Abstract
Let R be a semilocal principal ideal domain. Two algebraic objects over R in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of R and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over (not necessarily commutative) R-orders is always a finite power of 2, and under further assumptions, e.g., that the order is hereditary, this number is 1. The same result is also shown for related objects, e.g., systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings.
| Original language | English |
|---|---|
| Pages (from-to) | 7999-8035 |
| Number of pages | 37 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 369 |
| Issue number | 11 |
| DOIs | |
| State | Published - 2017 |
| Externally published | Yes |
Bibliographical note
Funding Information:Received by the editors April 7, 2015 and, in revised form, December 15, 2015. 2010 Mathematics Subject Classification. Primary 11E39, 11E41, 16H10. Key words and phrases. Quadratic form, hermitian form, algebraic patching, weak approximation, genus, order, hereditary order, sesquilinear form, hermitian category. The second-named author performed this research at EPFL, the Hebrew University of Jerusalem and the University of British Columbia (in this order), where he was supported by an SNFS grant #IZK0Z2 151061, an ERC grant #226135, and the UBC Mathematics Department, respectively.
Publisher Copyright:
© 2017 Amerian Mathematial Soiety.
Keywords
- Algebraic patching
- Genus
- Hereditary order
- Hermitian category
- Hermitian form
- Order
- Quadratic form
- Sesquilinear form
- Weak approximation
ASJC Scopus subject areas
- Mathematics (all)
- Applied Mathematics