Tangents to Chow groups: on a question of Green-Griffiths. (English) Zbl 1398.14010

Dribus, Benjamin F.; Hoffman, J. W.; Yang, Sen

Boll. Unione Mat. Ital. 11, No. 2, 205-244 (2018).

From Summary and Introduction: “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 M. Green and P. Griffiths [On the tangent space to the space of algebraic cycles on a smooth algebraic variety. Princeton, NJ: Princeton University Press (2005; Zbl 1076.14016)] 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 …In the present paper, we demonstrate that a satisfactory answer to this question requires reframing the entire picture in terms of Bass-Thomason nonconnective \(K\)-theory, the negative cyclic homology of C. Weibel [Proc. Am. Math. Soc. 124, No. 6, 1655–1662 (1996; Zbl 0855.19002)] and Keller, and the relative Chern character as treated by G. Cortiñas et al. [Ann. Math. (2) 167, No. 2, 549–573 (2008; Zbl 1191.19003); Math. Ann. 344, No. 4, 891–922 (2009; Zbl 1189.19002)].”
Recall that Green and Griffiths [loc. cit.] used Milnor \(K\)-groups instead of Quillen \(K\)-groups.
The approach of the authors is to develop what they call a coniveau machine for an infinitesimal thickening \(X_A\) of a nonsingular variety \(X\). It maps a Bloch-Gersten-Quillen type complex of sheaves on \(X\) in several stages to a similar complex on \(X\) involving cyclic homology on \(X_A\).
This requires a lot of machinery. There are many helpful discussions of relations with the literature.

Reviewer: Wilberd van der Kallen (Utrecht)

MSC:

14C15	(Equivariant) Chow groups and rings; motives
19E15	Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
14C25	Algebraic cycles
19D55	\(K\)-theory and homology; cyclic homology and cohomology
19L10	Riemann-Roch theorems, Chern characters
14B15	Local cohomology and algebraic geometry
55S25	\(K\)-theory operations and generalized cohomology operations in algebraic topology

Keywords:

Chow groups; algebraic \(K\)-theory; algebraic cycles; cyclic homology; Chern character; local cohomology; \(K\)-theory operations

Citations:

Zbl 1076.14016; Zbl 0855.19002; Zbl 1191.19003; Zbl 1189.19002

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