Lemma 114.6.1. 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 $R$.

## 114.6 Formally smooth ring maps

**Proof.**
See Algebra, Proposition 10.138.13.
$\square$

Remark 114.6.2. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 87.6.3 which became unused because of a rearrangement of the material.

Remark 114.6.3. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 87.6.3 which became unused because of a rearrangement of the material.

Remark 114.6.4. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 87.6.3 which became unused because of a rearrangement of the material.

Remark 114.6.5. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Proposition 87.6.3 which became unused because of a rearrangement of the material.

Remark 114.6.6. This tag used to refer to an equation in the proof of Algebraization of Formal Spaces, Lemma 87.9.1 which became unused because of a rearrangement of the material.

Lemma 114.6.7. 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 (87.2.0.2) 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 87.5.3. 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

Namely, suppose that $c$, $B$, $n$, $\psi _ n$ are as in the statement above. Then we see that

On the other hand, we have

Hence the assumptions of Algebraization of Formal Spaces, Lemma 87.5.3 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

we see that we have the congruence of $\varphi $ and $\psi _ n$ module $I^{m - c}C$ as desired. $\square$

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