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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.

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Notes

  1. In [10] this condition is called strict effaceability (definition 5.1.8, page 28) but we drop the adjective “strict” since this is the only such property used here.

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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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Authors and Affiliations

  1. Mathematics Department, William Carey University, Hattiesburg, MS, USA

    Benjamin F. Dribus

  2. Mathematics Department, Louisiana State University, Baton Rouge, LA, USA

    J. W. Hoffman

  3. Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

    Sen Yang

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  1. Benjamin F. Dribus
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  2. J. W. Hoffman
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Correspondence to J. W. Hoffman.

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Dribus, B.F., Hoffman, J.W. & Yang, S. Tangents to Chow groups: on a question of Green–Griffiths. Boll Unione Mat Ital 11, 205–244 (2018). https://doi.org/10.1007/s40574-017-0123-3

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