Relative cotangent and tangent spaces

Let $f\text{\hspace{0.17em}⁣}:X\to S$ be a morphism of schemes, let $x\in X$ be a point with image $s=f(x)\in S$, and let ${\mathrm{\Omega }}_{X\mathrm{/}S}$ be the sheaf of relative differentials (Sheaf of relative Kähler differentials), an ${\mathcal{O}}_X$-module.

Relative cotangent space. The relative cotangent space of $X$ over $S$ at $x$ is the $\kappa (x)$-vector space

${\mathrm{\Omega }}_{X\mathrm{/}S,x}{\otimes }_{{\mathcal{O}}_{X,x}}\kappa (x),$

where ${\mathcal{O}}_{X,x}$ is the local ring and $\kappa (x)={\mathcal{O}}_{X,x}\mathrm{/}{\mathfrak{m}}_x$ is the residue field of $x$ (The residue field at a point of an affine scheme) and the tensor product is formed along the residue map ${\mathcal{O}}_{X,x}\to \kappa (x)$; equivalently it is the fibre of the ${\mathcal{O}}_X$-module ${\mathrm{\Omega }}_{X\mathrm{/}S}$ at $x$ in the sense of the tensor product with the residue field. Its elements are written $\omega \otimes 1$ and, for a local section $a$ of ${\mathcal{O}}_X$ near $x$, ${\mathrm{d}}_{X\mathrm{/}S}(a)\otimes 1$ is the relative cotangent vector of $a$ at $x$.

Relative tangent space. The relative tangent space of $X$ over $S$ at $x$ is the $\kappa (x)$-linear dual

$T_{X\mathrm{/}S,x}:={\mathrm{Hom}}_{\kappa (x)}({\mathrm{\Omega }}_{X\mathrm{/}S,x}{\otimes }_{{\mathcal{O}}_{X,x}}\kappa (x),\kappa (x))\mathrm{.}$

Residue-field dependence. The residue field map $\kappa (s)\to \kappa (x)$ induced by $f$ is part of the data: the $\kappa (x)$-module ${\mathrm{\Omega }}_{X\mathrm{/}S,x}\otimes \kappa (x)$ is a vector space over $\kappa (x)$, and the $\kappa (s)$-structure obtained by restriction of scalars along $\kappa (s)\to \kappa (x)$ is used whenever the base field is fixed. No finiteness hypothesis is imposed: the spaces above may be infinite-dimensional over their residue fields, and the notation applies to any point of any morphism of schemes, including non-closed points and points of relative dimension $0$.