nLab
Bousfield lattice

Contents

Definition

Given a well-generated tensor triangulated category (T,,1)(T,\otimes, 1), let the Bousfield class of an object XX, denoted X\langle X\rangle, be the class {Yobj(T):XY=0}\{ Y\in obj(T): X\otimes Y=0\}. It was proven by Ohkawa that if TT is the stable homotopy category, (𝒮,,S)(\mathcal{S},\wedge,S), then the collection of all Bousfield classes is a set of cardinality at most 2\beth_2. It was proven more generally by Iyengar and Krause that such a collection is always a set and not a proper class when TT is well-generated. This set has a partial ordering on it and the structure of a complete lattice. This lattice is called the Bousfield lattice of TT, denoted B TB_T.

Note that, perhaps by some abuse of notation, B TLoc(T)B_T\subseteq\mathbf{Loc}(T), the collection of localizing subcategories, since every Bousfield class is a localizing subcategory. However, the question of whether or not every localizing subcategory is a Bousfield class is still open in general.

The Distributive Lattice of the Bousfield Lattice

Within B TB_T, there is a distributive lattice DL TDL_T that is precisely all Bousfield idempotent objects of TT. That is, DL TDL_T is precisely the objects XX such that XX=X\langle X\otimes X\rangle=\langle X\rangle. Because B TB_T is an affine quantale, it follows that DL TDL_T is a frame. In particular, DL TDL_T is a distributive lattice, so by the work of Stone, it corresponds to a coherent space. For more on this, see Stone duality.

References

  • S. B. Iyengar and H. Krause, The Bousfield Lattice of a Triangulated Category and Stratification (arXiv:1105.1799)

Revised on March 5, 2012 20:55:03 by Mike Shulman (71.136.234.110)