Holmstrom Additive Chow groups

Additive Chow groups

AG (Algebraic geometry)

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Additive Chow groups

Key authors: Bloch, Esnault, Rülling, Park, Krishna, Levine.

Krishna and Levine: Additive Chow groups of schemes

Bloch: Additive Chow groups and algebraic cycles

Bloch and Esnault: An additive version of higher Chow groups, The additive dilogarithm

Various articles of Jinhyun Park

Kay Rülling’s thesis

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Additive Chow groups

Classical (higher) Chow groups do not distinguish between a scheme and the associated reduced scheme. Additive (higher) Chow groups is a cohomology theory which should give a better notion of motivic cohomology for non-reduced schemes (for example truncated polynomial algebras such as the dual numbers). In particular, one hopes for an Atiyah-Hirzebruch spectral sequence from additive higher Chow groups to algebraic K-theory.

Remark: As far as I can see, additive Chow groups have nothing to do with Additive K-theory

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Additive Chow groups

arXiv:0909.3155 Moving Lemma for additive Chow groups and applications from arXiv Front: math.AG by Amalendu Krishna, Jinhyun Park We prove moving lemma for additive higher Chow groups of smooth projective varieties. As applications, we prove the very general contravariance property of additive higher Chow groups. Using the moving lemma, we establish the structure of graded-commutative differential graded algebra (CDGA) on these groups.

arXiv:1208.6455 Somekawa’s $K$-groups and additive higher Chow groups from arXiv Front: math.KT by Toshiro Hiranouchi We introduce the Milnor type $K$-group attached to some algebraic groups including Witt groups over a perfect field as an extension of Somekawa’s $K$-group. We give a description of this $K$-group associated to the additive group and the multiplicative group by the space of the absolute Kähler differentials, and relate also our Somekawa $K$-group for the additive group and the Jacobian variety of a curve with a complex determined by the residue maps and the trace maps. The same arguments work for the Mackey product of the additive higher Chow group and the higher Chow group for a scheme. This gives vanishing results of additive Chow groups on zero-cycles.

nLab page on Additive Chow groups