The Rising Sea

Update

Unfortunately this website was compromised in the recent wave of attacks on WordPress blogs, but hopefully everything is now OK. I’d like to thank Graham Leuschke for sounding the alarm.

In other news, I’ve been thinking lately about duality in Orlov’s singularity category (or, for the commutative algebraists, the stable category of Cohen-Macaulay modules) and the product of these musings is now available on the arXiv. The paper originated in an attempt to understand the geometric version of a paper of Henning Krause and Jue Le which uses compactly generated triangulated categories to study Auslander-Reiten translation. This made me curious about trace maps for the Serre functor in singularity categories, and it turns out that using the machinery of duality in compactly generated triangulated categories it was possible to recover the rather beautiful formula of the string theorists Anton Kapustin and Yi Li in the case of isolated hypersurface singularities.

For reasons of length the connection to the work of Krause and Le is almost completely suppressed in the paper, although this point of view is what led me to the main arguments. I still hope to write a sequel treating the global case using homotopy categories of sheaves, as for example developed in my thesis and joint work with Shokrollah Salarian, in which the “compactly generated” point of view is made explicit.

Talk slides

I have the good fortune of being in Barcelona, at the Centre de Recerca Matematica, for the month of November. Yesterday I spoke on my thesis research “The mock homotopy category of projectives and Grothendieck duality” as part of a workshop on derived categories being organised by Leovigildo Alonso Tarrío, Ana Jeremías López and Amnon Neeman. The slides of the talk are now available here (PDF).

Triangulated Category Notes

I’ve improved some statements in Triangulated Categories (TRC). For the curious: the changes are in the section on Localisation Sequences, where some inelegant writing exposed my lack of understanding at the time.

Notes Update

Lately I’ve been occupied with research, so there still isn’t much new in the notes. In this update the referencing (which was shocking before) has been improved a little and there are some extra historical remarks in DCOQS and DTC. If you’re reading through any of the following notes you may benefit from several minor improvements:

• Derived Categories (DTC)
• Derived Categories of Sheaves (DCOS)
• Derived Categories of Quasi-coherent Sheaves (DCOQS)
• Matsumura (MAT)

Anybody interested in derived categories of sheaves should also check out Lipman’s notes which have been recently updated with new material.

Notes Update

This month there are only some minor additions and corrections to existing notes:

• Derived Categories (DTC) In Section 5.1 on “Split Direct Limits” I made use of a complicated result with a long proof. This has now been omitted in favour of a weaker result with a much easier proof, that is nonetheless sufficient for the purposes of DTC.

• Abelian Categories (AC) The section on Finiteness Conditions has been improved. In particular there are now notes showing how the abstract conditions agree with the usual ones for modules over a ring. I don’t know the original reference, but I learnt this material from excellent webnotes of Paul Smith.

• Modules Over a Scheme (MOS) Now contains a classification of the quasi-coherent sheaves satisfying the various finiteness conditions of AC. If the scheme is noetherian this is essentially trivial, but on an arbitrary concentrated scheme things still work pretty much as you would expect. The finitely generated objects are the coherent sheaves, and the finitely presented objects are the locally finitely presented sheaves.

• Derived Categories of Sheaves (DCOS) One new section on Tor sheaves. Of course one cannot define Tor sheaves using projective resolutions, so the approach using the derived tensor product is the most convenient.

• Triangulated Categories (TRC) There is a new section on Localisation sequences, as defined by Verdier and used to great advantage in a recent paper of Krause on the stable derived category. Dually one has Colocalisation sequences and also Recollements. The reader will learn nothing here they cannot find elsewhere.

Notes Update

There have been various small changes to my notes, so if you’re using several of them you should update to the newest versions to avoid incorrect references. The only major addition this time is in DCOQS:

• Derived Categories of Quasi-coherent Sheaves (DCOQS) Added a section on “invertible complexes” which are complexes in the derived category of sheaves that are units under the derived tensor. Actually it is not too difficult to check that the derived picard group of a scheme is trivial, in that the only elements are shifts of invertible sheaves.

Study Software

A few people have already downloaded my simple application Study for managing mathematics notes. An unadvertised feature of the current version is the ability to create cross-references between results in different files. If you download any of my mathematics notes you will probably notice “coloured” links of the form (MRS, Proposition 6). Keeping such references updated by hand would require hundreds of changes for every minor change to my Modules over Ringed Spaces notes.

Clearly I do not keep these references updated by hand. I insert a latex command of the form \sef{MRS}{prop_someresult} in my LaTeX file. When the file is latexed an Applescript passes the text to the application Study which resolves the acronym “MRS” to a certain LaTeX file and replaces the \sef command with the appropriate text (MRS, Proposition 6). In fact these are working hyperlinks to the referenced result, usable in any viewer (such as Acrobat) which understands such things, provided you have a local copy of the referenced file.

I have not yet documented this feature properly because I’m not sure if anyone will actually use it, and it hasn’t been extensively tested. Any Mac users who would like to help test this feature, please send me an email (see the About page).

Notes Update

Nothing of great excitement in this month’s notes update. I’ve made some minor additions and changes to existing notes:

• Derived Categories of Quasi-coherent Sheaves (DCOQS) The section on the projection formula has been updated with a few results from SGA that I’ve updated to modern standards (i.e. some boundedness hypotheses were removed). This includes the following useful fact: if two perfect complexes are isomorphic on stalks then they are isomorphic on a neighborhood of the point (both isomorphisms are in the respective derived categories).

  I have also included a proof that on a quasi-compact semi-separated scheme every quasi-coherent sheaf can be written as a quotient of a flat quasi-coherent sheaf. This fact is known and is a special case of a published result of Alonso, Jeremias and Lipman.

• Spectral Sequences (SS) I have made a couple of minor corrections and improved the exposition in a few places. Much thanks to Rongmin Lu for pointing out most of these errors.

Notes Update

Another month, another update. Today I’ve added one new note on ample families of sheaves and updated two other notes with substantial new content:

• New Notes: Ample Families (AMF) If you grew up on Hartshorne’s book like I did, then you may have only encountered ample sheaves on a noetherian scheme. In fact the original definition in EGA is given for arbitrary schemes, and in the first section we basically just translate EGA II 4.5.2. Then we define an ample family of sheaves following SGA 6.II 2.2.3 and in particular observe that the tensor powers of an ample family generate the category of quasi-coherent sheaves (which in general has a pretty uninspired set of generators, so this is nice). It is shown in SGA (but not in our notes) that any nonsingular variety over a field has an ample family of sheaves, so this notion of an “ample family” is quite useful.

• Derived Categories of Sheaves (DCOS) I’ve fleshed out the section on the derived Hom and Tensor adjunction with some technical details that become crucial later (for example, the adjunction commutes with restriction). There is also an explicit description of what happens to a map under the adjunction, with a very careful treatment of some complicated sign issues.

  After working through these details one comes to understand that there is a simple technique for reducing all the complicated statements on the level of the derived adjunction to statements on the level of the adjunction on complexes, which one can check very efficiently using the explicit description alluded to above. Firstly, in every statement about the derived structures one expects a compatibility diagram relating the derived and underived structures. Usually the compatibility morphisms are isomorphisms for hoinjective/hoflat complexes, so one can reduce to these cases and then using the compatibility diagram reduce to the level of complexes.

  There is also a detailed study of the units and counits of this adjunction. By playing around with these morphisms one obtains “for free” several useful morphisms, such as the “double dual” morphism, and the “tensor inside Hom” morphism.

• Derived Categories of Quasi-Coherent Sheaves (DCOQS) On the level of sheaves of modules one defines derived inverse image and tensor using hoflat complexes, and to prove there are enough of these one uses the “extension by zero” construction. Unfortunately this produces bad (i.e. non-quasi-coherent) sheaves in general, so there is something of a problem when one comes to define the derived inverse image and tensor product for quasi-coherent sheaves. This is treated in detail in a new section of the notes.

  The section on perfect complexes has also been expanded, so that it now includes the proof that on a nice scheme compact = perfect.

  Finally we use the study of adjunctions in DCOS to define the derived projection morphism and “friends” such as the “double dual” and “tensor inside Hom” morphisms. Generally these are isomorphisms when enough things are perfect (an odd sentence). Essentially we give a modern treatment of the derived dual as studied in SGA, back in the days when unbounded complexes were more fearsome.

  Anything not from SGA is probably from Neeman’s paper on Grothendieck duality, even if individual attributions aren’t yet in the notes.

Notes Update

Today I’ve posted one new set of notes and made various changes to other notes. The major changes:

• New notes: Derived Categories of Quasi-coherent Sheaves (DCOQS) In algebraic geometry it is the derived categories of quasi-coherent (or coherent) sheaves that are usually of interest. For the unbounded derived category a lot of nice results are known, for example:
  • Neeman’s unbounded Grothendieck duality theorem.
  • The fact that perfect complexes are precisely the compact objects.
  • The equality of the derived category of quasi-coherent sheaves with the derived category of sheaves with quasi-coherent cohomology (without any notherian assumptions).

  There is some material that is not written down carefully in the literature, so someone new to the subject might find these notes useful.

• I’ve added a section to Higher Direct Images of Sheaves (HDIS) on the uniqueness of cohomology and higher direct image. That is, if you take the derived functor of global sections we check it doesn’t matter if you use quasi-coherent sheaves or general sheaves as your domain. This is a technical matter that crops up all over the place.

A few other notes have had typos and minor errors corrected. The list of acronyms for those who care: MRS, COS, CON, AC, TRC, MOS, DTC, DCOS.