Difference between revisions of "Zariski tangent space"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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"> | + | <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 |
Zariski tangent space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zariski_tangent_space&oldid=13053