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…