Johnson, FEA;
(2024)
Swan modules over Laurent polynomials.
Illinois Journal of Mathematics
, 68
(1)
45 -58.
10.1215/00192082-11081225.
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Abstract
Let Ω = P n , m ( Z [ C p ] ) and A = P n , m ( Z ) , where p is a positive integer and P n , m ( R ) is the R-algebra P n , m ( R ) = R [ t 1 , t − 1 1 , … , t n , t − 1 n ] ⊗ R R [ x 1 , … , x m ] . A Swan module is an extension module of the form 0 → I ( k ) → X → A ( k ) → 0 , where I is the kernel of the augmentation homomorphism ϵ : Ω → A . We show that, when p is prime, every such projective Swan module is free; this is false if p is not prime and n + m > 0 . The proof relies on the fact that when R is the ring of algebraic integers in Q ( ζ p ) and F p is the field with p elements, then the canonical homomorphism GL k ( P n , m ( R ) ) ↠ GL k ( P n , m ( F p ) ) is surjective for all k ≥ 1 .
| Type: | Article |
|---|---|
| Title: | Swan modules over Laurent polynomials |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1215/00192082-11081225 |
| Publisher version: | https://doi.org/10.1215/00192082-11081225 |
| Language: | English |
| Additional information: | Copyright 2024, the University of Illinois Urbana-Champaign. All rights reserved. Republished by permission of the publisher. www.dukeupress.edu |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10174674 |
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