Twists of symmetric bundles

Cassou-Noguès, Ph. and Erez, B. and Taylor, M.J. (2007) Twists of symmetric bundles. Proceedings of the London Mathematical Society. ISSN 1460-244x (Unpublished)

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Abstract

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. Fröhlich. This arises from twisting by a cocycle obtained from an orthogonal representation. A simple important example of this twisting procedure is the bilinear trace form of an étale algebra, which is obtained by twisting the standard/sum-of-squares form by the orthogonal representation attached to the algebra. 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 Fröhlich's formula holds. Namely let $(X, G)$ be a torsor with quotient $Y$, let $E$ be a symmetric bundle over $Y$, let $\rho : G \rightarrow {\bf O}(E)$ be an orthogonal representation and let $E_{\rho, X}$ be the corresponding twist of $E$. Then we verify up to degree $3$ that the formula $w_t (E_{\rho,X}) sp_t (\rho ) = w_t (E) w_t (\rho)$ holds. Here $sp_t (\rho )$ and $w_t (\rho )$ are respectively the spinor invariant and the Stiefel--Whitney class of $\rho$. The case of genuinely tamely ramified actions is geometrically more involved and leads us to introduce an invariant of ramification, which in a sense gives a decomposition in terms of representations of the inertia groups of the invariant introduced by Serre for curves.The comparison result in the tamely ramified case proceeds by reduction to the case of a torsor. The reduction is carried out by means of a partial normalisation procedure, which we had introduced in a previous paper. An important lemma of Esnault, Kahn and Viehweg allows us to express the difference between the invariants of bundles before and after the normalisation procedure in terms of Chern classes of certain sub-bundles. As noted elsewhere, this result can be best understood in the context of symmetric complexes and their invariants. Our results are new even for bundles over curves and they allow us to weaken the regularity assumptions that we had to impose in previous work of ours.

Item Type:	Article
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 11 Number theory MSC 2010, the AMS's Mathematics Subject Classification > 14 Algebraic geometry
Depositing User:	Ms Lucy van Russelt
Date Deposited:	21 May 2007
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/801

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