The scheme-theoretic tangent cone at a point

Let $X$ be a locally Noetherian scheme and $x\in X$. Write $A={\mathcal{O}}_{X,x}$, let ${\mathfrak{m}}_x$ be its maximal ideal, and put $\kappa (x)=A\mathrm{/}{\mathfrak{m}}_x$, as in The intrinsic cotangent space. The associated graded ring for the maximal-ideal filtration is

$${\mathrm{gr}}_{{\mathfrak{m}}_x}(A)=\underset{n\ge 0}{⨁}{\mathfrak{m}}_x^n\mathrm{/}{\mathfrak{m}}_x^{n+1}$$

with multiplication induced from $A$ (The associated graded ring and associated graded module of an ideal-adic filtration).

For every $n$, multiplication by ${\mathfrak{m}}_x$ sends ${\mathfrak{m}}_x^n$ into ${\mathfrak{m}}_x^{n+1}$, so it acts trivially on the degree-$n$ quotient. Thus the scalar action on each graded piece factors canonically through $\kappa (x)$, and the degree-zero piece is $A\mathrm{/}{\mathfrak{m}}_x=\kappa (x)$. This makes the associated graded ring a graded $\kappa (x)$-algebra without choosing a coefficient field in $A$.

The scheme-theoretic tangent cone of $X$ at $x$ is the affine $\kappa (x)$-scheme

$${\mathrm{Cone}}_x(X):=\mathrm{Spec}\text{\hspace{0.17em}⁣}({\mathrm{gr}}_{{\mathfrak{m}}_x}(A))⟶\mathrm{Spec}\kappa (x)\mathrm{.}$$

We use ${\mathrm{Cone}}_x(X)$ to distinguish this scheme from the cotangent space denoted $C_{x}X$ in the preceding definition. Here Spec carries its affine scheme structure (The underlying space of an affine spectrum, Affine schemes and their coordinate rings), and the map to $\mathrm{Spec}\kappa (x)$ is the structure morphism of a scheme over $\kappa (x)$ (Schemes and morphisms over a base) induced by the degree-zero inclusion. The grading is retained as part of the cone presentation. The reduction $({\mathrm{Cone}}_x(X){)}_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is a closed subscheme that can differ from ${\mathrm{Cone}}_x(X)$; the definition uses the full associated graded ring, without quotienting by its nilpotents (The reduction of a scheme).

If $A$ is already a field, then ${\mathfrak{m}}_x=0$, every positive graded piece vanishes, and ${\mathrm{Cone}}_x(X)=\mathrm{Spec}\kappa (x)$.

For the closed point $x=(ϵ)$ of the dual-numbers scheme $D_k=\mathrm{Spec}(k[ϵ]\mathrm{/}({ϵ}^2))$ (The affine scheme of dual numbers), every element $a+bϵ$ with $a\mathrm{\ne }0$ is a unit, so ${\mathcal{O}}_{D_k,x}=k[ϵ]\mathrm{/}({ϵ}^2)$ and ${\mathfrak{m}}_x=(ϵ)$. Its associated graded pieces are $k$ in degree $0$, $kϵ$ in degree $1$, and zero in every degree $n\ge 2$; the degree-one class squares to zero. Hence

$${\mathrm{gr}}_{(ϵ)}{\mathcal{O}}_{D_k,x}\cong k[ϵ]\mathrm{/}({ϵ}^2),{\mathrm{Cone}}_x(D_k)\cong D_k\mathrm{.}$$

The nilradical is $(ϵ)$, so the reduction is $\mathrm{Spec}k$. Thus ${\mathrm{Cone}}_x(D_k)$ and its reduction have the same one-point topological space but different structure sheaves.