Base change and composition of standard smooth presentations

Notation. Fix a standard smooth presentation $S\cong (R[x_1,\dots ,x_n]\mathrm{/}(f_1,\dots ,f_c){)}_g$ with leading $c\times c$ minor $h$ a unit of $S$ [F1], put $I:=(f_1,\dots ,f_c)\subseteq R[x_1,\dots ,x_n]$ and $A:=R[x_1,\dots ,x_n]\mathrm{/}I$, so that $S=A_g$. Fix also a standard smooth $S$-presentation $T\cong (S[y_1,\dots ,y_m]\mathrm{/}(f_1^{\mathrm{\prime }\mathrm{\prime }},\dots ,f_d^{\mathrm{\prime }\mathrm{\prime }}){)}_{g^{\mathrm{\prime }\mathrm{\prime }}}$ with leading $d\times d$ minor $h^{\mathrm{\prime }\mathrm{\prime }}$ a unit of $T$ [F1]. Finally fix a ring map $R\to R^{\mathrm{\prime }}$.

Base change of presentations. By [F2] applied to the presentation of step 1.1 and the ring map $R\to R^{\mathrm{\prime }}$ there is an $R^{\mathrm{\prime }}$-algebra isomorphism $R^{\mathrm{\prime }}{\otimes }_{R}S\cong (R^{\mathrm{\prime }}[x_1,\dots ,x_n]\mathrm{/}(f_1^{\mathrm{\prime }},\dots ,f_c^{\mathrm{\prime }}){)}_{g^{\mathrm{\prime }}}$, where $f_j^{\mathrm{\prime }},g^{\mathrm{\prime }}$ are the images of $f_j,g$; the target is a standard smooth $R^{\mathrm{\prime }}$-presentation with the same $n,c$ and relative dimension $n-c$, and the image $h^{\mathrm{\prime }}$ of $h$ is a unit. This is the first assertion of clause 1.

The polynomial presentation of $S[y_1,\dots ,y_m]$. The coefficient extension $A[y_1,\dots ,y_m]\cong R[x_1,\dots ,x_n,y_1,\dots ,y_m]\mathrm{/}IR[x_1,\dots ,x_n,y_1,\dots ,y_m]$ holds by [F5], since $A=R[x]\mathrm{/}I$ and $R[x_1,\dots ,x_n][y_1,\dots ,y_m]=R[x_1,\dots ,x_n,y_1,\dots ,y_m]$ by [F4]; combining it with $A_g[y_1,\dots ,y_m]\cong (A[y_1,\dots ,y_m]{)}_g$ from [F4] and with $S=A_g$ gives $S[y_1,\dots ,y_m]\cong (R[x_1,\dots ,x_n,y_1,\dots ,y_m]\mathrm{/}(f_1,\dots ,f_c){)}_g\mathrm{.}$ We use this isomorphism to read the presentation of $T$ in the polynomial ring over $R$.

Base change at a prime. Finite presentation is preserved by base change: tensoring $R[z_1,\dots ,z_a]\mathrm{/}(r_1,\dots ,r_b)$ with $R^{\mathrm{\prime }}$ gives $R^{\mathrm{\prime }}[z_1,\dots ,z_a]\mathrm{/}(r_1^{\mathrm{\prime }},\dots ,r_b^{\mathrm{\prime }})$ by [F4, F5], where the primes denote coefficient images. Suppose $R\to S$ is standard smooth at $\mathfrak{q}\in \mathrm{Spec}S$, witnessed by an element $u\mathrm{\notin }\mathfrak{q}$ with $S_u$ standard smooth over $R$ [F1]; by [F2] the base change $R^{\mathrm{\prime }}{\otimes }_R{S}_u$ is standard smooth over $R^{\mathrm{\prime }}$. Let $Q\subseteq R^{\mathrm{\prime }}{\otimes }_{R}S$ be a prime with $Q\cap S=\mathfrak{q}$, i.e. lying over $\mathfrak{q}$; then $u\mathrm{\notin }Q$, since $u\in Q$ would give $u\in Q\cap S=\mathfrak{q}$. Hence $Q$ lies in the principal open $D(u)$ of $\mathrm{Spec}(R^{\mathrm{\prime }}{\otimes }_{R}S)$, and the localisation $(R^{\mathrm{\prime }}{\otimes }_{R}S{)}_u$ — which is $R^{\mathrm{\prime }}{\otimes }_R{S}_u$ by [F4] — is standard smooth over $R^{\mathrm{\prime }}$ [F6]. Therefore $R^{\mathrm{\prime }}\to R^{\mathrm{\prime }}{\otimes }_{R}S$ is standard smooth at $Q$; as $Q$ was an arbitrary prime over $\mathfrak{q}$, this gives the pointwise form of clause 1, and taking the witnessing chart at every prime of $S$ gives stability of local standard smoothness under base change.

Clearing denominators and the composite presentation. By step 2.2 the $S$-presentation of $T$ is a presentation in the ring $(R[x,y]\mathrm{/}(f_1,\dots ,f_c){)}_g$, with $y=(y_1,\dots ,y_m)$; write $f_k^{\mathrm{\prime }\mathrm{\prime }}={\sum }_{\alpha }\overline{a_{k\alpha }}{y}^{\alpha }$ and $g^{\mathrm{\prime }\mathrm{\prime }}={\sum }_{\beta }\overline{c_{\beta }}{y}^{\beta }$ with coefficients in $S=A_g$, and choose representatives $a_{k\alpha }=g^{-N_{k\alpha }}{b}_{k\alpha }$ and $c_{\beta }=g^{-M_{\beta }}{d}_{\beta }$ with $b_{k\alpha },d_{\beta }\in R[x_1,\dots ,x_n]$ and $N_{k\alpha },M_{\beta }\ge 0$. Put $N:=\max (\{0\}\cup \{N_{k\alpha }\})$, $M:=\max (\{0\}\cup \{M_{\beta }\})$ (so empty families give $0$) and define $F_k:={\sum }_{\alpha }{b}_{k\alpha }{g}^{N-N_{k\alpha }}{y}^{\alpha }\in R[x,y],H:={\sum }_{\beta }{d}_{\beta }{g}^{M-M_{\beta }}{y}^{\beta }\in R[x,y]\mathrm{.}$ Multiplying the displayed identities by $g^N$ and $g^M$ shows $F_k=g^N{f}_k^{\mathrm{\prime }\mathrm{\prime }}$ and $H=g^M{g}^{\mathrm{\prime }\mathrm{\prime }}$ in $S[y]$. Since $g$ is a unit of $S[y{]}_{gH}$, the ideals $(F_1,\dots ,F_d)$ and $(f_1^{\mathrm{\prime }\mathrm{\prime }},\dots ,f_d^{\mathrm{\prime }\mathrm{\prime }})$ coincide there, and $H$ is a unit multiple of $g^{\mathrm{\prime }\mathrm{\prime }}$; by [F7] localising at $g$ and then at $g^{\mathrm{\prime }\mathrm{\prime }}$ is localising at $gH$. Hence $T=(S[y]\mathrm{/}(f_1^{\mathrm{\prime }\mathrm{\prime }},\dots ,f_d^{\mathrm{\prime }\mathrm{\prime }}){)}_{g^{\mathrm{\prime }\mathrm{\prime }}}\cong (R[x_1,\dots ,x_n,y_1,\dots ,y_m]\mathrm{/}(f_1,\dots ,f_c,F_1,\dots ,F_d){)}_{gH},$ the composite presentation of $T$ over $R$.

The Jacobian minor of the composite. In the ring $S[y]$, the sum formula and coefficient linearity of [F3] give $\mathrm{\partial }{F}_k\mathrm{/}\mathrm{\partial }{y}_l={\sum }_{\alpha }{b}_{k\alpha }{g}^{N-N_{k\alpha }}\mathrm{\partial }(y^{\alpha })\mathrm{/}\mathrm{\partial }{y}_l=g^N\mathrm{\partial }{f}_k^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}\mathrm{\partial }{y}_l$, because the coefficients $g^{N-N_{k\alpha }}{b}_{k\alpha }$ represent $g^N{g}^{-N_{k\alpha }}{b}_{k\alpha }=g^N\overline{a_{k\alpha }}$ and $\mathrm{\partial }\mathrm{/}\mathrm{\partial }{y}_l$ is $S$-linear on $S[y]$. Hence the leading $d\times d$ block $(\mathrm{\partial }{F}_k\mathrm{/}\mathrm{\partial }{y}_l{)}_{1\le k,l\le d}$ has determinant $g^{Nd}{h}^{\mathrm{\prime }\mathrm{\prime }}$, a unit of $T$ because $g$ and $h^{\mathrm{\prime }\mathrm{\prime }}$ are units. Moreover $\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{y}_l=0$ for all $j,l$, since $f_j\in R[x_1,\dots ,x_n]$ does not involve the $y$'s [F3]. Therefore the $(c+d)\times (c+d)$ minor of the Jacobian matrix of $(f_1,\dots ,f_c,F_1,\dots ,F_d)$ on the columns $x_1,\dots ,x_c$ and $y_1,\dots ,y_d$ is block triangular with diagonal blocks $(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{x}_i{)}_{1\le i,j\le c}$ and $(\mathrm{\partial }{F}_k\mathrm{/}\mathrm{\partial }{y}_l{)}_{1\le k,l\le d}$, so its determinant is $h\cdot g^{Nd}{h}^{\mathrm{\prime }\mathrm{\prime }},$ which is a unit of $T$ because $h$ maps to a unit of $S$ and hence of $T$, and $g$ and $h^{\mathrm{\prime }\mathrm{\prime }}$ are units of $T$.

The composite is standard smooth. By step 3.2 the algebra $T$ is presented over $R$ as $(R[x_1,\dots ,x_n,y_1,\dots ,y_m]\mathrm{/}(f_1,\dots ,f_c,F_1,\dots ,F_d){)}_{gH}$ with $n+m$ variables and $c+d$ equations, and by step 4.1 the displayed $(c+d)\times (c+d)$ minor of the Jacobian matrix is a unit of $T$; moreover the invertible minor may be assumed leading after permuting variables, so this is a standard smooth $R$-presentation [F1]. Its relative dimension is $(n+m)-(c+d)=(n-c)+(m-d)$, the sum of the relative dimensions of the two given presentations. This proves the first assertion of clause 2.

Composition at a point. Finite presentation is preserved by composition: from $S=R[z_1,\dots ,z_a]\mathrm{/}(r_1,\dots ,r_b)$ and $T=S[w_1,\dots ,w_e]\mathrm{/}(s_1,\dots ,s_l)$, lift the finitely many coefficients of the $s_i$ to $R[z]$; then $T=R[z,w]\mathrm{/}(r_1,\dots ,r_b,{\tilde{s}}_1,\dots ,{\tilde{s}}_l)$ by [F4, F5]. Thus the finite-presentation prerequisite in [F1] holds for the composite. Suppose $R\to S$ is standard smooth at $\mathfrak{q}$ and $S\to T$ is standard smooth at $\mathfrak{n}$ with $\mathfrak{n}\cap S=\mathfrak{q}$. Choose $u\mathrm{\notin }\mathfrak{q}$ with $S_u$ standard smooth over $R$ and $v\mathrm{\notin }\mathfrak{n}$ with $T_v$ standard smooth over $S$ [F1]. Since $u\mathrm{\notin }\mathfrak{q}=\mathfrak{n}\cap S$, both $u$ and $v$ lie outside $\mathfrak{n}$, so $uv\mathrm{\notin }\mathfrak{n}$. Base change of the standard smooth $S$-presentation of $T_v$ along $S\to S_u$ gives the standard smooth $S_u$-algebra $S_u{\otimes }_S{T}_v\cong (T_v{)}_u=T_{uv}$ by step 2.1 and [F4, F7]. Applying step 5.1 to $S_u$ over $R$ and $T_{uv}$ over $S_u$ exhibits $T_{uv}$ as standard smooth over $R$; since $uv\mathrm{\notin }\mathfrak{n}$, this witnesses that $R\to T$ is standard smooth at $\mathfrak{n}$ [F1]. As $\mathfrak{n}$ was arbitrary, a composite of locally standard smooth maps is locally standard smooth, which completes clause 2.