Zariski tangent space

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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