The Rising Sea

Notes Update

January 31, 2006 on 9:25 pm | In Note Updates | No Comments

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.

Spectral Sequences

January 7, 2006 on 7:25 am | In Note Updates | No Comments

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”.

And here is another list of spectral sequence notes that I didn’t use for various reasons, but might still be useful:

• 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”.