a finite free $A_ n$-module for each $n$ and $A = \mathop{\mathrm{lim}}\nolimits A_ n$ we see that $P^ i$ is finite free of the same rank as $P_1^ i$ for each $i$. This means that $K$ is pseudo-coherent. It also follows that $K \otimes _ A^\mathbf {L} A_ n$ is represented by $P^\bullet \otimes _ A A_ n = P_ n^\bullet $ which proves the final assertion. $\square$


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