One of my plans for this blog is write up some of my notes for talks I am about to give. Ideally, this will serve both as a means of collecting my thoughts and to let potential audience members read about it ahead of time so they can follow it better/remember what I said after the fact.
Last month, I went to University of Utah for the Derived Categories minicourse. The talks I enjoyed most were three by Arend Bayer on Bridgeland stability, and so I am giving a talk or two to my research group on the subject.
This post will presume the very basics of derived/triangulated categories. The best reference for this stuff is Bridgeland’s paper, which is very clearly written but has the downside of not explaining ‘well-known’ results.
T-structures and hearts
The basic question to motivate this discussion is as follows: When I construct the derived category of some abelian category , what extra structure does the derived category have that ‘remembers’ how it was built from ?
For one, I have the distinguished subcategory consisting of objects of concentrated in degree 0. Also, I can talk about what integers the homology of a given derived object are bounded by. It is the latter concept the following definition is patterned after.
Def. A t-structure on a triangulated category is a full subcategory s.t.
2) Every is part of an exact triangle , such that .
A t-structure is called bounded if , there are integers such that .
Note 1: is automatically closed under extension, which means that any exact triangle with first and third objects in also has its second object in . Virtually every subcategory of a derived category I will talk about today has this property.
Note 2: The absence of morphisms from to implies that the factorization in 2) is unique up to isomorphism. (Proof is a fun application of octahedral axiom)
Example: If , then let be complexes with trivial homology in negative degrees. Then is complexes with trivial homology in non-negative degrees, and this t-structure is bounded. Thus, bounded t-structures abstract this trait that bounded derived categories have. In fact, as we are about to see, all bounded t-structures arise in this way.
Once we have a t-structure, we can isolate which part of the category behaves like ‘objects in concentrated in degree o’, since it should be ‘the complexes with trivial homology outside degree zero’. This idea motivates the next definition.
Def. The heart of a t-structure is .
Wonderful fact: The heart of a t-structure is always an abelian category, with short exact sequences coming from exact triangles contained in the heart. Sadly, the proof of this is tedious and unenlightening.
Furthermore, if a t-structure on is bounded with heart , then , and the t-structure coincides with the one in the example. Thus, not only do hearts abstract the concept of ‘complexes concentrated in degree 0’, but every heart arises in that way (for bounded t-structures).
Lovely! We have very sharp abstractions of the extra structure that a derived category retains that point to the abelian category it was made from. What good is this? I know two good answers:
1) One of the most important questions in derived categories is asking when two abelian categories and are derived-equivalent. This is the type of question answered by results like the Beilinson equivalence or the derived Mckay correspondence. However, if we were to somehow have perfect understanding of hearts inside triangulated categories, all such equivalences would be trivial: and are derived-equivalent iff is a heart in .
2) Similar, it is a big question to understand auto-equivalences of a derived category; these are philosophically ‘hidden symmetries’ of the abelian category. An auto-equivalence clearly must take hearts to hearts, and so it permutes the set of hearts. Bridgeland has outlined a program for cataloging the auto-equivalences of the derived category of a variety, which starts by understanding how it acts on a manifold closely related to the set of hearts.
New Hearts from Old
With any luck, you now believe that hearts are a good thing to look at a little more. A first step is finding a more straightforward characterization/definition of them.
Thm. A full subcategory of a triangulated category is the heart of a bounded t-structure if and only if:
2) , there exists a diagram
such that each of the triangles is exact (with diagonal arrow a map of degree 1), with .
Note: Again, the vanishing of ‘s implies that the factorization in 2) is unique.
The factorization in 2) isn’t so weird… every object in a bounded derived category has a lowest non-zero homology group, which gives a map to into appropriate shifting of the heart. This map is part of an exact triangle, and its ‘kernel’ no longer has any homology in that degree. Then, just repeat, peeling off each successive homology.
It is really the ability to factor objects in the derived category into pieces that makes hearts work. This outlook helps us find new hearts by finding new ways of factoring. The simplest example of this is that of a torsion pair.
Def. A torsion pair for an abelian category is a pair of full subcategories such that:
2) , there exists and an exact triangle .
This definition should look almost boringly similar to previous ones. Arguably, the most interesting part is the choice of terminology, which suggests an interesting example. Let , the quasi-coherent sheaves on some nice variety . Let be the subcategory consising of torsion sheaves, and let be the subcategory of torsion-free sheaves. Then is a torsion pair for .
The utility and relevance of torsion pairs is given by the following theorem:
Thm. Let be the heart of a bounded t-structure on , and let be a torsion pair on . Then
is also a heart for .
One should think of this as a ‘fractional shift’ of . Specifically, now also admits a torsion pair , and the new heart corresponding to this pair is just . The overall idea is that knowing how to factor objects within the heart allows the creation of new hearts in between the old heart and its shift.
The next step will be to look at abelian categories that factor a whole lot, and give us a continuous family of new hearts. To do this, we will need to define ‘semistable bundles’, which will take a bit of time to setup. Hence, I think I will stop this post here, and start a new post with the elements of semistable bundles and then go into stability conditions. Appropriately, this also mirrors how I anticipate the talk going (having to stop and have a second talk later).