Lemma 15.13.4. Let A = \mathop{\mathrm{lim}}\nolimits A_ n be a limit of an inverse system (A_ n) of rings. Suppose given A_ n-modules M_ n and A_{n + 1}-module maps M_{n + 1} \to M_ n. Assume
the transition maps A_{n + 1} \to A_ n are surjective with locally nilpotent kernels,
M_1 is a finite projective A_1-module,
M_ n is a finite flat A_ n-module, and
the maps induce isomorphisms M_{n + 1} \otimes _{A_{n + 1}} A_ n \to M_ n.
Then M = \mathop{\mathrm{lim}}\nolimits M_ n is a finite projective A-module and M \otimes _ A A_ n \to M_ n is an isomorphism for all n.
Proof.
By Lemma 15.11.3 the pair (A, \mathop{\mathrm{Ker}}(A \to A_1)) is henselian. By Lemma 15.13.1 we can choose a finite projective A-module P and an isomorphism P \otimes _ A A_1 \to M_1. Since P is projective, we can successively lift the A-module map P \to M_1 to A-module maps P \to M_2, P \to M_3, and so on. Thus we obtain a map
P \longrightarrow M
Since P is finite projective, we can write A^{\oplus m} = P \oplus Q for some m \geq 0 and A-module Q. Since A = \mathop{\mathrm{lim}}\nolimits A_ n we conclude that P = \mathop{\mathrm{lim}}\nolimits P \otimes _ A A_ n. Hence, in order to show that the displayed A-module map is an isomorphism, it suffices to show that the maps P \otimes _ A A_ n \to M_ n are isomorphisms. From Lemma 15.3.4 we see that M_ n is a finite projective module. By Lemma 15.3.5 the maps P \otimes _ A A_ n \to M_ n are isomorphisms.
\square
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