We propose a definition of the Chow group of a rigid tensor triangulated category. The basic idea is to allow “generalized” cycles, with nonintegral coefficients. The precise choice of relations is open to some fine-tuning. Hypothesis 1. Let K be an essentially small tensor triangulated category. Let us assume that its triangular spectrum in the sense of [1], Spc(K) = { P ⊂ K ∣∣P is prime}, is ...

A smooth intersection Y of two quadrics in $$\mathbb {P}^{2g+1}$$ has Hodge level 1. We show that such varieties have a multiplicative Chow–Künneth decomposition, the sense Shen–Vial. As consequence, certain tautological subring Chow ring powers injects into cohomology.

We introduce "Lawson homology groups" LrH2r+i(Xf2i) associated to an arbitrary projective algebraic variety X over an algebraically closed field k of characteristic p > 0 and a prime I ^ p. Our work is directly inspired by recent work of Blaine Lawson (cf. [5, 6]), consisting in part of an algebraization of Lawson's geometric ideas and analytic arguments. The Lawson homology group LQH^X, Z/) is...