Tangents to Chow groups: on a question of Green–Griffiths

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Abstract

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch–Gersten–Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their “tangent sequences,” expressed in terms of absolute Kähler differentials. More generally, we place Green and Griffiths’ concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective K-theory, negative cyclic homology, and the relative algebraic Chern character.

Keywords

Chow groups Algebraic K-theory Algebraic cycles Cyclic homology Chern character Local cohomology K-theory operations 

Mathematics Subject Classification

14C15 19E15 14C25 19D55 19L10 14B15 55S25 

Notes

Acknowledgements

We would like to thank Marco Schlichting for his generous assistance in this investigation. J. W. Hoffman was supported in part by NSA Grant 115-60-5012 and NSF Grant OISE-1318015.

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Copyright information

© Unione Matematica Italiana 2017

Authors and Affiliations

  1. 1.Mathematics DepartmentWilliam Carey UniversityHattiesburgUSA
  2. 2.Mathematics DepartmentLouisiana State UniversityBaton RougeUSA
  3. 3.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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