Abstract
Let R be a semilocal Dedekind domain with fraction field F. It is shown that two hereditary R-orders in central simple F-algebras that become isomorphic after tensoring with F and with some faithfully flat etale R-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for Hermitian forms over hereditary R-orders with involution. The results can be restated by means of etale cohomology and can be viewed as variations of the Grothendieck-Serre conjecture on principal homogeneous spaces of reductive group schemes. The relationship with Bruhat-Tits theory is also discussed.
| Original language | English |
|---|---|
| Pages (from-to) | 573-584 |
| Number of pages | 12 |
| Journal | St. Petersburg Mathematical Journal |
| Volume | 31 |
| Issue number | 4 |
| DOIs | |
| State | Published - 2020 |
Bibliographical note
Funding Information:2010 Mathematics Subject Classification. 16H10, 16W10, 11E57, 11E72. Key words and phrases. Hereditary order, maximal order, Dedekind domain, group scheme, reductive group, involution, central simple algebra. This research was supported by a Swiss National Science Foundation grant #200021 163188.
Publisher Copyright:
© 2020 American Mathematical Society.
Keywords
- Central simple algebra
- Dedekind domain
- Group scheme
- Hereditary order
- Involution
- Maximal order
- Reductive group
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
- Applied Mathematics