Equivariant pretheories and invariants of torsors

TL;DR

This paper introduces equivariant pretheories, generalizes invariants of G-torsors, and extends spectral sequences, providing new tools for understanding torsors via algebraic and cohomological invariants.

Contribution

It defines equivariant pretheories with coefficients in Rost cycle modules and generalizes the Karpenko-Merkurjev theorem for G-torsors.

Findings

Introduces equivariant pretheories including Chow groups, K-theory, and cobordism.

Provides a spectral sequence for equivariant (co)homology theories.

Associates graded rings as invariants of G-torsors, encoding motivic J-invariant and Tits algebra indexes.

Abstract

In the present paper we introduce and study the notion of an equivariant pretheory: basic examples include equivariant Chow groups, equivariant K-theory and equivariant algebraic cobordism. To extend this set of examples we define an equivariant (co)homology theory with coefficients in a Rost cycle module and provide a version of Merkurjev's (equivariant K-theory) spectral sequence for such a theory. As an application we generalize the theorem of Karpenko-Merkurjev on G-torsors and rational cycles; to every G-torsor E and a G-equivariant pretheory we associate a graded ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information concerning the motivic J-invariant of E and in the case of Grothendieck's K_0 -- indexes of the respective Tits algebras.