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Difference between revisions of "Zariski tangent space"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski,   "The concept of a simple point of an abstract algebraic variety" ''Trans. Amer. Math. Soc.'' , '''62''' (1947) pp. 1–52</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Samuel,   "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, "The concept of a simple point of an abstract algebraic variety" ''Trans. Amer. Math. Soc.'' , '''62''' (1947) pp. 1–52 {{MR|0021694}} {{ZBL|0031.26101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P. Samuel, "Méthodes d'algèbre abstraite en géométrie algébrique" , Springer (1955) {{MR|0072531}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR></table>
  
  
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977) pp. Sect. IV.2</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>

Revision as of 21:57, 30 March 2012

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].

References

[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


Comments

References

[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: 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