The Rising Sea

Notes Update

July 10, 2006 on 12:19 am | In Note Updates | No Comments

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.