We give a decomposition of the Chow motive of an isotropic projective homogeneous variety of a semisimple algebraic group in terms of twisted motives of simpler projective homogeneous varieties. As an application, we prove a generalization of Rost's nilpotence theorem.where M(Y )(1) = M(Y ) ⊗ Z(1) and d = dim X . This result has been generalized by
Given an algebraic group G defined over a (not necessarily algebraically closed) field F and a commutative ring R we associate the subcategory M(G, R) of the category of Chow motives with coefficients in R, that is, the Tate pseudo-abelian closure of the category of motives of projective homogeneous G-varieties. We show that M(G, R) is a symmetric tensor category, i.e., the motive of the product of two projective homogeneous G-varieties is a direct sum of twisted motives of projective homogeneous G-varieties. We also study the problem of uniqueness of a direct sum decomposition of objects in M(G, R). We prove that the KrullSchmidt theorem holds in many cases.
Let K be a 2-dimensional global field of characteristic = 2, and let V be a divisorial set of places of K. We show that for a given n 5, the set of K-isomorphism classes of spinor groups G = Spin n (q) of nondegenerate n-dimensional quadratic forms over K that have good reduction at all v ∈ V , is finite. This result yields some other finiteness properties, such as the finiteness of the genus gen K (G) and the properness of the global-to-local map in Galois cohomology. The proof relies on the finiteness of the unramified cohomology groups H i (K, µ2)V for i 1 established in the paper. The results for spinor groups are then extended to some unitary groups and to groups of type G2.
Let D be a finite-dimensional central division algebra over a field K . We define the genus gen(D) of D to be the collection of classes [D ] ∈ Br(K ), where D is a central division K -algebra having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K . This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273-293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic Kgroups having the same isomorphism/isogeny classes of maximal K -tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups.Keywords Division algebra · Maximal field · Brauer group · Linear algebraic group · Maximal torus Communicated by Efim Zelmanov.
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