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 ⊗RL 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.
These videos were produced by Syracuse University Department of Mathematics as part of ICRA 2016.
