Purity and quasi-split torsors over Prüfer bases
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913217751547904 |
|---|---|
| author | Guo, Ning Liu, Fei |
| author_facet | Guo, Ning Liu, Fei |
| contents | We establish an analogue of the Zariski--Nagata purity theorem for finite étale covers on smooth schemes over Prüfer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. We derive an Auslander--Buchsbaum formula for general local rings, which provides a useful tool for studying the algebraic structures involved in our work. Through analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from EGAIV4 on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck--Serre conjecture for torsors under a quasi-split reductive group scheme over schemes smooth over Prüfer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi-split reductive group schemes in our Prüferian context, inspired by the recent work of Cesnavicius. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13206 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Purity and quasi-split torsors over Prüfer bases Guo, Ning Liu, Fei Algebraic Geometry Primary 14F22, Secondary 14F20, 14G22, 16K50 We establish an analogue of the Zariski--Nagata purity theorem for finite étale covers on smooth schemes over Prüfer rings by demonstrating Auslander's flatness criterion in this non-Noetherian context. We derive an Auslander--Buchsbaum formula for general local rings, which provides a useful tool for studying the algebraic structures involved in our work. Through analysis of reflexive sheaves, we prove various purity theorems for torsors under certain group algebraic spaces, such as the reductive ones. Specifically, using results from EGAIV4 on parafactoriality on smooth schemes over normal bases, we prove the purity for cohomology groups of multiplicative type groups at this level of generality. Subsequently, we leverage the aforementioned purity results to resolve the Grothendieck--Serre conjecture for torsors under a quasi-split reductive group scheme over schemes smooth over Prüfer rings. Along the way, we also prove a version of the Nisnevich purity conjecture for quasi-split reductive group schemes in our Prüferian context, inspired by the recent work of Cesnavicius. |
| title | Purity and quasi-split torsors over Prüfer bases |
| topic | Algebraic Geometry Primary 14F22, Secondary 14F20, 14G22, 16K50 |
| url | https://arxiv.org/abs/2301.13206 |