Let $k$ be a characteristic zero field, $V \subset \mathbb{A}^n\_k$ an open subscheme, $G$ a split reductive group over $k$ and $T$ a $G$-torsor over $V$ (in the etale, equivalently fppf topology). Suppose that $T$ extends along codimension $1$ points of $\mathbb{A}^n\_k$, i.e. the torsor class $[T\_{k(\mathbb{A}^n)}] \in \mathrm{H}^1(k(\mathbb{A}^n),G)$ lies in the image of $\mathrm{H}^1(\mathcal{O}\_{\mathbb{A}^n,Z},G)\rightarrow \mathrm{H}^1(k(\mathbb{A}^n),G)$ for all closed irreducible $Z\subset \mathbb{A}^n$ of codimension $1$. (This is true, for example, if $\mathbb{A}^n\setminus V$ has codimension $\geq 2$ in $\mathbb{A}^n$.)



> 
> **Question**: does there exist a $G$-torsor $T'$ on $\mathbb{A}^n$ such
> that $T$ and $T'|\_V$ are locally isomorphic in the Zariski topology?
> 
> 
> 


Some remarks:


1. By known cases of the Grothendieck-Serre conjecture [Panin], it suffices to show that there exists a $G$-torsor $T'$ on $\mathbb{A}^n$ with $[T\_{k(\mathbb{A}^n)}]=[T'\_{k(\mathbb{A}^n)}]$ in $\mathrm{H}^1(k(\mathbb{A}^n),G)$.
2. We cannot require that $T$ literally extends to a $G$-torsor on $\mathbb{A}^n$; indeed there exists a $\mathrm{GL}\_2$-torsor on $\mathbb{A}^3\setminus\{0\}$ that does not extend to $\mathbb{A}^3$.
3. The question has a negative answer if we allow $V$ to be an open subset in an arbitrary smooth affine $k$-variety [Antieau--Williams].
4. The question is trivial for $G = \mathrm{GL}\_n$ or $\mathrm{Sp}\_n$, since every torsor under such a group is Zariski locally trivial.


*Panin, Ivan A.*, [**Proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing a field**](http://dx.doi.org/10.1070/IM8982), Izv. Math. 84, No. 4, 780-795 (2020); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 84, No. 4, 169-186 (2020). [ZBL1458.14024](https://zbmath.org/?q=an:1458.14024).


*Antieau, Benjamin; Williams, Ben*, [**Topology and purity for torsors**](http://www.emis.de/journals/DMJDMV/vol-20/09.html), Doc. Math. 20, 333-355 (2015). [ZBL1349.14068](https://zbmath.org/?q=an:1349.14068).



Extending $G$-torsors on open subsets of affine space
In case this might be useful to anyone, it turns out that results of Colliot--Thelene can be used to resolve a closely related question: see Theorem 6.1 in the following preprint (apologies for the self-promotion):


[https://www.dpmms.cam.ac.uk/~jcsl5/ADEpaper.pdf](https://www.dpmms.cam.ac.uk/%7Ejcsl5/ADEpaper.pdf)
(version 5 January 2022)


It seems that a very similar proof establishes the claim of the question.


