# Good introductory references on algebraic stacks?

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.

• Kai Behrend, Introduction to algebraic stacks seems to be really interesting! See also here or here – Watson Oct 3 '18 at 14:26

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

Algebraic Spaces and Stacks, M. Olsson, AMS Colloquium Publications, volume 62, 2016. ISBN 978-1-4704-2798-6

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-Bailly's Champs Algébriques 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.

• Stacks generalize sheaves, fibered categories (equivalently pseudofunctors) generalize presheaves (contravariant functors [into Sets]). The key key idea behind stacks is not only the generalization of functors, but also a generalization of glueing. I disagree with the idea that algebraic spaces should be learned first for the following reason: all of the 2-categorical "stuff" is equivalent to the 1-categorical "stuff" when we restrict ourselves to 1-categories. – Harry Gindi Nov 15 '09 at 6:41
• Sorry for the nitpick: a scheme is a covariant functor from commutative rings to sets, not from the opposite category rings^op. – Axel Boldt Dec 31 '14 at 23:22
• It's probably worth pointing out that Olsson's book will be out soon. – Hoot May 9 '16 at 5:53

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.

• A nice sequel to that is Barnet-Lamb minor thesis at people.brandeis.edu/~tbl/minor-thesis.pdf The problem with Vistoli's notes is that they are a really good introduction to stacks, but do not mention at all the algebraic stack part. Barnet-Lamb follows closely Vistoli, so you can just skip many parts with a glance, and treats the problem of what does it mean to be an algebraic stack and why this stuff appears naturally in moduli problems. One word of caution: this is literally filled with typos, so pay attention to what you read. – Andrea Ferretti Feb 24 '10 at 12:57
• It's also worthwhile to point out that Vistoli's (fantastic) notes are nothing more than a corrected version of his contribution (Part 1 = first 4 Chapters) to Fantechi et alli's "FGA Explained", quoted by the OP. – Pedro Lauridsen Ribeiro May 9 '16 at 3:21

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.

• It's still not published... Just wonder when this great book will be finished... – WWK Feb 15 '16 at 16:18
• From here : « Andrew Kresch just told me that they gave up on the project ». However, there is a link here. – Watson Sep 1 '18 at 13:22

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

It's already more then 1400 pages long!

• plus. In there is a guide to the literature, which should be helpful. – GMRA Oct 24 '09 at 0:08
• Uhm... 1400 pages for an introductory reference seems a bit too much... – Andrea Ferretti Dec 24 '09 at 17:02
• Good point :) It is an introduction, and much, much more. It seems to be a good introduction, as well as a good reference book for people who know this stuff well. – GMRA Dec 24 '09 at 17:13

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

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.

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.

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.

• That looks like a nice introduction. The one complaint I have is that the definition of 2-functor given in the appendix seems to be rather inconvenient - one usually includes the composition compatibility as part of the datum. – S. Carnahan Oct 28 '11 at 18:30

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.

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

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.

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.

• I think the introduction is far too concise to be a good introductory reference. – Andrea Ferretti Feb 24 '10 at 12:53

Here are some other references:

Besides the references already given, I like Dan Edidin's

Notes on the construction of the moduli space of curves

https://arxiv.org/abs/math/9805101

I quote :

"In section 3 we return to curves and outline Deligne and Mumford's proof that the stack of stable curves is a smooth and irreducible Deligne-Mumford stack which is proper over $$\operatorname{Spec} \mathbb Z$$".

It starts from the basics but still explains a key result in the history of the subject. Highly recommended !

there should be a video-lecture of Kai Behrend "algebraic stacks" which I once watched. I can no more find it quickly. But i think one could find it on the homepage of Cambridge