Lemma 88.5.4. Let I = (a) be a principal ideal of a Noetherian ring A. Let B be an object of (88.2.0.2). Assume given an integer c \geq 0 such that \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}^\wedge , N) is annihilated by a^ c for all B-modules N. Let C be an I-adically complete A-algebra such that a is a nonzerodivisor on C. Let n > 2c. For any A-algebra map \psi _ n : B \to C/a^ nC there exists an A-algebra map \varphi : B \to C such that \psi _ n \bmod a^{n - c}C = \varphi \bmod a^{n - c}C.
Proof. Consider the obstruction class
o(\psi _ n) \in \mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}^\wedge , a^ nC/a^{2n}C)
of Remark 88.5.2. Since a is a nonzerodivisor on C the map a^ c : a^ nC/a^{2n}C \to a^ nC/a^{2n}C is isomorphic to the map a^ nC/a^{2n}C \to a^{n - c}C/a^{2n - c}C in the category of C-modules. Hence by our assumption on \mathop{N\! L}\nolimits _{B/A}^\wedge we conclude that the class o(\psi _ n) maps to zero in
\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}^\wedge , a^{n - c}C/a^{2n - c}C)
and a fortiori in
\mathop{\mathrm{Ext}}\nolimits ^1_ B(\mathop{N\! L}\nolimits _{B/A}^\wedge , a^{n - c}C/a^{2n - 2c}C)
By the discussion in Remark 88.5.2 we obtain a map
\psi _{2n - 2c} : B \to C/a^{2n - 2c}C
which agrees with \psi _ n modulo a^{n - c}C. Observe that 2n - 2c > n because n > 2c.
We may repeat this procedure. Starting with n_0 = n and \psi ^0 = \psi _ n we end up getting a strictly increasing sequence of integers
n_0 < n_1 < n_2 < \ldots
and A-algebra homorphisms \psi ^ i : B \to C/a^{n_ i}C such that \psi ^{i + 1} and \psi ^ i agree modulo a^{n_ i - c}C. Since C is I-adically complete we can take \varphi to be the limit of the maps \psi ^ i \bmod a^{n_ i - c}C : B \to C/a^{n_ i - c}C and the lemma follows. \square
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