BOUSFIELD LOCALIZATION FOR REPRESENTATION THEORISTS

JEREMY RICKARD

Bousfield localization is a technique that has been used extensively in algebraic topology, specifically in stable homotopy theory, over the last quarter century. Its name derives from the fundamental work of Bousfield [3], although this is based on earlier work of Brown [4] and Adams [1]. In abstract terms, Bousfield localization deals with the inclusion of a thick subcategory (i.e., a triangulated subcategory closed under direct summands) into a triangulated category and the existence of adjoint functors to such an inclusion. In the original topological setting, the triangulated category was typically the stable homotopy category, and the thick subcategory was defined by the vanishing of some homology theory, but the techniques involved work much more generally. In particular, the triangulated category can be one of interest to representation theorists, such as a stable module category or the derived category of a module category: since the techniques rely heavily on limiting procedures, however, one is forced to work with infinite dimensional modules.

In this pair of talks, I shall discuss Bousfield localization from a fairly abstract point of view, using examples from representation theory. In my first talk, I will discuss ‘finite localization’, which is the simplest case, and which has been sufficient for most of the existing applications in representation theory: some of these applications will be discussed in more detail in the talks of Dave Benson. For finite localization, one deals with a triangulated category and a thick subcategory that are both ‘compactly generated’: this means that every nonzero object has a nonzero map from some ‘compact’ (or ‘small’) object (i.e., an object X such that Hom (X,?) commutes with infinite direct sums or, in other words, every map from X to an infinite direct sum factors through some finite subsum). For example, the compact objects of the derived category D(Mod(R)) of the module category of a ring R are the objects isomorphic to bounded complexes of finitely generated projective R-modules, and since every nonzero object of D(Mod(R)) has nonzero homology in some degree, it has a nonzero map from the compact object R[n] for some n: hence D(Mod(R)) is compactly generated.

In my second talk I shall deal with more general versions of Bousfield localization, where the condition of compact generation is weakened.

Much of what I know about Bousfield localization I learned from the papers of Neeman [2, 5, 6, 7, 8, 9], who has enthusiastically argued that these techniques are much underused outside algebraic topology and has shown that they can be used to simplify substantially the proofs of certain theorems of algebraic geometry and algebraic K-theory. My talks will owe a lot to this work of Neeman, though I will mostly restrict my attention to examples from representation theory.

References

[1] J. F. Adams, A variant of E. H. Brown’s representability theorem, Topology 10 (1971), 185–198.

[2] M. Bökstedt, A. Neeman, Homotopy limits in triangulated categories, Compositio Math. 86 (1993), 209–234.

[3] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257–281.

[4] E. H. Brown, Jr., Abstract homotopy theory, Trans. Amer. Math. Soc. 119 (1965), 79–85.

[5] A. Neeman, The Brown representability theorem and phantomless triangulated categories, J. Algebra 151 (1992), 118–155.

[6] A. Neeman, The chromatic tower for D(R), Topology 31 (1992), 519–532.

[7] A. Neeman, The connection between the K-theory localisation theorem of Thomason, Trobaugh and Yao, and the smashing subcategories of Bousfield and Ravenel, Ann. Scient. de l’ENS 25 (1992), 547–566.

[8] A. Neeman, The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Amer. Math. Soc. 9 (1996), 205–236.

[9] A. Neeman, On a theorem of Brown and Adams, Topology 36 (1997), 619–645.

School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, ENGLAND

  E-mail address: j.rickard@bristol.ac.uk