# 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 .

How to Cite This Entry:
Zariski tangent space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zariski_tangent_space&oldid=13053
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article