Burnside Form Rings and the K-Theory of Forms
The Burnside form ring Z is the initial object and tensor unit in the category of form rings; therefore, its Grothendieck-Witt ring GW0(Z), since it acts on GWi(R; ) for any i ≥ 0 and any form ring (R; A), is of fundamental importance in the study of the K-theory of forms. We show that GW0(Z) is iso...
Saved in:
| Main Author: | |
|---|---|
| Format: | Dissertation |
| Language: | English |
| Published: |
ProQuest Dissertations & Theses
01-01-2021
|
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | The Burnside form ring Z is the initial object and tensor unit in the category of form rings; therefore, its Grothendieck-Witt ring GW0(Z), since it acts on GWi(R; ) for any i ≥ 0 and any form ring (R; A), is of fundamental importance in the study of the K-theory of forms. We show that GW0(Z) is isomorphic to Z3 as an abelian group, and also give its ring structure. Using an extension of scalars construction defined by a universal property, one can define a Burnside form ring R for any commutative ring R. After calculating GW0(Z), the remainder of the thesis calculates GW0(R) when R is a finite field. Along the way, we calculate GW0(R) for any ring R with 2 invertible and finitely generated projective R-modules free, and we define a determinant map which generalises the classical determinant map on symmetric bilinear forms. |
|---|