Lemma 10.137.20. Let $R$ be a ring and let $I \subset R$ be an ideal. Let $R/I \to \overline{S}$ be a smooth ring map. Then there exists elements $\overline{g}_ i \in \overline{S}$ which generate the unit ideal of $\overline{S}$ such that each $\overline{S}_{g_ i} \cong S_ i/IS_ i$ for some (standard) smooth ring $S_ i$ over $R$.

Proof. By Lemma 10.137.10 we find a collection of elements $\overline{g}_ i \in \overline{S}$ which generate the unit ideal of $\overline{S}$ such that each $\overline{S}_{g_ i}$ is standard smooth over $R/I$. Hence we may assume that $\overline{S}$ is standard smooth over $R/I$. Write $\overline{S} = (R/I)[x_1, \ldots , x_ n]/(\overline{f}_1, \ldots , \overline{f}_ c)$ as in Definition 10.137.6. Choose $f_1, \ldots , f_ c \in R[x_1, \ldots , x_ n]$ lifting $\overline{f}_1, \ldots , \overline{f}_ c$. Set $S = R[x_1, \ldots , x_ n, x_{n + 1}]/(f_1, \ldots , f_ c, x_{n + 1}\Delta - 1)$ where $\Delta = \det (\frac{\partial f_ j}{\partial x_ i})_{i, j = 1, \ldots , c}$ as in Example 10.137.8. This proves the lemma. $\square$

Comment #773 by Keenan Kidwell on

This doesn't actually make any difference, but to be consistent with 00T8 and the definition of standard smooth in terms of Jacobian determinants, the matrix of which $\Delta$ is the determinant should be $(\partial f_j/\partial x_i)_{i,j=1,\ldots,c}$.

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