Symplectic \(K_ 2\) of Laurent polynomials, associated Kac-Moody groups and Witt rings
In this note, we will show the following theorems. Here F denotes an arbitrary field, \(F^*\) the multiplicative group of F, and \(F[\xi,\xi^{-1}]\) the ring of Laurent polynomials in \(\xi\) with coefficients in F. Let W(F) be the Witt ring over F with the maximal ideal I(F) consisting of classes with even rank. Then there are two exact sequences as follows: \[ 1\to I^ 3(F)\to K_ 2Sp(F)\to K_ 2(F)\to 1, \] \[ 1\to N(F)\to K_ 2(F)\to K_ 2Sp(F)\to I^ 2(F)\to 1, \] where \(N(F)=<\{u,-1\}| u\in F^*>\subseteq K_ 2(F)\) and N(F) is the 2-torsion part of \(K_ 2(F)\). We will study an \(F[\xi,\xi^{-1}]\)-version. As is well-known, the tame symbol induces \(K_ 2(F[\xi,\xi^{-1}])\simeq K_ 2(F)\oplus F^*.\) Then, we obtain the following. Theorem A. There are the following exact sequences: \[ 1\to I^ 3(F)\oplus I^ 2(F)\to K_ 2Sp(F[\xi,\xi^{-1}])\to K_ 2(F[\xi,\xi^{-1}])\to 1, \] \[ 1\to N(F)\oplus \{\pm 1\in F\}\to K_ 2(F[\xi,\xi^{-1}])\to K_ 2Sp(F[\xi,\xi^{-1}])\to I^ 2(F)\oplus I(F)\to 1. \] Theorem B. \(K_ 2Sp_{2n}(F[\xi,\xi^{-1}])\simeq K_ 2Sp(F)\oplus P(F)\) for all \(n\geq 1.\) To prove these two theorems, we will study the associated Kac-Moody groups. Let \(K_ 2(\tilde C_ n,F)\) be the \(K_ 2\)-group associated with a (universal) Kac-Moody group of type \(\tilde C_ n\). Then we obtain the following. Theorem C. \(K_ 2(\tilde C_ n,F)\simeq K_ 2Sp(F)\oplus I^ 2(F)\) for all \(n\geq 1\).
scientific article; zbMATH DE number 4202541