JournalsjemsVol. 24, No. 6pp. 2169–2189

Homogeneous spaces, algebraic KK-theory and cohomological dimension of fields

  • Diego Izquierdo

    Max-Planck-Institut für Mathematik, Bonn, Germany; Institut Polytechnique de Paris, Palaiseau, France
  • Giancarlo Lucchini Arteche

    Universidad de Chile, Santiago, Chile
Homogeneous spaces, algebraic $K$-theory and cohomological dimension of fields cover
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Abstract

Let qq be a non-negative integer. We prove that a perfect field KK has cohomological dimension at most q+1q + 1 if, and only if, for any finite extension LL of KK and for any homogeneous space ZZ under a smooth linear connected algebraic group over LL, the qq-th Milnor KK-theory group of LL is spanned by the images of the norms coming from finite extensions of LL over which ZZ has a rational point. We also prove a variant of this result for imperfect fields.

Cite this article

Diego Izquierdo, Giancarlo Lucchini Arteche, Homogeneous spaces, algebraic KK-theory and cohomological dimension of fields. J. Eur. Math. Soc. 24 (2022), no. 6, pp. 2169–2189

DOI 10.4171/JEMS/1129