Automorphism of principal bundles and sections of the adjoint bundle

Question

I am studying the paper of Biswas and Ramanan An infinitesimal study of the moduli space of hitching pairs and at some point they make a statement that I don't know if it is true because they don't give any kind of reference. The authors consider a smooth projective curve $C$ over $\mathbb{C}$, and algebraic group $G$. Let $P$ be a principal $G$ bundle over $C$ and let $P(\epsilon):=P\times\operatorname{Spec}(\mathbb{C}[\epsilon])$ where $\epsilon^2=0$ the $G$-bundle over $C\times\operatorname{Spec}(\mathbb{C}[\epsilon])$ obtained by the pullback through the projection $C\times\operatorname{Spec}(\mathbb{C}[\epsilon])\rightarrow C$ on the first factor. They claim that is equivalent to give an automorphism of $P(\epsilon)$ which induces the identity over the closed point that to give a section of the adjoint bundle $ad P$ (the fibers are the lie algebra of $G$). For a section $s$ of $ad P$, the corresponding automorphism is denoted by $1+\epsilon s$. Can anybody explain to me why this is true or at least give some reference for this (a priori, well-known) fact?

Answer 1

Question: "Can anybody explain to me why this is true or at least give some reference for this (a priori, well-known) fact?"

Answer: Let $A$ be a commutative ring and $E$ a finite rank projective $A$-module and let $B:=A[\epsilon], E[\epsilon]:=A[\epsilon]\otimes_A E$. You may check that a $B$-linear endomorphism $\phi$ of $B\otimes_A E$ "inducing the identity" must be on the following form:

$$\phi(x+y\epsilon)=x+(s(x)+y)\epsilon.$$

Written in "matrix notation" you get

\begin{align*} \phi:= \begin{pmatrix} 1 & 0 \\ s & 1 \end{pmatrix} \end{align*}

where $s\in End_A(E)$. The inverse $\phi^{-1}$ is given by the matrix

\begin{align*} \psi:=\phi^{-1}:= \begin{pmatrix} 1 & 0 \\ -s & 1 \end{pmatrix} \end{align*}

and you may check that $\phi \circ \psi = \psi \circ \phi=Identity$, hence $\phi$ and $\phi$ are automorphisms of the $B$-module $B\otimes_A E$. The element $s\in End_A(E)$ can be an arbitrary element, and this gives a 1-1 correspondence between automorphisms of the $B$-module $B\otimes_A E$ inducing the identity on $A\cong A[\epsilon]/(\epsilon)$ and the endomorphisms $End_A(E)$. You need $E$ to be a finite rank projective $A$-module for this to hold ($E$ must be torsion free) - the fact that $E$ is torsion free implies the above description of the matrices $\phi, \psi$. This "proves" the case of principal $GL_n$-bundles.

If $X$ is a scheme and $E$ is a locally trivial finite rank sheaf on $X$ you get for any $s\in End_{\mathcal{O}_X}(E)$ an endomorphism

$$\phi_s: E[\epsilon] \rightarrow E[\epsilon]$$

defined locally by (let $U \subseteq X$ be an open subscheme)

$$\phi_s(U)(x+y\epsilon):=x+(s_U(x)+y)\epsilon.$$

Here $s_U$ is the restriction of $s$ to $U$. This gives a 1-1 correspondence between the $\mathcal{O}_{X[\epsilon]}$-automorphisms of $E[\epsilon]$ inducing the identity on $X:=V(\epsilon) \subseteq X[\epsilon]$ and $End_{\mathcal{O}_X}(E)$. The result is in fact more general:

Theorem If $E$ is a torsion free $\mathcal{O}_X$-module, there is a 1-1 correspondence between the set of $\mathcal{O}_{X[\epsilon]}$-automorphisms of $E[\epsilon]$ inducing the identity on $X \subseteq X[\epsilon]$ and $\mathcal{O}_X$-linear endomorphisms $\phi: E \rightarrow E$.

Note: If a more general principal fiber bundle $\pi:E \rightarrow X$ has sheaf of sections $sh(\pi)$ that is a finite rank locally trivial $\mathcal{O}_X$-module, I believe the same argument applies.