Chow groups of tensor triangulated categoriesChow groups of tensor triangulated categories
Faculty of Sciences. Mathematics and Computer Science
Journal of pure and applied algebra. - Amsterdam
220(2016):4, p. 1343-1381
University of Antwerp
We recall P. Balmer's definition of tensor triangular Chow group for a tensor triangulated category K and explore some of its properties. We give a proof that for a suitably nice scheme X it recovers the usual notion of Chow group from algebraic geometry when we put K = D-perf (X) Furthermore, we identify a class of functors for which tensor triangular Chow groups behave functorially and show that (for suitably nice schemes) proper push-forward and fiat pull-back of algebraic cycles can be interpreted as being induced by the derived inverse and direct image functors between the bounded derived categories of the involved schemes. We also compute some examples for derived and stable categories from modular representation theory, where we obtain tensor triangular cycle groups with torsion coefficients. This illustrates our point of view that tensor triangular cycles are elements of a certain Grothendieck group, rather than Z-linear combinations of closed subspaces of some topological space. (C) 2015 Elsevier B.V. All rights reserved.