Let $ f: X \to Y$ be a morphism of schemes and let $ G=\operatorname{Aut}(X)$ denote the functor sending a $ Y$ -scheme $ Z$ to the group $ \operatorname{Aut}_Z(X \times_Y Z)$ of automorphisms of $ X \times_Y Z$ over $ Z$ . Let us assume $ X$ and $ Y$ are such that $ G$ is a Lie group over $ Y$ . Let us denote the structure morphism by $ g: G \to Y$ .

Let $ e \in G$ be the identity element in $ G$ so that the Lie algebra $ \operatorname{Lie}(G) = (\mathcal{T}_{G/Y})_e$ . I saw on the Stacks page that there exists a natural identification $ $ (\mathcal{T}_{G/Y})_e = (\mathcal{T}_{G_s/s})_e$ $ where $ s=g(e) \in Y$ , and $ G_s = G \times_Y k(s)$ is the fiber over $ s \in Y$ .

The idea of this makes sense since tangent vector should be defined fiberwise .

However, what does $ (\mathcal{T}_{G_s/s})_e$ represent in terms of automorphisms of $ Z$ ? Is $ (\mathcal{T}_{G_s/s})_e$ the Lie Algebra of the Lie group $ \operatorname{Aut}(X \times_Y k(s))$ as a scheme over $ k(s) \cong k$ ?