On the K-theory coniveau epimorphism for products of Severi–Brauer varieties

Abstract

For X a product of Severi-Brauer varieties, we conjecture: if the Chow ring of X is generated by Chern classes, then the canonical epimorphism from the Chow ring of X to the graded ring associated to the coniveau filtration of the Grothendieck ring of X is an isomorphism. We show this conjecture is equivalent to: if G is a split semisimple algebraic group of type AC, B is a Borel subgroup of G and E is a standard generic G-torsor, then the canonical epimorphism from the Chow ring of E/B to the graded ring associated with the coniveau filtration of the Grothendieck ring of E/B is an isomorphism. In certain cases we verify this conjecture. Notation and Conventions. We fix a field k throughout. All of our objects are defined over k unless stated otherwise. Sometimes we use k as an index when no confusion will occur. For any field F , we fix an algebraic closure F . A variety X is a separated scheme of finite type over a field. Let X = X1 × · · · ×Xr be a product of varieties with projections πi : X → Xi. Let F1, ...,Fr be sheaves of modules on X1, ..., Xr. We use F1 · · · Fr for the external product π∗ 1F1⊗ · · ·⊗π∗ rFr. For a ring R with a Z-indexed descending filtration F • ν , (e.g. ν = γ or τ as in Section 2), we write grνR for the corresponding quotient F i ν/F i+1 ν . We write grνR = ⊕ i∈Z gr i νR for the associated graded ring. A semisimple algebraic group G is of type AC if its Dynkin diagram is a union of diagrams of type A and type C. Similarly a semisimple group G is of type AA if its Dynkin diagram is a union of diagrams of type A. For an index set I, two elements i, j ∈ I, we write δij for the function which is 0 when i 6= j and 1 if i = j. Given two r-tuples of integers, say I, J , we write I < J if the ith component of I is less than the ith component of J for any 1 ≤ i ≤ r.