Higher Tate traces of Chow motives (Q6629500)

Let \(Mot_F(\Lambda)\) denote the category of Chow motives over the field \(F\) with coefficient ring \(\Lambda\) and let \(Mot_F^{KS}(\Lambda)\) denote the full subcategory of \(Mot_F(\Lambda)\) generated by direct summands of Tate shifts of motives of geometrically split varieties satisfying the Rost nilpotence principle. In this paper, the authors define the notion of Tate trace of a motive \(M \in Mot_F^{KS}(\mathbb{Z}/p\mathbb{Z})\) as a pure Tate motive that is isomorphic to a pure Tate summand of maximal rank of \(M\). This turns out to be an important invariant to classify Chow motives (with coefficients in \(\mathbb{Z}/p\mathbb{Z}\)) of projective homogeneous varieties of \(p\)-inner semi-simple algebraic groups.\N\NThe main result of the paper is the following. Let \(Mot_F^{I}(\mathbb{Z}/p\mathbb{Z})\) denote the full subcategory of \(Mot_F^{KS}(\mathbb{Z}/p\mathbb{Z})\) generated by twists of upper motives of some irreducible geometrically split variety satisfying Rost nilpotence principle. The authors show that two motives \(M,N \in Mot_F^{I}(\mathbb{Z}/p\mathbb{Z})\) are isomorphic if and only if \(M_L\) and \(N_L\) have isomorphic Tate traces for all field extensions \(L/F\).\N\NAs an immediate corollary, one obtains a new proof of Vishik's criterion for motivic equivalence for quadrics. The authors also show many interesting applications such as proving a cancellation property in \(Mot_F^{I}(\mathbb{Z}/p\mathbb{Z})\) and relating motivic isomorphisms of two projective homogeneous varieties for inner semi-simple groups of same type to their Tits indices over field extensions.