Given an object X in a compactly generated tensor triangulated category C (such as the derived category of a ring, or the stable homotopy category), the Bousfield class of X is the collection of objects that tensor with X to zero.  The set of Bousfield classes forms a lattice, called the Bousfield lattice BL(C).  First, we look at examples of when a functor F: C--> D induces a lattice map BL(C) --> BL(D), and will describe several lattice quotients and lattice isomorphisms.  Second, we will focus on homological algebra: a ring map f: R --> S induces, via extension of scalars, a functor D(R) --> D(S), and this induces a map on Bousfield lattices.  Third, we specialize to specific maps between some interesting non-Noetherian rings.

Slides from this talk are available here. 

A map of rings induces adjoint functors between derived categories of modules.  We investigate the induced operations on Bousfield lattices, focusing on a particular setting in which these operations give nice quotient lattices.  We apply this theory to a toy example involving maps between several non-Noetherian rings, and establish a splitting of the Bousfield lattices into a product.  We also apply the theory to the case of a surjection onto a Noetherian ring, and are able to pull back some of the Noetherian structure to the non-Noetherian setting.

Given W and X in a tensor triangulated category, we say W is X-acyclic if W tensors with X to zero.  Two objects X and Y are called Bousfield equivalent if they have the same acyclics.  In this talk we give a (non-constructive) proof that there exist objects in the derived category of graded modules over a certain graded non-Noetherian ring Lambda that are not Bousfield equivalent to any module.  This contrasts with the Noetherian case, and has consequences for subcategory classification.