The Rising Sea

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.

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

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.

Spectral Sequences

The Rising Sea does Spectral Sequences (SS). New to the notes section is a streamlined exposition of the basic properties of spectral sequences needed in algebraic geometry. Since it might save someone else the trouble, I’ve listed here some of the good references I’ve found:

• Grothendieck’s EGA III is the main source for my notes. My presentation of the spectral sequence associated to a filtration is different, however, because EGA builds everything out of connecting morphisms which seems unnecessarily mysterious. You can find scans of EGA III here.
• Cartan & Eilenberg’s “Homological algebra” is still a good reference for several points.
• Weibel’s book “An introduction to homological algebra”.

• Chow’s “You could have invented spectral sequences“.
• McCleary’s book “A user’s guide to spectral sequences”. If you google for it you can find an online version.
• Barry Mitchell’s paper “Spectral sequences for the layman” in The American Mathematical Monthly, Vol.76, No.6, 599-605.
• Gelfand and Manin’s “Methods of Homological Algebra”.

Notes Update

• Derived Categories (DTC) and Derived Categories Part II (DTC2) In both of these notes some material has been added and various things have been rearranged..

• New Notes: Derived Categories of Sheaves (DCOS) I’ve added a basic treatment of the derived category of sheaves of modules on a ringed space.

Notes Update

• New Notes: Concentrated Schemes (CON) I’ve added a brief note on quasi-compact quasi-separated schemes (basically just a translation of EGA).

• Modules Over a Scheme (MOS) Has been updated with the more general proof of extension of quasi-coherent sheaves given in EGA, which is then used to prove that the category of quasi-coherent sheaves over a concentrated (=quasi-compact quasi-separated) scheme is grothendieck abelian.

• Cohomology of Sheaves (COS) Removed the noetherian hypotheses used in Hartshorne and replaced them with concentratedness, in line with the modern literature. Other related improvements.

• Higher Direct Image of Sheaves (HDIS) Removed noetherian hypotheses of Hartshorne, so that we now show the higher direct image of any concentrated morphism preserves quasi-coherence.