Question

Are there any good introductory texts on algebraic stacks? I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also browsed through FGA explained (Fantechi et al.). Although I find the level good, it is somewhat incomplete and I would want to see more basic examples. Unfortunately I don't read french.

Answer 1

You might take also a look at this: http://staff.science.uva.nl/~heinloth/SeminarStacks.html

especially the references and more especially the last two paper of them.

Answer 2

I always found Algebraic Stacks by Tomas Gomez to be a very readable quick introduction. It is virtually without proofs but explains on 34 pages the most relevant definitions and constructions and discusses the example of vector bundle in some detail. He has both the definition of a stack as a sheaf of groupoids and as a category fibred in groupoids in it.

Answer 3

Linked below is a note written by Kai Behrend whose first section gives a concise introduction to stacks, building them directly out of (lax) functors from the category of affine schemes.

http://www.math.ubc.ca/~behrend/cet.ps

Answer 4

Vistoli's notes on descent, grothendieck topologies, fibered categories, and stacks at http://homepage.sns.it/vistoli/descent.pdf are not only just a really good introduction to algebraic stacks, they're some of the best notes I've ever read on any subject. What I really liked is that he took the time to not identify f*g* with (gf)*, which makes the proofs longer, but absolutely rigorous.

He starts with a review of category theory and classical scheme theory, then builds up grothendieck (pre)topologies, then builds up the notion of a fibered category, which is a generalization of a presheaf, then defines stacks in terms of fibered categories and descent. What's really great about this approach is that once you see how fibered categories work, Lurie's approach to higher topos theory ((infty,1)-categories generalize categories fibered in groupoids) makes a good deal more sense. I can't recommend it enough.

Answer 5

I have to follow Alberto's answer with Deligne and Mumford's paper on irreducibility of the moduli of curves.

Answer 6

It might not be the best reference for a systematic study of stacks and some of the terminology is old, but Mumford's "Picard Groups of Moduli Problems" (1965) might be a nice complement. It explains why stacks came to be and does a few calculations to show their usefulness.

Answer 7

There was an MSRI summer school on stacks and deformation theory a few years ago. The video of all the talks are online, at the workshop's webpage. There are several copies of notes around, I believe they are on Ravi Vakil's webpage somewhere.

Answer 8

There is an open-source textbook on stacks being created. You can find it here

It's already more then 1400 pages long!

Answer 9

Dennis Gaitsgory is currently running a graduate seminar with a website here. There are quite a few notes and references on there about algebraic stacks. You should first look at the notes from the second and third talks.

Answer 10

I find the "review of algebraic stacks and Artin's method" in chapter 1 of Faltings, Chai "Degeneration of abelian varieties" very nice.

Answer 11

I am not sure if the book I am about to suggest is the half-finished text you are hinting at, but there is a book in progress by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch. You can find a link to it here:

Book

It is the most complete reference on algebraic stacks in English that I am aware of. It also has the advantage of being addressed to the beginner.

I think that beyond the basic things, anything deeper you learn about stacks typically involves specific stacks, with certain applications or questions in mind.

Answer 12

Anton live-texed notes to Martin's Olsson's course on stacks a few years ago. They are online here.

My general advice is to learn algebraic spaces first. The point is that the new things you need to learn for stacks fall into two categories (which are mostly disjoint): 1) making local, functorial, and non-topological definitions (e.g. what it means for a morphism to be smooth or flat or locally finitely presented) and 2) 2-categorical stuff (e.g. what is a 2-fiber product). You don't need to do things 2-categorically for algebraic spaces, so it makes sense to learn them first. I believe it really clarifies things to learn these separately.

Also, the formal notion of a stack is a generalization of functor. If you are not used to thinking of schemes functorially (e.g. as a functor from rings^op to sets) it will be hard to wrap your head around the notion of a stack. the The intermediate step of learning to think about geometry in terms of functors of points is crucial.

Knutson's book Algebraic Spaces is very good for the EGA-style content, and its introduction will point you to many nice applications of algebraic spaces that are worth learning and will motivate you to learn the EGA-style stuff. Laumon and Moret-Baily's Champs Algebriques is nice and contains more theorems that just the EGA style stuff.

Its hard to point you any other particular reference without knowing what your goal in learning stacks is.