Picard groups of Witt rings
For an arbitrary Witt ring R it is shown that the Picard group Pic(R) is isomorphic to the ideal class group. Exact sequences of K-theory then may be used to improve earlier results on the ideal class group (which required R to have only finitely many orderings). Analysis of one such sequence shows that if s is the reduced stability index of R and e is the exponent of Pic(R) then \(e=\max \{1,2^{s-2}\}\) whenever e or s is finite. In particular \(Pic(R)=1\) (that is, all invertible ideals are principal) if and only if \(s\leq 2\). A complete computation of Pic(R) is also given for R with only finitely many orderings. This involves K- theory and Marshall's classification of finitely generated reduced Witt rings.
scientific article; zbMATH DE number 4150277