Invariants of degree 3 and torsion in the Chow group of a versal flag

Abstract

Abstract. We prove that the group of normalized cohomological invariants of degree 3 modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group G is isomorphic to the torsion part of the Chow group of codimension 2 cycles of the respective versal G-flag. In particular, if G is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of G. As an application, we construct nontrivial cohomological classes for indecomposable central simple algebras.

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

“…Let H be a split semisimple group over F . In [9] a semi-decomposable invariant of H was introduced. We write Sdec(H) for the subgroup of semi-decomposable invaraints.…”

“…The first and, so far, the only example of nontrivial semi-decomposable invariants comes from a semisimple group SO 4 ≃ (SL 2 × SL 2 )/µ, where µ = {(λ 1 , λ 2 ) ∈ µ 2 × µ 2 | λ 1 λ 2 = 1}; see [9,Example 3.1]. Indeed, this invariant is given by φ := a b, c → (a) ∪ [b, c], where φ is a 4-dimensional quadratic form with trivial discriminant over a field extension K/F and [b, c] is the class of a quaternion algebra in the Brauer group Br(K); see [6,Example 20.3].…”

“…Recently, a new type of degree 3 invariant of a split semisimple group, called a semidecomposable invariant was introduced by Merkurjev-Neshitov-Zainoulline [9]. This invariant is locally decomposable, so any decomposable invariant is semi-decomposable.…”

“…This invariant is locally decomposable, so any decomposable invariant is semi-decomposable. In their paper, authors established a relation between nontrivial semi-decomposable invariants and the torsion in the codimension 2 Chow groups of a generic flag variety 2 S. BAEK and showed that there is no nontrivial semi-decomposable invariant of a split simple group in [9,Theorem].…”

“…In the proof of the Theorem we compute the torsion in codimension 2 cycles of a product of Severi-Brauer varieties associated to quaternion algebras and relate it to the torsion of the generic variety of (SL 2 ) n /µ. We then combine it with the group of indecomposable invariants of (SL 2 ) n /µ and (GL 2 ) n /µ using the short exact sequence in [9,Theorem] and its extension to reductive groups (Proposition 2.1).…”