The Rising Sea

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”.

Notes Update

December 21, 2005 on 7:25 am | In Note Updates | No Comments

• 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

November 19, 2005 on 7:22 am | In Note Updates | No Comments

• 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.