Lemma 115.6.7. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $t$ be the minimal number of generators for $I$. Let $C$ be a Noetherian $I$-adically complete $A$-algebra. There exists an integer $d \geq 0$ depending only on $I \subset A \to C$ with the following property: given
$c \geq 0$ and $B$ in Algebraization of Formal Spaces, Equation (88.2.0.2) such that for $a \in I^ c$ multiplication by $a$ on $\mathop{N\! L}\nolimits _{B/A}^\wedge $ is zero in $D(B)$,
an integer $n > 2t\max (c, d)$,
an $A/I^ n$-algebra map $\psi _ n : B/I^ nB \to C/I^ nC$,
there exists a map $\varphi : B \to C$ of $A$-algebras such that $\psi _ n \bmod I^{m - c} = \varphi \bmod I^{m - c}$ with $m = \lfloor \frac{n}{t} \rfloor $.
Proof.
This lemma has been obsoleted by the stronger Algebraization of Formal Spaces, Lemma 88.5.3. In fact, we will deduce the lemma from it.
Let $I \subset A \to C$ be given as in the statement above. Denote $d(\text{Gr}_ I(C))$ and $q(\text{Gr}_ I(C))$ the integers found in Local Cohomology, Section 51.22. Observe that $t$ is an upper bound for the minimal number of generators of $IC$ and hence we have $d(\text{Gr}_ I(C)) + 1 \leq t$, see discussion in Local Cohomology, Section 51.22. We may and do assume $t \geq 1$ since otherwise the lemma does not say anything. We claim that the lemma is true with
\[ d = q(\text{Gr}_ I(C)) \]
Namely, suppose that $c$, $B$, $n$, $\psi _ n$ are as in the statement above. Then we see that
\[ n > 2t\max (c, d) \Rightarrow n \geq 2tc + 1 \Rightarrow n \geq 2(d(\text{Gr}_ I(C)) + 1)c + 1 \]
On the other hand, we have
\[ n > 2t\max (c, d) \Rightarrow n > t(c + d) \Rightarrow n \geq q(C) + tc \geq q(\text{Gr}_ I(C)) + (d(\text{Gr}_ I(C)) + 1)c \]
Hence the assumptions of Algebraization of Formal Spaces, Lemma 88.5.3 are satisfied and we obtain an $A$-algebra homomorphism $\varphi : B \to C$ which is congruent with $\psi _ n$ module $I^{n - (d(\text{Gr}_ I(C)) + 1)c}C$. Since
\begin{align*} n - (d(\text{Gr}_ I(C)) + 1)c & = \frac{n}{t} + \frac{(t - 1)n}{t} - (d(\text{Gr}_ I(C)) + 1)c \\ & \geq \frac{n}{t} + \frac{(d(\text{Gr}_ I(C))n}{t} - (d(\text{Gr}_ I(C)) + 1)c \\ & > \frac{n}{t} + \frac{d(\text{Gr}_ I(C))2tc}{t} - (d(\text{Gr}_ I(C)) + 1)c \\ & = \frac{n}{t} + 2d(\text{Gr}_ I(C))c - (d(\text{Gr}_ I(C)) + 1)c \\ & = \frac{n}{t} + d(\text{Gr}_ I(C))c - c \\ & \geq m - c \end{align*}
we see that we have the congruence of $\varphi $ and $\psi _ n$ module $I^{m - c}C$ as desired.
$\square$
Comments (2)
Comment #5883 by Johan on
Comment #5921 by Johan on