Bousfield lattices and derived categories

John Palmieri — Wed, 12 Dec 2007 18:46:41 +0000

Let R be a commutative ring, and look at its unbounded derived category from the viewpoint of stable homotopy theory. In particular, we can define its Bousfield lattice: first, two objects X and Y are Bousfield equivalent if . This defines an equivalence relation on the objects of ; the equivalence class of an object X is called its Bousfield class and is written . We can define a partial ordering on these Bousfield classes, by setting if and only if . (Therefore, is the largest equivalence class and is the smallest.) The Bousfield lattice is the resulting partially ordered “set”.

Question 1. Is the Bousfield lattice a set?

It is known to be a set if the ring R is countable or noetherian, but not in general.

Question 2. Is every object in Bousfield equivalent to a module?

This is known in the noetherian case. Indeed, the noetherian case is pretty well understood, by work of Amnon Neeman: if R is noetherian, then is isomorphic to the lattice of subsets of Spec(R). For any prime ideal in R, define an R-module to be ; then any object X in the derived category is Bousfield equivalent to , which in turn is Bousfield equivalent to , where supp(X) is the set of primes for which .

Question 3. What does the Bousfield lattice look like for non-noetherian rings?

This is likely to be quite complicated, but perhaps some examples could be understood. Neeman has shown that the Bousfield lattice for the ring has cardinality at least . Note that this ring has exactly one prime ideal, so the lattice of subsets of Spec has two elements, and so this is very different from the noetherian situation…

Intro

John Palmieri — Tue, 29 May 2007 23:18:37 +0000

In a perfect world, this blog would be a vibrant discussion of the cohomology of the mod p Steenrod algebra A, . We’ll see what happens…

By the way, I’m doing this on http://www.wordpress.com/ because it allows (crude) : see this page for instructions.

Ext over A

Bousfield lattices and derived categories

Intro