Relative differential-rank condition

Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a morphism of schemes with sheaf of relative differentials ${\mathrm{\Omega }}_{X\mathrm{/}S}$ (Sheaf of relative Kähler differentials), and let $n\ge 0$ be an integer.

Locally free of constant rank $n$. An ${\mathcal{O}}_X$-module $\mathcal{F}$ is locally free of constant rank $n$ on an open subscheme $U\subseteq X$ when every point $x\in U$ has an open neighbourhood $W\subseteq U$ together with an isomorphism of ${\mathcal{O}}_W$-modules $\mathcal{F}{\mathrm{\mid }}_W\cong {\mathcal{O}}_W^{\oplus n}\mathrm{.}$ Equivalently, $\mathcal{F}{\mathrm{\mid }}_U$ is a locally free ${\mathcal{O}}_U$-module whose rank function $x\mapsto {\mathrm{rk}}_{{\mathcal{O}}_{X,x}}{\mathcal{F}}_x$ is constant equal to $n$ on $U$; the locally free rank is locally constant, so if $U$ is nonempty and $\mathcal{F}{\mathrm{\mid }}_U$ is locally free of constant rank $n$, then $n$ is determined by $U$ and $\mathcal{F}$. On $U=\mathrm{\varnothing }$ the condition holds for every $n$ and determines no rank. No quasi-coherence, finiteness or flatness hypothesis on $f$ is built into this definition; the hypothesis is placed on the module $\mathcal{F}={\mathrm{\Omega }}_{X\mathrm{/}S}$ alone.

Differential rank. The morphism $f$ has differential rank $n$ on the open subscheme $U\subseteq X$ when the restriction ${\mathrm{\Omega }}_{X\mathrm{/}S}{\mathrm{\mid }}_U$ is locally free of constant rank $n$ on $U$ in the sense above. Thus differential rank $0$ on $U$ means that ${\mathrm{\Omega }}_{X\mathrm{/}S}$ vanishes locally on $U$, and differential rank $n$ for $n>0$ means that the module of relative differentials is locally standard of rank $n$ over $U$.

This condition alone does not define smoothness. Differential rank $n$ is a statement about the first-order infinitesimal structure of $f$; it is not a smoothness criterion. In the source treatment the relative-dimension notion smooth of relative dimension $n$ is defined as smoothness together with finiteness and local freeness of constant rank $n$ of ${\mathrm{\Omega }}_{X\mathrm{/}S}$, and it is equivalently described by the four hypotheses: locally of finite presentation, flat, all nonempty fibres equidimensional of dimension $n$, and ${\mathrm{\Omega }}_{X\mathrm{/}S}$ finite locally free of rank $n$. None of these four hypotheses beyond the last is built into the definition above, and the comparison of the rank condition with flatness and fibre conditions belongs to the smooth-morphism development of the library rather than to this definition. In particular, no item on this page may conclude smoothness from differential rank alone.

Consistency with standard smooth presentations. The condition is not empty: if $A$ is a commutative ring and $B$ is an $A$-algebra admitting a standard smooth presentation of relative dimension $n$ (Standard smooth presentations and locally standard smooth maps), that is $B\cong (A[x_1,\dots ,x_N]/(f_1,\dots ,f_c){)}_g$ with $N-c=n$ and with a $c\times c$ minor of the Jacobian matrix $\left(\mathrm{\partial }{f}_j\mathrm{/}\mathrm{\partial }{x}_i\right)$ invertible in $B$, then ${\mathrm{\Omega }}_{B\mathrm{/}A}$ is a free $B$-module of rank $n$, as follows. By Jacobian presentation of Ω applied to the presentation before inverting $g$, the module ${\mathrm{\Omega }}_{(P\mathrm{/}I)\mathrm{/}A}$ for $P=A[x_1,\dots ,x_N]$ and $I=(f_1,\dots ,f_c)$ is the cokernel of the $B^{\mathrm{\prime }}$-linear map $B^{\mathrm{\prime }c}\to B^{\mathrm{\prime }N}$ (with $B^{\mathrm{\prime }}=P\mathrm{/}I$) given by the transpose of the row-oriented $c\times N$ Jacobian matrix; localising at $g$, which commutes with $\mathrm{\Omega }$ and with forming the cokernel (Kähler differentials commute with localization), presents ${\mathrm{\Omega }}_{B\mathrm{/}A}$ as the cokernel of this transposed Jacobian over $B$. Reordering the variables so that the invertible minor occupies the first $c$ columns of the row-oriented Jacobian, write its transpose in vertical blocks $\left(\begin{array}{c}C\\ D\end{array}\right)$, with $C\in {\mathrm{G}\mathrm{L}}_c(B)$ and $D\in {\mathrm{Mat}}_{N-c,c}(B)$. Then the map $B^c\to B^N$, $u\mapsto (Cu,Du)$ has image $\{(u^{\mathrm{\prime }},v^{\mathrm{\prime }}):u^{\mathrm{\prime }}\in B^c,v^{\mathrm{\prime }}=D{C}^{-1}{u}^{\mathrm{\prime }}\}$, and the $B$-linear map $B^N⟶B^{N-c},(u^{\mathrm{\prime }},v^{\mathrm{\prime }})⟼v^{\mathrm{\prime }}-D{C}^{-1}{u}^{\mathrm{\prime }},$ vanishes on this image and restricts to the identity on the complementary coordinates; hence it induces an isomorphism from the cokernel to $B^{N-c}$. So ${\mathrm{\Omega }}_{B\mathrm{/}A}$ is free of rank $N-c=n$, and the morphism $\mathrm{Spec}B\to \mathrm{Spec}A$ has differential rank $n$ on its whole chart, with no smoothness hypothesis needed for this computation.