Tag - Homological algebra

Let R be a commutative Noetherian ring. Denote by D^-(R) the derived category of cochain complexes X of finitely generated R-modules with Hⁱ(X)=0 for i>>0. Then D^-(R) has a structure of a tensor triangulated category with tensor product ⊗_R^L and unit R. In this series of lectures, we study thick tensor ideals of D^-(R), i.e., thick subcategories closed under the tensor action by each object in D^-(R), and investigate the Balmer spectrum Spc D^-(R) of D^-(R), i.e., the set of prime thick tensor ideals of D^-(R). Here is a plan.
  •   We give a complete classification of the (co)compactly generated thick tensor ideals of D^-(R), establishing a generalized version of the Hopkins--Neeman smash nilpotence theorem.
  •   We construct a pair of maps between the Balmer spectrum Spc D^-(R) and the prime spectrum Spec R, and explore their topological properties.
  •   We compare several classes of thick tensor ideals of D^-(R), relating them to specialization-closed subsets of Spec R and Thomason subsets of Spc D^-(R).

If time permits, I would like to talk about the case where R is a discrete valuation ring. My lectures are based on joint work with Hiroki Matsui.