Relative dimension of a smooth morphism at a point

Let $f:X\to S$ be a morphism of schemes that is smooth at a point $x\in X$ (Smooth morphism of schemes), and put $s=f(x)$. For a scheme $Y$ and a point $y\in Y$ the local dimension of $Y$ at $y$, written ${\dim }_{y}Y$, is the infimum of the Krull dimensions of the open neighbourhoods of $y$ in $Y$ (Krull dimension of a nonzero ring). If $Y$ is locally of finite type over a field, this equals the largest dimension of an irreducible component through $y$ in any finite-type affine neighbourhood of $y$: nonempty principal opens of a finite-type domain have the same fraction field and hence the same dimension by Affine-domain dimension equals transcendence degree, while components not containing $y$ may be removed after shrinking that affine neighbourhood. The component formula need not hold for arbitrary locally Noetherian schemes: at the generic point of the spectrum of a discrete valuation ring, a principal open is a field of dimension zero although the whole space is an irreducible component of dimension one. This is the Stacks convention for ${\dim }_y$; it is not the height of the local ring ${\mathcal{O}}_{Y,y}$, which can be strictly smaller for a point lying on a lower-dimensional component (A local ring is a nonzero commutative ring with a unique maximal ideal).

Let $K\mathrm{/}\kappa (s)$ be a field extension and let $X_{s,K}=X_s{\times }_{\mathrm{Spec}\kappa (s)}\mathrm{Spec}K$ be the base-changed fibre; it is the geometric fibre of Geometric fibres and geometric points when $K$ is an algebraic closure of $\kappa (s)$. We say $f$ has relative dimension $n$ at $x$ when ${\dim }_y{X}_{s,K}=n$ for every field extension $K\mathrm{/}\kappa (s)$ and every point $y\in X_{s,K}$ lying over the image of $x$ in $X_s$ (the quantifier convention of Geometrically regular algebras and geometrically regular fibres). We say $f$ has pure relative dimension $n$ when it is smooth with relative dimension $n$ at every point of $X$, that is, when $f$ is smooth and every geometric fibre of $f$ is pure $n$-dimensional in the sense that its local dimension equals $n$ at each of its points.

Only the smooth case is used on this page. There, the condition holds for a unique $n$ at each point $x$: the fibre is geometrically regular at the points over $x$ by the definition of smoothness, and a computation with a standard smooth chart — $m$ variables and $c$ independent equations, hence $m-c$ free parameters — gives the common value $m-c$ of the local dimensions of the geometric fibre at the points lying over $x$, independent of the field extension $K$. On the model chart ${\mathbb{A}}_S^n\to S$ the relative dimension is $n$ everywhere, and for $n=0$ the morphism is étale; the chart computation is carried out in the standard-form and Jacobian results on this page. The integer is allowed to depend on the point $x$; pure relative dimension means that it does not, and the empty source case is vacuous.