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…

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