Between the genus and the $\Gamma$-genus of an integral quadratic $\Gamma$-form

Abstract

Let \Gamma be a finite group and (V,q) be a regular quadratic \Gamma-form defined over an integral domain $\mathcal{O}_S$ of a global function field (of odd characteristic). We use flat cohomology to classify the quadratic \Gamma-forms defined over $\mathcal{O}_S$ that are locally \Gamma-isomorphic for the flat topology to (V,q) and compare between the genus c(q) and the \Gamma-genus c_{\Gamma}(q) of q. We show that c_{\Gamma}(q) should not inject in c(q). The suggested obstruction arises from the failure of the Witt cancellation theorem for $\mathcal{O}_S$.