Chow groups of tensor triangulated categories

TL;DR

This paper explores tensor triangular Chow groups, showing they generalize classical Chow groups for schemes, behave functorially under certain conditions, and can be computed in derived categories from algebraic geometry and modular representation theory.

Contribution

It demonstrates that tensor triangular Chow groups extend classical notions, behave functorially, and can be explicitly computed in various derived and stable categories.

Findings

Recovers classical Chow groups for schemes from tensor triangular Chow groups.

Shows functoriality of tensor triangular Chow groups under specific functors.

Provides explicit computations in derived categories from algebraic geometry and modular representation theory.

Abstract

We recall P. Balmer's definition of tensor triangular Chow group for a tensor triangulated category $\mathcal{K}$ 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 $\mathcal{K} = \mathrm{D^{perf}}(X)$. 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 flat 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 $\mathbb{Z}$-linear combinations of closed subspaces of some topological space.