Hi,


let $X$ be an algebraic variety over a field $k$. Let $G$ be an algebraic group over $k$ and $P$, $Q$ $G$-torsors over $X$. Assume I have a morphism $\phi:P\to Q$ of $G$ torsors. Let $A\to k$ be a ring extension with square zero kernel $I$, $X\_A$ a variety over $Spec(A)$ such that $X\_A\otimes k\cong X$ and assume that $P$, $Q$ lifts to $X\_A$ as $G\_A$ torsors. What are the obstructions to lift $\phi $ to a morphism of $G\_A$ torsors $\phi\_A: P\_A\to Q\_A$ ?


thanks



lifting morphism of torsors
Assuming that $G\_A$ is smooth over $A$ and that $X\_A$ is flat over $A$, the "naive" obstruction to lifting $\phi$ is an element in $H^1(X,\mathfrak{g}\_P)$, where $\mathfrak{g}\_P$ is the locally free $\mathcal{O}\_X$-module obtained from the trivial bundle $\mathfrak{g}\otimes\_k \mathcal{O}\_X$ by "twisting" via the adjoint action of $G$ on $\mathfrak{g}$ with respect to $P$ (or $Q$, since they are isomorphic). I will say more precisely what this means in a moment.


The simplest way to think of this is to consider the graph $\Gamma\_\phi$ as a closed subscheme of $P\times\_X Q$. Consider the diagonal action of $G$ on $P\times\_X Q$, $g\cdot(p,q) = (gp,gq)$. Then $\Gamma\_\phi$ is $G$-equivariant. Equivalently, $\Gamma\_\phi$ is the pullback of a closed subscheme of the quotient $P\times^G Q := (P\times\_X Q)/\Delta(G)$, which is itself a smooth scheme over $X$, i.e., $\rho:P\times^G Q \to X$ is smooth.


In fact, $\Gamma\_\phi$ is the pullback of the image of a section $\widetilde{\phi}:X\to P\times^G Q$. This establishes a one-to-one correspondence between $G$-equivariant morphisms $\phi:P\to Q$ and sections $\widetilde{\phi}$ of $P\times^G Q$. So now your original problem, deforming $\phi$ to $\phi\_A$, is equivalent to the problem of deforming the section $\widetilde{\phi}$ to a section $\widetilde{\phi}\_A$ of $P\_A \times^{G\_A} Q\_A$. Since $P\times^G Q$ is smooth over $X$, the obstruction to such a deformation is an element in $H^1(X,\widetilde{\phi}^\* T\_\rho)$. And $\widetilde{\phi}^\*T\_\rho$ equals the locally free sheaf $\mathfrak{g}\_P$ from above. The fastest way to see this is to check that they give isomorphic descent data with respect to the fppf cover $P\to X$.


