Triangulated categories and applications (11w5009)
Paul Balmer (University of California Los Angeles)
J. Daniel Christensen (University of Western Ontario)
Amnon Neeman (Australian National University)
Here are some subjects where triangulated categories currently are of particular interest. We have already successfully contacted experts from each of these domains.
1. Classification of triangulated subcategories.
This subject is emblematic of the cross-fertilization that triangulated categories can create between different areas. In the 1970s, 80s and 90s there was a string of results showing that it was valuable to study the thick subcategories of a triangulated category. For the stable homotopy category this came from the work of Ravenel, Devinatz, Hopkins and Smith, for the derived category of perfect complexes on a scheme the key papers were by Hopkins, Neeman and Thomason, while for the stable module category of a finite group the relevant articles were by Benson, Carlson and Rickard. More recent examples (post 2000) include the articles of Friedlander and Pevtsova for finite group schemes, published 2005 and 2007, and the recent preprint of Takahashi, which deals with maximal Cohen-Macauley modules.
The first attempt to present a general theory, which covers all the above results, were the two articles by Balmer (2005 and 2007). These told us how to associate a ringed space to every tensor triangulated category. In 2008, Benson, Iyengar and Krause developed an ``unbounded'' version. In this context it is fascinating to try to see what the general theory says about tensor triangulated categories other than the ones already studied in the literature; the recent PhD thesis of Dell'Ambrogio, which studies the Kasparov category of C*-algebras, is the first interesting example in this direction. But the theory is so new that much work is needed to express its full potential.
2. Homotopy theoretic methods in other triangulated categories.
For at least 20 years people have known that techniques and questions from stable homotopy theory can be fruitfully adapted to a wide range of other triangulated categories; see for example Margolis' 1991 book. Slightly more recent examples include papers by Beligiannis, Benson, Christensen, Gnacadja, Keller, Neeman and Strickland on the relationship between Brown representability and phantom maps (1997 to 2001), and the 1997 foundational monograph by Hovey, Palmieri and Strickland. Since 2000 there has been an explosion of articles: we mention especially the articles by Rosicky (2005) and Neeman (2009) on Brown representability, work of Benson, Chebolu, Christensen and Minac (2007-2009) on the generating hypothesis in the stable module category of a finite group, and work of Hovey, Lockridge and Puninski (2007) on the generating hypothesis in the derived category of a ring. This is a rich topic for further study, because it leads to interactions between diverse areas, and the interplay often leads to novel results.
3. Model categories.
Many triangulated categories arise as homotopy categories of model categories; see Hovey's book. It is natural to wonder how many model categories, up to Quillen equivalence, give rise to a given triangulated category. Schlichting's Inventiones article, in 2002, gave an example of a triangulated category with two inequivalent models. The 2007 Annals paper of Schwede showed that the stable homotopy category of spectra has a unique model (building on earlier work of Schwede and Shipley). The 2007 Inventionnes paper of Muro, Schwede and Strickland describes a triangulated category which has no model. More recently there has been the 2009 article by Dugger and Shipley (following on Schlichting's result), and results by Roitzheim (2007-2009) for the K-local category. There are also many open questions in this area, including the question of finding algebraic models for various naturally arising triangulated categories.
4. Derived categories as a tool in algebraic geometry.
4.a. Grothendieck duality.
Grothendieck's duality theory dates back to the 1960s. Recent work in the field includes creating a non-commutative analog, as well as ``infinite'' versions. The non-commutative analog was introduced in Yekutieli's PhD thesis in 1990, and then studied extensively by several people, among them Van den Bergh, Yekutieli and Zhang. The infinite-dimensional version is very new; it began with two 2005 articles by Jorgensen and by Krause. It then continued in a 2006 article by Iyengar and Krause, a 2008 Inventiones article and a 2009 Annals article by Neeman, as well as several articles by Murfet and by Salarian, all still in preprint form. This subject is in its formative stage.
The theory of motives seeks to give us a better understanding of the homological invariants of algebraic varieties, and some insight into the comparison maps between them. The theory has so far led to at least one spectacular result, namely Voevodsky's proof of the Milnor conjecture. Furthermore, the proof of the more general Bloch-Kato conjecture, due to Rost and Voevodsky, is still the subject of intense activity. The list of open conjectures that remain, where the theory of motives might well prove useful, is truly immense. Triangulated categories feature prominently in this area, via the derived categories of motives and the A^1-homotopy categories.
4.c. Derived categories as invariants of algebraic varieties.
A 2005 theorem by Balmer tells us that, if we consider the derived category of a scheme together with its tensor product, then one can reconstruct the scheme from this data. But a 1981 article by Mukai shows that, if we forget the tensor product, it is possible for two different schemes to have the same derived category. Much work has gone into understanding Mukai's phenomenon. It is known that if two smooth, projective varieties of general type have equivalent derived categories then their canonical rings coincide; hence they are birational. For several years the hope has been that derived categories will give useful invariants for studying the minimal model program. Perhaps the most tantalizing open problem is the conjecture that two varieties, which can be obtained from each other by a flop, must have equivalent derived categories. Bondal and Orlov gave the first result in this direction. In a 2002 Inventiones article Bridgeland proved the conjecture for smooth 3-fold flops, and Kawamata (2002) and Chen (2002) generalized to 3-fold flops with restricted singularities. In general the conjecture remains wide open. Much work has also gone into varieties not of general type, with the majority of the known results being about either abelian varieties or Calabi-Yau manifolds. For Calabi-Yaus this pertains to Kontsevich's homological mirror symmetry; this is another area in which triangulated categories play a pivotal role.
4.d. Derived algebraic geometry.
In this field one generalizes the notion of a scheme to that of a ``derived scheme''. The idea is that, by developing algebraic geometry in a more general framework, it will be possible to apply it to new and emerging fields. A scheme X can be viewed as a functor taking a commutative ring R to the set of R-valued points of X. In derived algebraic geometry we allow functors taking differential graded algebras (rather than mere rings) to higher categories (in place of sets). The subject of derived algebraic geometry is still very much in its formative state as well, with major foundational work by Toen and Vezzosi (2005 to 2008) and later by Lurie (mostly in preprint form). The range of potential applications is diverse and includes anything from homotopy theory to the geometric Langlands program.
5. Representation theory of finite groups.
We give two illustrations of the way triangulated categories enter into this field. The Brouï¿½ Abelian Defect Conjecture predicts a derived equivalence between a block and its Brauer correspondent; remarkably those two blocks are neither isomorphic as algebras nor Morita equivalent in general. The use of triangulated categories is essential. Secondly, in modular representation theory, one studies non-projective representations of the group, which is the same thing as studying the stable module category - this happens to be a tensor triangulated category. Determination of its Picard group, for instance, led to the Inventiones and Annals papers by Carlson and Thï¿½venaz in 2004-2005.
6. Noncommutative topology.
The study of C*-algebras considered as generalized (non-commutative) topological spaces has led to Kasparov's KK-theory. The ongoing work of Meyer and his coauthors has brought to light the importance of triangulated category methods, for instance in connection with the celebrated Baum-Connes conjecture. This connects with more classical areas like relative homological algebra, developed for triangulated categories by Beligiannis and by Christensen around year 2000.
Disclaimer: Our survey aims at illustrating the broad power of triangulated categories, not at being exhaustive. For reasons of space, we regretfully had to skip the very interesting work of many people.