Since the last update about a week ago we have started adding some very basic material on algebraic stacks. It should be possible to lay out the basics of theory of algebraic stacks in the rest of the summer. Mainly I mean material on types of morphisms of algebraic stacks (how to define them, how do you check for a property, how are they related, etc), perhaps a little bit on quasi-coherent sheaves on algebraic stacks, more examples of algebraic stacks, etc. It will probably turn out that during this process the existing material on algebraic spaces will have to be expanded upon and improved.
What are some more exciting topics that I hope to get to in the not too distant future? Here are some possibilities:
- Artin’s axioms, starting with a chapter on Schlessinger’s paper.
- Olsson’s paper on the equivalence of the different definitions on proper.
- Translating some of the more interesting results on algebraic spaces (regarding points, decent spaces, etc) into results on algebraic stacks.
- A proof of Artin’s trick for algebraic stacks to prove that [U/R] is an algebraic stack if s, t : R —> U are flat and locally of finite presentation.
- Diagonal unramified <=> DM
- Prove the correct version of the Keel-Mori theorem in our setting.
- Discuss the lisse-etale site (I may somehow want to avoid this — perhaps you can always avoid working with this beast?)
- Treatment of simplicial schemes (as an example of simplicial objects) and relation to algebraic stacks. Formulate possible extension to higher algebraic stacks (e.g., hands on version of algebraic 2-stacks with example moduli of stacky curves)
- Gabber’s lemma and more on finding nice maps from schemes to algebraic stacks.
- Tricks with limits. Quasi-coherent sheaves are directed colimits of finite type quasi-coherent sheaves if X is…
- Gerbs and relation to H^2.
- Stacky curves. What are 1-dimensional stacks like?
- General Neron desingularization (needed for 1).
Of course it will take a long time to add all of these topics. But it probably is a good idea to take one or two of these topics and keep them in mind while developing the general theory, so as to have a good reason for building theory. Moreover, many of these topics require additional material on schemes and algebraic spaces. For example, 4 requires a bit about Hilbert spaces (only relative dimension 0 however).
Finally two questions:
- A while back David Rydh suggested we give a name to algebraic stacks with locally quasi-finite diagonal. Should we? If so any suggestions?
- And what to call an algebraic stack with finite diagonal? (Maybe these should have been called separated and the ones with only proper diagonal something else. Too late to change now since it has been used like this for a while now in the literature.)
Maybe we could call an algebraic stack with locally quasi-finite diagonal “space-like”? And then the algebraic stacks with finite diagonal are space-like separated algebraic stacks. Hmm… not sure.