Grothendieck–Serre conjecture
Let be an unramified regular local ring, let be its fraction field, and let be a reductive group scheme over . If a -torsor over becomes trivial after base change to , then it is already trivial over ; equivalently, every generically trivial -torsor over is trivial.
References
Primary source
Additional references
- The unramified Grothendieck--Serre conjecture — arXiv — Fei Liu
Progress summary
A new unrefereed manuscript claims the conjecture in the unramified case, but the full conjecture remains open.
The Grothendieck–Serre conjecture predicts that torsors for reductive groups over regular local rings are trivial whenever they become trivial over the fraction field. The latest result covers unramified regular local rings, not the full conjecture.
Known results
- Česnavičius, 2020: the split case over unramified regular local rings.
- Guo, 2020: the conjecture over valuation rings.
- A 2022 result: the simply connected, totally isotropic case over unramified regular local rings.
- Panin and Stavrova, 2024: the constant-group case in mixed characteristic.
September 2026 unramified-case claim
Fei Liu’s manuscript claims the conjecture for unramified regular local rings. This is a substantial advance, but the manuscript is unrefereed and does not settle the general case.
Current status (as of September 2026): the unramified case is claimed in an unrefereed manuscript, while the full Grothendieck–Serre conjecture remains open.
Sources
- arxiv.org
- arxiv.org
- quantamagazine.org
- openai.com
- openai.com
- cdn.openai.com
- scientificamerican.com
- quantamagazine.org
- openai.com
- cdn.openai.com
- arxiv.org
- mathstodon.xyz
- mathstodon.xyz
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- arxiv.org
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- arxiv.org
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