[Lemma 4.2, Bhatt-Algebraize]
Lemma 15.99.3. 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
the transition maps $A_{n + 1} \to A_ n$ are surjective with locally nilpotent kernels,
either all $K_ n$ are perfect or there exists an integer $n_0$ such that $K_{n_0}$ is perfect and the kernels $A_{n + 1} \to A_ n$ are nilpotent for $n \geq n_0$, and
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 15.99.1. Consider a surjective map $A \to \kappa $ whose kernel is a maximal ideal $\mathfrak m$. 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
\[ K \otimes _ A^\mathbf {L} \kappa = K \otimes _ A^\mathbf {L} A_1 \otimes _{A_1}^\mathbf {L} \kappa = K_1 \otimes _{A_1}^\mathbf {L} \kappa \]
Note that $K_1$ is perfect by assumption (2). Hence $K_1$ has finite tor dimension by Lemma 15.76.2. Thus there exist $a, b \in \mathbf{Z}$ such that $H^ i(K \otimes _ A^\mathbf {L} \kappa ) = 0$ for all $i \not\in [a, b]$. By Lemma 15.79.2 we conclude that $K$ is perfect.
$\square$
Comments (3)
Comment #10539 by Stacks Project on
Comment #11380 by Adrien Morin on
Comment #11519 by Stacks Project on