Work

On Grothendieck-Serre conjecture in mixed characteristic for $\operatorname{SL}_{1,D}$

Year: 2025

Type: article

Abstract: Let $R$ be an unramified regular local ring of mixed characteristic, $D$ an Azumaya $R$-algebra, $K$ the fraction field of $R$, $\operatorname{Nrd}\colon D^{\times} \to R^{\times}$ the reduced norm homomorphism. Let $a \in R^{\times}$ be a unit. Suppose the equation $\operatorname{Nrd}=a$ has a solution over $K$, then it has a solution over $R$. Particularly, we prove the following. Let $R$ be as above and $a$, $b$, $c$ be units in $R$. Consider the equation $T^2_1-aT^2_2-bT^2_3+abT^2_4=c$. If it has a solution over $K$, then it has a solution over $R$. Similar results are proved for regular local rings, which are geometrically regular over a discrete valuation ring. These results extend result proved in [23] to the mixed characteristic case.

Source: Izvestiya Mathematics

DOI: 10.4213/im9661e

Author: Ivan Alexandrovich Panin

Institutions: Russian Academy of Sciences, St. Petersburg Department of Steklov Institute of Mathematics, Steklov Mathematical Institute

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: Ministry of Education and Science of the Russian Federation, Ministry of Science and Higher Education of the Russian Federation

Awards: G597338630, G6444257319, G8105343478, G92726623