Аннотация:
The Grothendieck–Serre conjecture predicts that every principal bundle under a reductive group scheme $G$ over a regular local ring $R$ is trivial if it is generically trivial. In other words,
\[
\text{the map \quad $H^1_{\text{\'et}}(R,G)\rightarrow H^1_{\text{\'et}}(\mathrm{Frac}\,R,G)$\quad has trivial kernel.}
\]
When $R$ contains a field, the conjecture was solved affirmatively, whereas when $R$ is of mixed characteristic, it is widely open.
Besides, a non-Noetherian variant of the conjecture, when $R$ is replaced by a valuation ring, was solved affirmatively.
In this report, beyond the historical summary, I briefly talk about the following recent progress on this conjecture.
(i) For an irreducible scheme $X$ smooth projective over a discrete valuation ring $R$ of mixed characteristic, every generically trivial principal bundle under a reductive connected $R$-group is Zariski-locally trivial. This is a joint work with Панин Иван Александрович.
(ii) For a scheme $X$ smooth of finite type over a valuation ring $V$, if $x\in X$ is not a maximal point of $V$-fibers of $X$ and $\mathrm{dim}\, \mathcal{O}_{X,x}\geq 2$, then for any $X$-torus $T$, we have the purity
\[
H^1_{ét}(\mathrm{Spec} \mathcal{O}_{X,x}, T)\simeq H^1_{ét}(\mathrm{Spec} \mathcal{O}_{X,x}\backslash\{x\},T),
\]
which leads to the Grothendieck–Serre for tori on $X$. This is a joint work with Fei Liu.
