Tangent sheaf

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in algebraic geometry

The sheaf on an algebraic variety or scheme over a field , whose sections over an open affine subspace are the -modules of -derivations of the ring . An equivalent definition is that be the sheaf of homomorphisms of the sheaf of differentials into the structure sheaf (see Derivations, module of).

For any rational -point , the stalk of the sheaf is identical to the Zariski tangent space to at , that is, to the vector -space , where is the maximal ideal of the local ring . Instead of the tangent sheaf one can use the sheaf of germs of sections of the vector bundle dual to (or the tangent bundle to ). In the case when is a smooth connected -scheme, is a locally free sheaf on of rank equal to the dimension of .


[1] 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
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Tangent sheaf. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.V. Dolgachev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article