Vector field on a smooth variety

Question

Let $X$ be a smooth affine variety. The tangent sheaf is defined as $\mathcal{Hom}(\Omega_X,\mathcal{O}_X)$, then what is a vector field on $X$? It's written here that a vector field is a derivation $A:\mathcal{O}_X\to\mathcal{O}_X$ such that $A(fg)=A(f)g+fA(g)$.

Why is that so? Tangent sheaf is a locally free sheaf, so it's in correspondence with a vector bundle, will this make sense if I look at it from this perspective? Sorry about my lack of rigour / understanding, I am trying to make sense of this.

Answer 1

An vector field is just a global section of the tangent sheaf, that is an element of $\Gamma(X,\mathcal{H}om(\Omega_X,\mathcal{O}_X))=\mathrm{Hom}(\Omega_X,\mathcal{O}_X)$.

Now, the object $\Omega_X$ satisfies a universal property : for any $\mathcal{O}_X$-module $\mathcal{F}$, there is a bijection between :

• morphisms of $\mathcal{O}_X$-modules : $\Omega_X\rightarrow\mathcal{F}$
• derivations of $\mathcal{F}$, that is maps $D:\mathcal{O}_X\rightarrow\mathcal{F}$ such that $D(fg)=D(f)g+fD(g)$.

Thus, a vector field is the same as a derivation of $\mathcal{O}_X$, in other words as a map $A:\mathcal{O}_X\rightarrow\mathcal{O}_X$ such tat $A(fg)=A(f)g+fA(g)$.