The intrinsic cotangent space

Let $X$ be a scheme and $x\in X$. Write $A={\mathcal{O}}_{X,x}$ and let ${\mathfrak{m}}_x$ be the maximal ideal of this local ring. The intrinsic Zariski cotangent space of $X$ at $x$ is $C_{x}X:={\mathfrak{m}}_x\mathrm{/}{\mathfrak{m}}_x^2\mathrm{.}$ It is a vector space over the residue field $\kappa (x)={\mathcal{O}}_{X,x}\mathrm{/}{\mathfrak{m}}_x$ (The residue field at a point of an affine scheme).

The scalar action is induced by multiplication in $A$: the class of $a\in A$ acts on the class of $b\in {\mathfrak{m}}_x$ by the class of $ab$. This action depends only on the residue class of $a$, since replacing $a$ by an element congruent modulo ${\mathfrak{m}}_x$ changes $ab$ by an element of ${\mathfrak{m}}_x^2$. Thus the action factors through the field $\kappa (x)$.

For a classical variety over an algebraically closed field at a closed $k$-rational point, the residue field is $k$ and this is the usual cotangent space $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$ of the local ring. The definition above also applies to nonclosed points of arbitrary schemes.

This is the cotangent space of the underlying scheme at the point. No identification with a relative cotangent space over a chosen base is included in this definition.

For a direct check of the extreme dimensions, at the origin of $\mathrm{Spec}k[t]$ the local ring is $k[t{]}_{(t)}$ and its maximal ideal is generated by $t$, so $C_{0}X=(t)\mathrm{/}(t^2)\cong k$ with basis the class of $t$. At the generic point $\eta$ of $\mathrm{Spec}k[t]$, the local ring is the field $k(t)$, so its maximal ideal is zero and $C_{\eta }X=0$.