Standard smooth presentations and locally standard smooth maps

Let $R$ be a commutative ring. A standard smooth presentation of an $R$-algebra $S$ consists of integers $n\ge c\ge 0$, elements $f_1,\dots ,f_c$ of the polynomial ring $P=R[x_1,\dots ,x_n]$ (The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials) and an element $g\in P$ such that $S\cong (R[x_1,\dots ,x_n]\mathrm{/}(f_1,\dots ,f_c){)}_g$ as $R$-algebras, and such that the Jacobian matrix of (Differentials of a polynomial quotient and the Jacobian cokernel) $(\frac{\mathrm{\partial }{f}_j}{\mathrm{\partial }{x}_i}{)}_{\begin{array}{c}1\le j\le c\\ 1\le i\le n\end{array}}$ has a $c\times c$ minor whose image in $S$ is a unit, that is, an invertible element. The integer $n-c\ge 0$ is the relative dimension of the presentation. The case $c=0$ is allowed and is exactly a localisation of a polynomial ring, $S\cong (R[x_1,\dots ,x_n]{)}_g$; the case $n=c=0$ presents $S=R[{]}_g$ for $g\in R$, that is a localisation of $R$ itself.

For a homomorphism $R\to S$ of commutative rings that is finitely presented as an $R$-algebra (Finitely presented modules and finitely presented algebras) and a prime $\mathfrak{q}\in \mathrm{Spec}S$, the map is standard smooth at $\mathfrak{q}$, or has a standard smooth presentation at $\mathfrak{q}$, when there is $h\in S\setminus \mathfrak{q}$ such that $S_h$ admits a standard smooth presentation over $R$. It is locally standard smooth when this holds at every prime of $S$; equivalently, when every point of $\mathrm{Spec}S$ has an affine open neighbourhood on which $S$ is presented by a single standard smooth presentation.

Two conventions are part of the definition. First, the invertible minor may be assumed to sit in the first $c$ columns: if a minor on columns $i_1<\cdots <i_c$ is a unit, the automorphism of $R[x_1,\dots ,x_n]$ permuting the variables so that these become the first $c$ variables carries the presentation to one whose minor in the first $c$ columns is that unit, and the relative dimension $n-c$ is unchanged. Second, a further principal localisation can be absorbed into the presentation: adjoining a variable $z$ with the single equation $zg-1$ to a presentation produces a standard smooth presentation of the localisation, the new equation contributing a diagonal entry that keeps a block minor invertible, and it changes $n$ and $c$ by the same amount, so that the relative dimension is again unchanged.