The Stacks project

114.6 Formally smooth ring maps

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 $R$.

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

  1. $c \geq 0$ 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 $D(B)$,

  2. an integer $n > 2t\max (c, d)$,

  3. 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

\[ d = q(\text{Gr}_ I(C)) \]

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 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

\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 (0)


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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07GD. Beware of the difference between the letter 'O' and the digit '0'.