Contou-Carrère symbol
From HandWiki
Revision as of 13:39, 21 December 2020 by imported>Jworkorg (over-write)
In mathematics, the Contou-Carrère symbol 〈a,b〉 is a Steinberg symbol defined on pairs of invertible elements of the ring of Laurent power series over an Artinian ring k, taking values in the group of units of k. It was introduced by (Contou-Carrère 1994).
Definition
If k is an Artinian local ring, then any invertible formal Laurent series a with coefficients in k can be written uniquely as
- [math]\displaystyle{ a=a_0t^{w(a)}\prod_{i\ne 0}(1-a_it^i) }[/math]
where w(a) is an integer, the elements ai are in k, and are in m if i is negative, and is a unit if i = 0.
The Contou-Carrère symbol 〈a,b〉 of a and b is defined to be
- [math]\displaystyle{ \langle a,b\rangle=(-1)^{w(a)w(b)}\frac{a_0^{w(b)}\prod_{i,j\gt 0}(1-a_i^{j/(i,j)}b_{-j}^{i/(i,j)})^{(i,j)}} {b_0^{w(a)}\prod_{i,j\gt 0}(1-b_i^{j/(i,j)}a_{-j}^{i/(i,j)})^{(i,j)}} }[/math]
References
- Contou-Carrère, Carlos (1994), "Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole modéré" (in French), Comptes Rendus de l'Académie des Sciences, Série I 318 (8): 743–746
![]() | Original source: https://en.wikipedia.org/wiki/Contou-Carrère symbol.
Read more |