On factorizations of generically étale double coverings (Q2028019)

Let \(X\) be a locally Noetherian scheme, \(B\) a quadratic \({\mathcal O}_X\)-algebra, \(Y =\mathrm{Spec}\ B\) and \(p: Y \rightarrow X\) a double covering. The authors define a genus of \(p\) as the class of an etale double covering \(y: Z \rightarrow Y\) with the property locally, \(q\) can already be defined over the base scheme \(X\), and that the canonical involution of the double covering \(p\) can be lifted to an involution of \(Z\). The genera of \(p\) form a group Gen\((Y|X)\). When specialized to the base scheme \(X = {\mathbb Z}\), the authors recover the genera of quadratic number fields (in the strict sense). They also give an interpretation of Hilbert's theory of genera of quadratic extensions of \({\mathbb Q}(i)\) in terms of their abstract genus theory.