Algebraic stacks from scratch - MathOverflow [closed]

Algebraic stacks from scratch

2010-01-23T20:04:08Z

I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and flat ring maps). However, I've never seriously studied Algebraic geomtry. Can anyone recommend a book that builds stacks directly on top of CRing in a (pseudo)functor of points approach? Typically, one builds up stacks segmentwise, first constructing Aff as the category of sheaves of sets on CRing with the canonical topology, which gives us CRing^op. Then, one constructs the Zariski topology on Aff, and from that constructs Sch, then one equips Sch with the étale topology and constructs algebraic stacks above that. (I assume that one gets Artin stacks if one replaces the étale topology there with the fppf topology?)

Does anyone know of a book/lecture notes/paper that takes this approach, where everything is just developed from scratch in the language of categories, stacks, and commutative algebra?

Edit: Some motivation: It seems like many of the techniques used to build the category of schemes in the first place are just less generalized versions of the constructions for algebraic stacks. So the idea is to develop all of algebraic geoemtry in "one fell swoop", so to speak.

Edit 2: As far as answers go, I'm not really interested in seeing value judgements about this approach. I know that it's at best a controversial approach, but I've seen all of the arguments against it before.

Edit 3: Part of the motivation for this question comes from a (possibly incorrect) footnote on Wikipedia:

One can always assume that U is an affine scheme. Doing so means that the theory of algebraic spaces is not dependent on the full theory of schemes, and can indeed be used as a (more general) replacement of that theory.

If this is true, then at least we can avoid most of the trouble Anton says we'll go through in his comment below. However, this being true seems to indicate that we should be able to do the same thing for algebraic stacks.

Edit 4: Since Felipe made his comment on this post, everyone has just been "voting up the comment". Since said comment was a question, I'll just post a response.

Mainly because I study category theory on my own time, and I've taken commutative algebra courses.

Now that that's over and done with, I've also added a bounty to this question.

Answer by Anton Geraschenko for Algebraic stacks from scratch

2010-01-23T21:36:40Z

Demazure and Gabriel's Introduction to algebraic geometry and algebraic groups develops standard algebraic geometry in terms of functors on CRing.

Martin Olsson's stacks course did something like what you're describing, first characterizing separated schemes among functors on Aff, then algebraic spaces among functors on Sch, then algebraic stacks among stacks over Sch. I took notes in that class which I think are pretty good: http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf

Answer by mdeland for Algebraic stacks from scratch

2010-01-23T22:32:59Z

You should read the following post: http://math.columbia.edu/~dejong/wordpress/?p=8

It partially explains why this approach was not taken in the stacks project, and probably isn't generally taken elsewhere.

Now I'll just quote "... any full discussion of the theory of algebraic stacks is going to mention affine schemes, schemes, and algebraic spaces. It will still be the case that the most interesting objects of study are algebraic varieties, and their moduli spaces.

...Sure you can define algebraic stacks without first defining any intermediate geometric objects. However, once this is done, there you are, and there is nothing that you can hold onto and relate the objects to… "

Answer by algori for Algebraic stacks from scratch

2010-01-23T23:50:26Z

There is a book project in progress by Kai Behrend, Brian Conrad, Dan Edidin, William Fulton, Barbara Fantechi, Lothar Göttsche and Andrew Kresch. It has not been completed yet and it is not clear when it will be, but I find the existing chapters quite useful.

http://www.math.uzh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1

Answer by Timo Schürg for Algebraic stacks from scratch

2010-01-24T21:14:01Z

Another good place to look are the notes of a course on stacks by Betrand Toen. Here's the link: http://www.math.univ-toulouse.fr/~toen/m2.html I think they pretty much do exactly what you are looking for. (Edit: new link http://ens.math.univ-montp2.fr/~toen/m2.html - Toen has moved)

Here's the quick summary: You will want to read section 1 of Cours 2 where the term geometric context is defined. It's basically a category with a Grothendieck topology with a fixed class of morphisms that you call geometric. The main example are commutative rings with the etale topology or the smooth topology. This induces coverings in the presheaf category in the standard way.

Then skip straight to Cours 5. Although you said that you are comfortable with descent this section is definitely worth a close look. It introduces a homotopy theory on the category of groupoids and shows that there always is a weakly equivalent groupoid such that your functor becomes strict. It then reformulates descent via homotopy limits. The upshot is a nice category of stacks, Definition 4.4.

Then jump straight to Cours 8, Definition 1.4. and you've got algebraic stacks. The only point where you will need schemes or algebraic spaces is for representable morphisms, but judging from the remark after the definition you can get around that as well.

Answer by David Steinberg for Algebraic stacks from scratch

2010-02-24T01:37:34Z

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 CRing.

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

Answer by Buschi Sergio for Algebraic stacks from scratch

2010-11-13T21:46:41Z

A old text on AG with functorial approach is Demazure-Gabriel (Introduction to algebraic geometry and algebraic groups) :

http://books.google.it/books?id=RDKRyP00aoMC&printsec=frontcover&dq=Algebraic+groups,+demazure&source=bl&ots=bINVNX5u5x&sig=PA40Oq8LOQdfBZgci6hFtXM-HGA&hl=it&ei=nQbfTJiCDs-fOqKP4esO&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBsQ6AEwAA#v=onepage&q&f=false