The tangent bundle to an algebraic variety $X$ is abstractly defined by the methods of synthetic differential geometry as the space of maps
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.
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 is
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.
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
e.g. (Costa, def. 1.1.8).
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
is a pullback.
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
Last revised on November 22, 2013 at 09:13:29. See the history of this page for a list of all contributions to it.