Base change of standard smooth presentations

Notation. Put $P:=R[x_1,\dots ,x_n]$, $I:=(f_1,\dots ,f_c)\subseteq P$, $A:=P\mathrm{/}I$, so that $S=A_g$; put $P^{\mathrm{\prime }}:=R^{\mathrm{\prime }}[x_1,\dots ,x_n]$, $I^{\mathrm{\prime }}:=(f_1^{\mathrm{\prime }},\dots ,f_c^{\mathrm{\prime }})\subseteq P^{\mathrm{\prime }}$ and $S^{\mathrm{\prime }}:=(P^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}{)}_{g^{\mathrm{\prime }}}$. All four are commutative rings by [F2] and [F8], and the coefficient-change map $P\to P^{\mathrm{\prime }}$, $x_i\mapsto x_i$, is a ring homomorphism by [F3] applied to $R\to R^{\mathrm{\prime }}$.

The map $\psi \text{\hspace{0.17em}⁣}:S\to S^{\mathrm{\prime }}$. The composite $P\to P^{\mathrm{\prime }}\to P^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}\to S^{\mathrm{\prime }}$ kills $I$ and carries $g$ to $g^{\mathrm{\prime }}$, which is a unit of $S^{\mathrm{\prime }}$ by [F8]; by [F5] it factors uniquely through $P\mathrm{/}I=A$, and by [F4] the resulting map $A\to S^{\mathrm{\prime }}$ factors uniquely through $A_g=S$. This gives a unique ring homomorphism $\psi \text{\hspace{0.17em}⁣}:S\to S^{\mathrm{\prime }}$ with $\psi (\bar{F}\mathrm{/}{g}^N)=\overline{F^{\mathrm{\prime }}}\mathrm{/}(g^{\mathrm{\prime }}{)}^N$ for $F\in P$, $N\ge 0$.

The map $\tau \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}\to R^{\mathrm{\prime }}{\otimes }_{R}S$. By [F7] the assignment $a\mapsto a\otimes 1$ is a ring homomorphism $R^{\mathrm{\prime }}\to R^{\mathrm{\prime }}{\otimes }_{R}S$, and $1\otimes \overline{x_i}\in R^{\mathrm{\prime }}{\otimes }_{R}S$ are elements; by [F3] there is a unique ring homomorphism $P^{\mathrm{\prime }}=R^{\mathrm{\prime }}[x_1,\dots ,x_n]\to R^{\mathrm{\prime }}{\otimes }_{R}S$ restricting to $a\mapsto a\otimes 1$ and sending $x_i\mapsto 1\otimes \overline{x_i}$. Its kernel contains $I^{\mathrm{\prime }}$, because $f_j^{\mathrm{\prime }}\mapsto 1\otimes \overline{f_j}=0$, and it sends $g^{\mathrm{\prime }}$ to $1\otimes \bar{g}$, which is a unit with inverse $1\otimes {\bar{g}}^{-1}$ in view of [F7] and $g\cdot g^{-1}$ invertible in $S$. Applying [F5] and then [F4] gives a unique ring homomorphism $\tau \text{\hspace{0.17em}⁣}:S^{\mathrm{\prime }}=(P^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}{)}_{g^{\mathrm{\prime }}}\to R^{\mathrm{\prime }}{\otimes }_{R}S$ over $R^{\mathrm{\prime }}$.

The map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:R^{\mathrm{\prime }}{\otimes }_{R}S\to S^{\mathrm{\prime }}$. The identity map of $R^{\mathrm{\prime }}$ and the map $\psi \text{\hspace{0.17em}⁣}:S\to S^{\mathrm{\prime }}$ of step 2.1 are $R$-algebra maps into $S^{\mathrm{\prime }}$ that agree on $R$; the latter is induced by the coefficient-change map $R\to R^{\mathrm{\prime }}$. Hence [F6] provides a unique $R^{\mathrm{\prime }}$-algebra homomorphism $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:R^{\mathrm{\prime }}{\otimes }_{R}S\to S^{\mathrm{\prime }}$ with $\mathrm{\Phi }(a\otimes 1)=a$ and $\mathrm{\Phi }(1\otimes y)=\psi (y)$, that is, $\mathrm{\Phi }(a\otimes y)=a\psi (y)$; on the elements $a\otimes \bar{F}\mathrm{/}{g}^N$ it is $\mathrm{\Phi }(a\otimes \bar{F}\mathrm{/}{g}^N)=a\overline{F^{\mathrm{\prime }}}\mathrm{/}(g^{\mathrm{\prime }}{)}^N$.

$\mathrm{\Phi }\circ \tau ={\mathrm{i}\mathrm{d}}_{S^{\mathrm{\prime }}}$ and $\tau \circ \mathrm{\Phi }={\mathrm{i}\mathrm{d}}_{R^{\mathrm{\prime }}{\otimes }_{R}S}$. Both composites are $R^{\mathrm{\prime }}$-algebra homomorphisms. The $R^{\mathrm{\prime }}$-algebra $S^{\mathrm{\prime }}$ is generated by the images of the $x_i$ and by $(g^{\mathrm{\prime }}{)}^{-1}$: every element of $(P^{\mathrm{\prime }}\mathrm{/}{I}^{\mathrm{\prime }}{)}_{g^{\mathrm{\prime }}}$ is a class $u\mathrm{/}(g^{\mathrm{\prime }}{)}^N$ with $u\in P^{\mathrm{\prime }}$, and $u$ is an $R^{\mathrm{\prime }}$-linear combination of monomials in the $x_i$. A ring homomorphism out of $S^{\mathrm{\prime }}$ is determined by its restriction to $R^{\mathrm{\prime }}$ and the images of the $x_i$, by [F3], [F5] and [F4] applied in that order, so $(\mathrm{\Phi }\circ \tau )(x_i^{\mathrm{\prime }})=x_i^{\mathrm{\prime }}$ and $(\mathrm{\Phi }\circ \tau )((g^{\mathrm{\prime }}{)}^{-1})=(g^{\mathrm{\prime }}{)}^{-1}$ force $\mathrm{\Phi }\circ \tau ={\mathrm{i}\mathrm{d}}_{S^{\mathrm{\prime }}}$. Similarly $R^{\mathrm{\prime }}{\otimes }_{R}S$ is generated as an $R^{\mathrm{\prime }}$-algebra by $1\otimes \overline{x_i}$ and $1\otimes {\bar{g}}^{-1}$, by [F7], and $\tau \circ \mathrm{\Phi }$ fixes these elements: $\tau (\mathrm{\Phi }(1\otimes \overline{x_i}))=\tau (x_i^{\mathrm{\prime }})=1\otimes \overline{x_i}$ and $\tau (\mathrm{\Phi }(1\otimes {\bar{g}}^{-1}))=\tau ((g^{\mathrm{\prime }}{)}^{-1})=1\otimes {\bar{g}}^{-1}$, the last because $\tau$ is a ring homomorphism sending $g^{\mathrm{\prime }}$ to $1\otimes \bar{g}$. Hence $\tau \circ \mathrm{\Phi }=\mathrm{i}\mathrm{d}$ as well, and $\mathrm{\Phi }$ is an isomorphism with inverse $\tau$.

The minor maps to a unit of $S^{\mathrm{\prime }}$. The element $h\in P$ maps to a unit of $S$ by hypothesis, hence $1\otimes h\in R^{\mathrm{\prime }}{\otimes }_{R}S$ is a unit with inverse $1\otimes h^{-1}$ by [F7], and $\mathrm{\Phi }$ carries it to $(1\otimes h)$'s image, namely $h^{\mathrm{\prime }}=\mathrm{\Phi }(1\otimes h)$; as a ring isomorphism $\mathrm{\Phi }$ carries units to units, so $h^{\mathrm{\prime }}$ is a unit of $S^{\mathrm{\prime }}$.

The Jacobian of the changed polynomials. By the monomial formula of [F9] the partial derivative ${\mathrm{\partial }}_i$ is linear over the coefficient ring, so ${\mathrm{\partial }}_i(f_j^{\mathrm{\prime }})$ is the image of ${\mathrm{\partial }}_i(f_j)$ under $P\to P^{\mathrm{\prime }}$ for all $i,j$; hence $h^{\mathrm{\prime }}=\det ({\mathrm{\partial }}_i{f}_j^{\mathrm{\prime }}{)}_{1\le i,j\le c}$ is the leading $c\times c$ minor of the Jacobian matrix of $f_1^{\mathrm{\prime }},\dots ,f_c^{\mathrm{\prime }}$. By step 5.1 its image in $S^{\mathrm{\prime }}$ is a unit, so $S^{\mathrm{\prime }}\cong (R^{\mathrm{\prime }}[x_1,\dots ,x_n]\mathrm{/}(f_1^{\mathrm{\prime }},\dots ,f_c^{\mathrm{\prime }}){)}_{g^{\mathrm{\prime }}}$ is a standard smooth presentation over $R^{\mathrm{\prime }}$ of relative dimension $n-c$, the same parameters as the given presentation. With step 4.1 this proves both assertions.