# Extending $G$-torsors on open subsets of affine space

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, Izv. Math. 84, No. 4, 780-795 (2020); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 84, No. 4, 169-186 (2020). ZBL1458.14024.

Antieau, Benjamin; Williams, Ben, Topology and purity for torsors, Doc. Math. 20, 333-355 (2015). ZBL1349.14068.

• I'm probably missing something, but the answer would be yes if your $G$-torsor is already trivial, locally for the Zariski topology, right? So, if $G$ is special, the answer is yes. (This is just a bit more general than your 4.) Jul 5 at 15:22
• Maybe you can use 3 to cook up an example. Indeed, let $X$ be smooth affine, and let $U\subset X$ such that $T\to U$ is a counterexample. Now, choose $X\to\mathbb{A}^n$ finite surjective and consider $U\to V$. Assuming the latter is finite (i.e., proper), you could maybe use Weil restriction of scalars along $U\to V$. Jul 5 at 15:24
• @AriyanJavanpeykar: the question is indeed trivial if $G$ is special. About your suggestion: it sounds interesting, but filling in the details does not seem easy. Also if $T$ is a $G$-torsor over $U$, then the Weil restriction to $V$ would be a $G'$ torsor, where $G'$ is the Weil restriction of $G_U$ along $U\rightarrow V$. I think $G'$ will not be split unless $U\rightarrow V$ is of a very degenerate form.
– Jef
Jul 5 at 16:39
• Ah yes, if you insist on $G$ being split, this probably won't work... Jul 9 at 18:58