Posted by: John Palmieri | December 12, 2007

Bousfield lattices and derived categories

Let R be a commutative ring, and look at its unbounded derived category \mathsf{D}(R) 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 X \underset{R}{\overset{L}{\otimes}} Z = 0 \Leftrightarrow Y \underset{R}{\overset{L}{\otimes}} Z = 0. This defines an equivalence relation on the objects of \mathsf{D}(R); the equivalence class of an object X is called its Bousfield class and is written \langle X \rangle. We can define a partial ordering on these Bousfield classes, by setting \langle X \rangle \geq \langle Y \rangle if and only if X \underset{R}{\overset{L}{\otimes}} Z = 0 \Rightarrow Y \underset{R}{\overset{L}{\otimes}} Z = 0. (Therefore, \langle R \rangle is the largest equivalence class and \langle 0 \rangle is the smallest.) The Bousfield lattice \mathbf{B}=\mathbf{B}(\mathsf{D}(R)) 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 \mathsf{D}(R) 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 \mathbf{B}(\mathsf{D}(R)) is isomorphic to the lattice of subsets of Spec(R). For any prime ideal \mathfrak{p} in R, define an R-module k(\mathfrak{p}) to be (R/\mathfrak{p})_{\mathfrak{p}}; then any object X in the derived category is Bousfield equivalent to \displaystyle \bigoplus_{\mathfrak{p}} k(\mathfrak{p}) \underset{R}{\overset{L}{\otimes}} X, which in turn is Bousfield equivalent to \displaystyle \bigoplus_{\mathfrak{p} \in \text{supp}\,X} k(\mathfrak{p}), where supp(X) is the set of primes \mathfrak{p} for which k(\mathfrak{p}) \underset{R}{\overset{L}{\otimes}} X \neq 0.

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 k[x_2, x_3, x_4, ...] / (x_i^i) has cardinality at least 2^{2^{\aleph_0}}. 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…

Posted by: John Palmieri | May 29, 2007

Intro

In a perfect world, this blog would be a vibrant discussion of the cohomology of the mod p Steenrod algebra A, \mathrm{Ext}_A^*(\mathbf{F}_p, \mathbf{F}_p). We’ll see what happens…

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

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