Rational equivalence on adjoint groups of type 𝐷_{𝑛} over fields of virtual cohomological dimension 2

Abstract

Let F F be a field of characteristic different from 2 2 and such that virtual cohomological dimension of F F is 2 2 . Let G G be a semisimple classical adjoint group of type D n D_n defined over F F . We show that G ( F ) / R = 0 G(F) / R = 0 , where R R denotes rational equivalence on G ( F ) G(F) . The analogous result for groups of type 1 A n {}^1A_n and B n B_n has been proved by Merkurjev, for groups of type 2 A 2 n {}^2A_{2n} by Voskresenskii-Klyachko and for general groups of type 2 A n {}^2A_n and C n C_n by Kulshreshta-Parimala. Combining the main theorem of this paper with the above mentioned results, we have G ( F ) / R G(F) / R is trivial, for any semisimple adjoint classical group G G defined over F F .