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

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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 provedin [23] to the mixed characteristic case. 

About the authors

Ivan Alexandrovich Panin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Author for correspondence.
Email: paniniv@gmail.com
Doctor of physico-mathematical sciences

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