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…

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