Zariski tangent space

to an algebraic variety or scheme $ X $ at a point $ x $

The vector space over the residue field $ k ( x ) $ of the point $ x $ that is dual to the space $ \mathfrak M _ {x} / \mathfrak M _ {x} ^ {2} $, where $ \mathfrak M $ is the maximal ideal of the local ring $ {\mathcal O} _ {X ,x } $ of $ x $ on $ X $. If $ X \subset A _ {k} ^ {n} $ is defined by a system of equations

$$ F _ \alpha = 0 , $$

where $ F _ \alpha \in k [ X _ {1} \dots X _ {n} ] $, then the Zariski tangent space at a rational point $ x = ( x _ {1} \dots x _ {n} ) $ is defined by the system of linear equations

$$ \sum _ { i=1 } ^ { n } \frac{\partial F _ \alpha }{\partial X _ {i} } ( x ) ( X _ {i} - x _ {i} ) = 0 . $$

A variety $ X $ is non-singular at a rational point $ x $ if and only if the dimension of the Zariski tangent space to $ X $ at $ x $ is equal to the dimension of $ X $. For a rational point $ x \in X $, the Zariski tangent space is dual to the space $ \Omega _ {X / k } ^ {1} \otimes k ( x ) $ — the stalk at $ x $ of the cotangent sheaf $ \Omega _ {X / k } ^ {1} $. An irreducible variety $ X $ over a perfect field $ k $ is smooth if and only if the sheaf $ \Omega _ {X / k } ^ {1} $ is locally free. The vector bundle $ T _ {X} = V ( \Omega _ {X /k } ^ {1} ) $ associated with $ \Omega _ {X / k } ^ {1} $ is called the tangent bundle of $ X $ over $ k $; it is functorially related to $ X $. Its sheaf of sections is called the tangent sheaf to $ X $. 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

References

[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001