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

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 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

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

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)