Ketu and the second invariant of a quadratic space
Let \(A\) be a commutative ring, in which 2 is invertible, and let \(W(A)\) denote the Witt ring of \(A\) with fundamental ideal \(I(A)\). The Milnor \(K\)-group \(K_ 2(A)\) carries a natural involution \(\alpha\to \alpha^ t\) induced from the involution on the Steinberg group \(St(A)\), which interchanges the indices of the generators. \textit{C. H. Giffen} [J. Algebra 44, 434-456 (1977; Zbl 0354.16012)], defined an invariant \(c_ 2: I^ 2(A)\to K_ 2(A)/ \text{Tr}(K_ 2 (A))\) where \(\text{Tr}\colon K_ 2(A)\to K_ 2(A)\) sends \(\alpha\) to \(\alpha- \alpha^ t\). The authors show that the Giffen-invariant is related to the Clifford- invariant \(Cliff: I^ 2(A)\to H^ 2_{\text{ét}} (\text{Spec } A, \mu_ 2^{\otimes_ 2})\) via the Chern-class map \(c_{22} \colon K_ 2(A)\to H^ 2_{\text{ét}} (\text{Spec } A, \mu_ 2^{\otimes_ 2})\) defined by Soulé and Shekhtman.
scientific article; zbMATH DE number 549705