Today I was able to add the result that the quotient stack [U/R] associated to a smooth groupoid in algebraic spaces is an algebraic stack. See Theorem Tag 04TK. Very satisfying!
There are several things that have to be done next:
- Work on the chapter Examples of Stacks, and start a parallel chapter Examples of Algebraic Stacks, where we discuss in detail some very basic examples of algebraic stacks.
- Add some foundational material on changing the base scheme and the underlying big site. This could then be used to define a 2-category of algebraic stacks which lumps all algebraic stacks together regardless of big site that was used to define them (but I’m not sure this would be useful, so I may not add this).
- Write a Chapter on properties of algebraic stacks
- Write a Chapter on morphisms of algebraic stacks
- Write about separation axioms for morphisms of algebraic stacks
- And so on and so forth.
Next week I will not have time to work on the stacks project, so you can start working on the topics above yourself!