Additive higher Chow groups of schemes

Amalendu Krishna 1 , 2  and Marc Levine 1 , 2
  • 1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India. e-mail: amal@math.tifr.res.in
  • 2 Northeastern University, Department of Mathematics, 360 Huntington Avenue, Boston, MA 02115, USA. e-mail: marc@neu.edu

Abstract

We show how to make the additive Chow groups of Bloch-Esnault, Rülling and Park into a graded module for Bloch's higher Chow groups, in the case of a smooth projective variety over a field. This yields a projective bundle formula as well as a blow-up formula for the additive Chow groups of a smooth projective variety.

In case the base field k admits resolution of singularities, these properties allow us to apply the technique of Guillén and Navarro Aznar to define the additive Chow groups “with log poles at infinity” for an arbitrary finite-type k-scheme X. This theory has all the usual properties of a Borel-Moore theory on finite type k-schemes: it is covariantly functorial for projective morphisms, contravariantly functorial for morphisms of smooth schemes, and has a projective bundle formula, homotopy property, and Mayer-Vietoris and localization sequences.

Finally, we show that the regulator map defined by Park from the additive Chow groups of 1-cycles to the modules of absolute Kähler differentials of an algebraically closed field of characteristic zero is surjective, giving evidence of a conjectured isomorphism between these two groups.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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