Zariski tangent space

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to an algebraic variety or scheme at a point

The vector space over the residue field of the point that is dual to the space , where is the maximal ideal of the local ring of on . If is defined by a system of equations

where , then the Zariski tangent space at a rational point is defined by the system of linear equations

A variety is non-singular at a rational point if and only if the dimension of the Zariski tangent space to at is equal to the dimension of . For a rational point , the Zariski tangent space is dual to the space — the stalk at of the cotangent sheaf . An irreducible variety over a perfect field is smooth if and only if the sheaf is locally free. The vector bundle associated with is called the tangent bundle of over ; it is functorially related to . Its sheaf of sections is called the tangent sheaf to . The Zariski tangent space was introduced by O. Zariski [1].


[1] O. Zariski, "The concept of a simple point of an abstract algebraic variety" Trans. Amer. Math. Soc. , 62 (1947) pp. 1–52 MR0021694 Zbl 0031.26101
[2] P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) MR0072531
[3] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001



[a1] R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001
How to Cite This Entry:
Zariski tangent space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article