nLab
tangent cone

Contents

Contents

Idea

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

DX, D \longrightarrow X \,,

where DD is the spectrum of the ring of dual numbers.

If XX 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 X𝔸 nX \hookrightarrow \mathbb{A}^n an affine algebraic variety defined by an ideal IR[x 1,,x n]I \hookrightarrow R[x_1, \cdots, x_n], hence XSpec(R[x 1,,x n]/I)X \simeq Spec(R[x_1, \cdots, x_n]/I), then its tangent cone is

CXSpec(R[x 1,,x n]/I *) C X \coloneqq Spec(R[x_1, \cdots, x_n]/I_\ast)

where I *I_\ast is the ideal obtained from II by truncating each polynomial fIR[x 1,,x n]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:XYf \colon X \longrightarrow Y of algebraic varieties over an algebraically closed field is an étale morphism if for all points xXx \in X it induces an isomorphism of tangent cones

C xXC f(x)Y. 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 Π inf\Pi_{inf} as those morphisms f:XYf \colon X \to Y such that the unit naturality square

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

is a pullback.

References

Reviews include

See also

Last revised on November 22, 2013 at 09:13:29. See the history of this page for a list of all contributions to it.