# Lie Algebra of Automorphism Group of a Scheme

Let $$X$$ be a scheme over an algebraically closed field $$k$$ and let $$\operatorname{Aut}(X)$$ denote the functor sending a $$k$$-scheme $$T$$ to the group $$\operatorname{Aut}_T(X \times_k T)$$ of automorphisms of $$X \times_k T$$ over $$T$$.

My goal is to have a better grasp of why $$\operatorname{Lie}(\operatorname{Aut}(X))= H^0(X, \mathcal{T} X)$$ and therefore I am trying to work through an example where I know both the group $$\operatorname{Aut}(X)$$ and $$\operatorname{Lie}(\operatorname{Aut}(X))$$.

Let $$X = \mathbb{P}_k^1$$ so that $$\operatorname{Aut}(X)= PGL(2,k)$$. Now I try to recover the fact that $$\operatorname{Aut}(X)= PGL(2,k)$$.

The global sections of $$X$$ are locally of the form $$a_0 + a_1 z + a_2 z^2$$ where $$z=v/u$$ is a choice of homogeneous coordinates on $$X$$.

Is it possible to go from this description of global sections to the group \$ \operatorname{Aut}(X)?

My first guess would be to use the exponential map but I don’t know how to apply it in such a concrete example.