The frame of smashing tensor-ideals

We prove that every flat tensor-idempotent in the module category Mod- of a tensor-triangulated category comes from a unique smashing ideal in . We deduce that the lattice of smashing ideals forms a frame.

Separable monoids in Dqc (X)

Suppose{({\mathscr{T}},\otimes,\mathds{1})}is a tensor triangulated category. In a number of recent articles Balmer defines and explores the notion of “separable tt-rings” in{{\mathscr{T}}}(in this paper we will call them “separable monoids”). The main result of this article is that, if{{\mathscr{T}}}is the derived quasicoherent category of a noetherian schemeX, then the only separable monoids are the pushforwards by étale maps of smashing Bousfield localizations of the structure sheaf.

Mixed motives over k[t]/(tm+1)

For a perfect field k, we use the techniques of Bondal-Kapranov and Hanamura to construct a tensor triangulated category of mixed motives over the truncated polynomial ring k[t]/(tm+1). The extension groups in this category are given by Bloch's higher Chow groups and the additive higher Chow groups. The main new ingredient is the moving lemma for additive higher Chow groups by the authors and its refinements.