Tangent sheaf

in algebraic geometry

The sheaf $\theta _ {X}$ on an algebraic variety or scheme $X$ over a field $k$, whose sections over an open affine subspace $U = \mathop{\rm Spec} ( A)$ are the $A$- modules of $k$- derivations $\mathop{\rm Der} _ {k} ( A, A)$ of the ring $A$. An equivalent definition is that $\theta _ {X}$ be the sheaf of homomorphisms $\mathop{\rm Hom} ( \Omega _ {X/k} ^ {1} , {\mathcal O} _ {X} )$ of the sheaf of differentials $\Omega _ {X/k} ^ {1}$ into the structure sheaf ${\mathcal O} _ {X}$( see Derivations, module of).

For any rational $k$- point $x \in X$, the stalk $\theta _ {X} ( x)$ of the sheaf $\theta _ {X}$ is identical to the Zariski tangent space $T _ {K,x}$ to $X$ at $x$, that is, to the vector $k$- space $\mathop{\rm Hom} _ {k} ( \mathfrak M _ {x} / \mathfrak M _ {x} ^ {2} , k)$, where $\mathfrak M _ {x}$ is the maximal ideal of the local ring ${\mathcal O} _ {K,x}$. Instead of the tangent sheaf $\theta _ {X}$ one can use the sheaf of germs of sections of the vector bundle $V ( \Omega _ {X/k} ^ {1} )$ dual to $\Omega _ {X} ^ {1}$( or the tangent bundle to $X$). In the case when $X$ is a smooth connected $k$- scheme, $\theta _ {X}$ is a locally free sheaf on $X$ of rank equal to the dimension of $X$.

References

[1]	I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[a1]	R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. Sect. IV.2 MR0463157 Zbl 0367.14001