Isotropic Torsors on Smooth Algebras over Prüfer Rings
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912417975369728 |
|---|---|
| author | Kundu, Arnab |
| author_facet | Kundu, Arnab |
| contents | The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group over a regular semilocal ring is itself trivial. Extending the work of Česnavičius and Fedorov, we prove a non-noetherian analogue of this conjecture for rings $A$ that are semilocalisations of smooth schemes over valuation rings of rank one, and for reductive $A$-group schemes $G$ that are totally isotropic. Roughly speaking, such group schemes are characterised by the existence of a parabolic subgroup of their adjoint quotients. Since quasi-split groups are totally isotropic, our result, in particular, generalises the Grothendieck--Serre result of Guo--Liu and the author's thesis. Our proof relies on a new instance of Gabber's presentation lemma, obtained by extending techniques developed in the author's thesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04760 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Isotropic Torsors on Smooth Algebras over Prüfer Rings Kundu, Arnab Algebraic Geometry The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group over a regular semilocal ring is itself trivial. Extending the work of Česnavičius and Fedorov, we prove a non-noetherian analogue of this conjecture for rings $A$ that are semilocalisations of smooth schemes over valuation rings of rank one, and for reductive $A$-group schemes $G$ that are totally isotropic. Roughly speaking, such group schemes are characterised by the existence of a parabolic subgroup of their adjoint quotients. Since quasi-split groups are totally isotropic, our result, in particular, generalises the Grothendieck--Serre result of Guo--Liu and the author's thesis. Our proof relies on a new instance of Gabber's presentation lemma, obtained by extending techniques developed in the author's thesis. |
| title | Isotropic Torsors on Smooth Algebras over Prüfer Rings |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2505.04760 |