The Rising Sea

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.