The Stacks project

Lemma 10.138.14. Let $R \to S$ be a smooth ring map. Then there exists a subring $R_0 \subset R$ of finite type over $\mathbf{Z}$ and a smooth ring map $R_0 \to S_0$ such that $S \cong R \otimes _{R_0} S_0$.

Proof. We are going to use that smooth is equivalent to finite presentation and formally smooth, see Proposition 10.138.13. Write $S = R[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$ and denote $I = (f_1, \ldots , f_ m)$. Choose a right inverse $\sigma : S \to R[x_1, \ldots , x_ n]/I^2$ to the projection to $S$ as in Lemma 10.138.5. Choose $h_ i \in R[x_1, \ldots , x_ n]$ such that $\sigma (x_ i \bmod I) = h_ i \bmod I^2$. Since $x_ i - h_ i \in I$, there exist $b_{ij} \in R[x_1, \ldots , x_ n]$ such that

\[ x_ i - h_ i = \sum \nolimits _ j b_{ij} f_ j \]

The fact that $\sigma $ is an $R$-algebra homomorphism $R[x_1, \ldots , x_ n]/I \to R[x_1, \ldots , x_ n]/I^2$ is equivalent to the condition that

\[ f_ j(h_1, \ldots , h_ n) = \sum \nolimits _{j_1 j_2} a_{j_1 j_2} f_{j_1} f_{j_2} \]

for certain $a_{kl} \in R[x_1, \ldots , x_ n]$. Let $R_0 \subset R$ be the subring generated over $\mathbf{Z}$ by all the coefficients of the polynomials $f_ j, h_ i, a_{kl}, b_{ij}$. Set $S_0 = R_0[x_1, \ldots , x_ n]/(f_1, \ldots , f_ m)$, with $I_0 = (f_1, \ldots , f_ m)$. Since the second displayed equation holds in $R_0[x_1, \ldots , x_ n]$ we can let $\sigma _0 : S_0 \to R_0[x_1, \ldots , x_ n]/I_0^2$ be the $R_0$-algebra map defined by the rule $x_ i \mapsto h_ i \bmod I_0^2$. Since the first displayed equation holds in $R_0[x_1, \ldots , x_ n]$ we see that $\sigma _0$ is a right inverse to the projection $R_0[x_1, \ldots , x_ n] / I_0^2 \to R_0[x_1, \ldots , x_ n] / I_0 = S_0$. Thus by Lemma 10.138.5 the ring $S_0$ is formally smooth over $R_0$. $\square$


Comments (2)

Comment #9094 by Christian Merten on

It is not clear to me why the constructed is a right inverse to the projection : To prove this, we need to check that for each . Since is a right inverse, we have and by the construction of , . But is not necessarily equal to .

This can be fixed in the following way: Since , there exist such that If we adjoin the (finitely many) coefficients of the to , the above relation also holds in and we may conclude.

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  • 5 comment(s) on Section 10.138: Formally smooth maps

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