Lattices over cyclic groups and Noether settings
| Autor: | Esther Beneish, Nick Ramsey |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Discrete mathematics
Algebra and Number Theory 010102 general mathematics Field (mathematics) Cyclic group 01 natural sciences Prime (order theory) Combinatorics Integer 0103 physical sciences Order (group theory) Maximal ideal 010307 mathematical physics Ideal (ring theory) 0101 mathematics Primitive root modulo n Mathematics |
| Zdroj: | Journal of Algebra. 452:212-226 |
| ISSN: | 0021-8693 |
| DOI: | 10.1016/j.jalgebra.2016.01.005 |
| Popis: | For each positive integer m, let ω m be a primitive mth root of 1. Given a prime p, let a be an integer which is a primitive root of 1 mod p . Let I p , m be the ideal in Z [ ω m ] generated by p and ω m − a . Let G be a cyclic group of order n such that the flasque class of any maximal projective ideal in Z G contains a maximal ideal U, of index p for some prime p such that n divides p − 1 , and such that, I p , n is principal. It is not known whether the number of groups satisfying this property is finite or infinite. We show that if F is a field of characteristic zero containing all nth roots of 1, then for any Z G -lattice M the fixed subfield of F ( M ) is stably rational over F. If F also contains primitive mth roots of 1 for all natural numbers m, then for any finite G-module T the fixed subfield of the Noether setting of the group G ′ = T ⋊ G is stably rational over F. |
| Databáze: | OpenAIRE |
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