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

Archita, M.; Preeti, R.

doi:10.1090/tran/8726

Rational equivalence on adjoint groups of type $D_n$ over fields of virtual cohomological dimension $2$
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by M. Archita and R. Preeti PDF

Trans. Amer. Math. Soc. 375 (2022), 7373-7384 Request permission

Abstract:

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

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