View source for Zariski tangent space

''to an algebraic variety or scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991201.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991202.png" />''

The vector space over the residue field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991203.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991204.png" /> that is dual to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991205.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991206.png" /> is the maximal ideal of the [[Local ring|local ring]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991207.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991208.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z0991209.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912010.png" /> is defined by a system of equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912012.png" />, then the Zariski tangent space at a rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912013.png" /> is defined by the system of linear equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912014.png" /></td> </tr></table>

A variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912015.png" /> is non-singular at a rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912016.png" /> if and only if the dimension of the Zariski tangent space to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912017.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912018.png" /> is equal to the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912019.png" />. For a rational point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912020.png" />, the Zariski tangent space is dual to the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912021.png" /> — the stalk at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912022.png" /> of the cotangent sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912023.png" />. An irreducible variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912024.png" /> over a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912025.png" /> is smooth if and only if the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912026.png" /> is locally free. The vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912027.png" /> associated with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912028.png" /> is called the tangent bundle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912029.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912030.png" />; it is functorially related to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912031.png" />. Its sheaf of sections is called the tangent sheaf to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099120/z09912032.png" />. The Zariski tangent space was introduced by O. Zariski [[#References|[1]]].

====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>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Hartshorne,   "Algebraic geometry" , Springer  (1977)  pp. Sect. IV.2</TD></TR></table>