Chow groups of products of Severi-Brauer varieties and invariants of degree 3

Abstract

We study the semi-decomposable invariants of a split semisimple group and their extension to a split reductive group by using the torsion in the codimension 2 Chow groups of a product of Severi-Brauer varieties. In particular, for any n ≥ 2 we completely determine the degree 3 invariants of a split semisimple group, the quotient of (SL 2 ) n by its maximal central subgroup, as well as of the corresponding split reductive group. We also provide an example illustrating that a modification of our method can be applied to find the semi-decomposable invariants of a split semisimple group of type A.

The $K$-Theory of Versal Flags and Cohomological Invariants of Degree 3

“…For d = 3 it coincides with the group of decomposable invariants for all simple groups [15]. It was also shown that these groups are different for G = SO 4 [15,Ex.3.1] and for some semisimple groups of type A (see [1]). The relationships between the subgroups I W ⊆ I W sc ∩ R(T ) ⊆ I W sc and the groups of cohomological invariants are explained in Section 5.…”

“…Note that by definition we have inclusions of abelian groups I W ⊆ I W sc ∩R(T ) ⊆ I W sc which all coincide if taken with Q-coefficients. However, there are examples of semisimple groups (see [15,Ex.3.1] and [1]) where both quotients (I W sc ∩ R(T ))/I W and I W sc /(I W sc ∩ R(T )) are non-trivial. Our Theorem 3.4 provides a complete list of generators (Definition 3.2) of the ideal I W sc ∩ R(T ) assuming the root system of G sc satisfies the generalized flatness condition (see Definition 2.9).…”

“…Following Merkurjev [14], an invariant is called decomposable if it is given by a cup-product of invariants of smaller degrees; the factor group of (normalized) invariants modulo decomposable is called the group of indecomposable invariants. For d = 3 the latter (denoted by Inv 3 ind (G)) has been computed for all simple split groups in [14] and [2]; for some semi-simple groups of type A in [13] and [1]; for adjoint semisimple groups in [12].…”

Degree three invariants for semisimple groups of types $B$, $C$, and $D$

“…The quotient group Inv 3 (G) norm / Inv 3 (G) dec is called the group of indecomposable invariants and is denoted by Inv 3 (G) ind . This group has been completely determined for all split simple groups in [10], [19], [4] and for some semisimple groups in [17], [1], [2], and [15].…”

Rost Multipliers of Lifted Kronecker Tensor Products

“…Degree three cohomological invariants of semisimple linear algebraic groups as given in [GMS] have been recently studied and computed in [Ba17], [GQ08], [Me16], [MNZ] and others. Rost multipliers played an important role in those computations.…”