The Stacks project

Lemma 15.97.1. 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 15.64.1. By Lemma 15.75.5 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 15.87.1 and Remark 15.87.6. 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. $\square$

Comments (1)

Comment #10030 by Adrien Morin on

It seems to me that there is a missing assumption needed in order to apply Lemma 0BCB. Indeed, lemma 0BCB requires that the complex upstairs already be pseudocoherent. This is automatic if the kernel of is not just locally nilpotent but nilpotent, by lemma 0H76. Thus a correct statement would either assume that each is pseudocoherent, or that is surjective with nilpotent kernel for all (or a mix of the two, meaning that if is surjective with nilpotent kernel for , then requiring that be pseudocoherent).

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