Lemma 115.6.1. Let R be a ring. Let S be a R-algebra. If S is of finite presentation and formally smooth over R then S is smooth over R.
115.6 Formally smooth ring maps
Proof. See Algebra, Proposition 10.138.13. \square
Remark 115.6.2. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 88.6.3 which became unused because of a rearrangement of the material.
Remark 115.6.3. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 88.6.3 which became unused because of a rearrangement of the material.
Remark 115.6.4. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 88.6.3 which became unused because of a rearrangement of the material.
Remark 115.6.5. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 88.6.3 which became unused because of a rearrangement of the material.
Remark 115.6.6. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Lemma 88.9.1 which became unused because of a rearrangement of the material.
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
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