Chow groups of tensor triangulated categories

Sebastian Klein
Tuesday, July 23, 2013 - from 17:30 to 17:55
Some interesting invariants of an algebraic variey $X$ are provided by its Chow groups, which are defined as the free abelian groups on irreducible subvarieties of $X$ of a fixed codimension, modulo rational equivalence. Following P. Balmer, we define and study generalized Chow groups of tensor triangulated categories $\mathcal{T}$, such that for $\mathcal{T} = \mathrm{D}^{\mathrm{perf}}(X)$, the derived category of perfect complexes on $X$, they coincide with the usual Chow groups of $X$ in the non-singular case. Whereas usual cycles on $X$ are $\mathbb{Z}$-linear combinations of subvarieties, for more general $\mathcal{T}$ they have coefficients in Grothendieck groups of local categories associated to $\mathcal{T}$.