The Stacks project

[Lemma 4.2, Bhatt-Algebraize]

Lemma 15.95.3. 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 a perfect object, 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 perfect object of $D(A)$ and $K \otimes _ A^\mathbf {L} A_ n \to K_ n$ is an isomorphism for all $n$.

Proof. We already know that $K$ is pseudo-coherent and that $K \otimes _ A^\mathbf {L} A_ n \to K_ n$ is an isomorphism for all $n$ by Lemma 15.95.1. Thus it suffices to show that $H^ i(K \otimes _ A^\mathbf {L} \kappa ) = 0$ for $i \ll 0$ and every surjective map $A \to \kappa $ whose kernel is a maximal ideal $\mathfrak m$, see Lemma 15.76.3. Any element of $A$ which maps to a unit in $A_1$ is a unit in $A$ by Algebra, Lemma 10.32.4 and hence $\mathop{\mathrm{Ker}}(A \to A_1)$ is contained in the Jacobson radical of $A$ by Algebra, Lemma 10.19.1. Hence $A \to \kappa $ factors as $A \to A_1 \to \kappa $. Hence

\[ K \otimes _ A^\mathbf {L} \kappa = K \otimes _ A^\mathbf {L} A_1 \otimes _{A_1}^\mathbf {L} \kappa = K_1 \otimes _{A_1}^\mathbf {L} \kappa \]

and we get what we want as $K_1$ has finite tor dimension by Lemma 15.73.2. $\square$


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