nLab tangent cone

Idea

The tangent bundle to an algebraic variety $X$ is abstractly defined by the methods of synthetic differential geometry as the space of maps

$$D \longrightarrow X \,,$$

where $D$ is the spectrum of the ring of dual numbers.

If $X$ is sufficiently regular, then this is the naive “Zariski tangent space”. More generally the correct construction is given by the tangent cone construction.

Definition

Definition

For $X \hookrightarrow \mathbb{A}^n$ an affine algebraic variety defined by an ideal $I \hookrightarrow R[x_1, \cdots, x_n]$, hence $X \simeq Spec(R[x_1, \cdots, x_n]/I)$, then its tangent cone at the origin is

$$C X \coloneqq Spec(R[x_1, \cdots, x_n]/I_\ast)$$

where $I_\ast$ is the ideal obtained from $I$ by truncating each polynomial $f \in I \hookrightarrow R[x_1, \cdots, x_n]$ to its homogeneous part of lowest monomial degree.

Properties

Relation to étale morphisms

A homomorphism $f \colon X \longrightarrow Y$ of algebraic varieties over an algebraically closed field is an étale morphism if for all points $x \in X$ it induces an isomorphism of tangent cones

$$C_x X \longrightarrow C_{f(x)} Y \,.$$

e.g. (Costa, def. 1.1.8).

Remark

This is the same kind of characterization as for local diffeomorphisms (see there) in terms of tangent spaces.

More generally, both cases are special cases of the definition of formally étale morphisms in terms of an infinitesimal shape modality $\Pi_{inf}$ as those morphisms $f \colon X \to Y$ such that the unit naturality square

$$\array{ X &\longrightarrow& \Pi_{inf}(X) \\ \downarrow && \downarrow \\ Y &\longrightarrow& \Pi_{inf}(Y) }$$

is a pullback.

References

Reviews include

• James Milne, p. 18 of Lectures on Étale Cohomology

• Edgar Costa, around def. 1.1.6 of Étale cohomology (pdf)

See also

• Igor R. Shafarevich, Basic algebraic geometry

• Wikipedia Tangent cone

• eom: tangent cone