Let $X$ be an abelian variety over a field, $\Omega_X$ the differential sheaf, $\mathscr{T}_X$ the tangent sheaf, i.e., $\mathscr{Hom}(\Omega_X, \mathscr{O}_X)$, and $T_{X,0}$ be the tangent space at $0$ of $X$. I want a natural isomorphism $\Omega_X \to T_{X,0}^* \otimes _k \mathscr{O}_X$. ($V^*$ is the dual space of a vector space $V$.)
The proposition 1.5. in http://page.mi.fu-berlin.de/elenalavanda/BMoonen.pdf proves this, but I don't understand.
He says that a vector field on $X$ (i.e., an element of $\Gamma (X, \mathscr{T}_X )$ ) can be identified with an automorphism $X_S \to X_S$ which reduces to the identity on $X$. ($S = k[x]/x^2$)
I don't understand this.
(This is a similar question to Making rigorous Mumford's argument about the sheaf of differentials on an Abelian Variety )
A morphism $\operatorname{Spec} S \to X$ is equivalent to a choice of a $k$-point and a tangent vector (see, for instance, here). An automorphism $X_S\to X_S$ which is the identity on $X$ gives a family of morphisms $\operatorname{Spec} S \to X_S \stackrel{red}{\to} X$ indexed by the points of $X$, which can be interpreted as a family of tangent vectors at each point $x\in X$, also known as a vector field.