Lemma 88.5.5. 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. Assume given an integer d \geq 0 such that C[a^\infty ] \cap a^ dC = 0. Let n > \max (2c, c + d). 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} = \varphi \bmod a^{n - c}.
Proof. Let C \to C' be the quotient of C by C[a^\infty ]. The A-algebra C' is I-adically complete by Algebra, Lemma 10.96.10 and the fact that \bigcap (C[a^\infty ] + a^ nC) = C[a^\infty ] because for n \geq d the sum C[a^\infty ] + a^ nC is direct. For m \geq d the diagram
\xymatrix{ 0 \ar[r] & C[a^\infty ] \ar[r] \ar[d] & C \ar[r] \ar[d] & C' \ar[r] \ar[d] & 0 \\ 0 \ar[r] & C[a^\infty ] \ar[r] & C/a^ m C \ar[r] & C'/a^ m C' \ar[r] & 0 }
has exact rows. Thus C is the fibre product of C' and C/a^ mC over C'/a^ mC' for all m \geq d. By Lemma 88.5.4 we can choose a homomorphism \varphi ' : B \to C' such that \varphi ' and \psi _ n agree as maps into C'/a^{n - c}C'. We obtain a homomorphism (\varphi ', \psi _ n \bmod a^{n - c}C) : B \to C' \times _{C'/a^{n - c}C'} C/a^{n - c}C. Since n - c \geq d this is the same thing as a homomorphism \varphi : B \to C. This finishes the proof. \square
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