Twists of symmetric bundles

We establish comparison results between the Hasse-Witt invariants w_t(E) of a symmetric bundle E over a scheme and the invariants of one of its twists E_{\alpha}. For general twists we describe the difference between w_t(E) and w_t(E_{\alpha}) up to terms of degree 3. Next we consider a special kind of twist, which has been studied by A. Fr\"ohlich. This arises from twisting by a cocycle obtained from an orthogonal representation. We show how to explicitly describe the twist for representations arising from very general tame actions. This involves the ``square root of the inverse different'' which Serre, Esnault, Kahn, Viehweg and ourselves had studied before. For torsors we show that, in our geometric set-up, Jardine's generalization of Frohlich's formula holds. The case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce a ramification invariant which generalises in higher dimension the invariant introduced by Serre for curves.