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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.

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closed as no longer relevant by Harry Gindi, Andy Putman, Ryan Budney, Yemon Choi, José Figueroa-O'Farrill Nov 13 '10 at 23:07

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That may too tall an order to fill. When you glue stuff together to enlarge your category, you always want to insist on representability of the diagonal (by objects in the smaller category) if you want to be able to reduce results about the larger category to the smaller category. But the diagonal of an algebraic stack might be representable by algebraic spaces (but not by schemes or affine schemes), so it's hard to imagine jumping to stacks from a category smaller than algebraic spaces. This argument suggests that you "have to" go through the steps (Aff)→(SepSch)→(Sch)→(AlgSp)→(AlgStacks). –  Anton Geraschenko Jan 24 '10 at 6:19
Dear Harry, The footnote in Wikipedia just reflects the fact that since any scheme is glued from affine schemes, if we are going to glue a scheme by an etale equivalence relation to make an algebraic space, we may as well combine the two gluings, and obtain our algebraic space by gluing affine schemes. –  Emerton Jan 24 '10 at 19:59
Pardon the curiosity but: "I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra. However, I've never seriously studied Algebraic geometry." How the hell does that happen? –  Felipe Voloch Jan 24 '10 at 21:17
Felipe, I don't think it's as surprising as all that. Many of the things on that list are historically associated with algebraic geometry, but don't depend on it either logically or conceptually. Sheaves are a case in point. Many people think they belong to alg geom, but they really don't: for a proof, see maths.gla.ac.uk/~tl/sheaves.pdf . Ditto stacks and Grothendieck topologies --- there's nothing inherently algebro-geometric about them. –  Tom Leinster Feb 1 '10 at 0:55
@Harry: that still sounds weird. I guess most questions requesting something specific don't need further answers after one satisfactory answer, but I don't remember seeing them closed. Maybe this is a recent change in policy. –  Omar Antolín-Camarena Nov 14 '10 at 14:48

6 Answers 6

up vote 9 down vote accepted

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.

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Given that it's in french, could you at least describe the procedure he goes through? You're saying that he skips all of the intermediates? –  Harry Gindi Jan 24 '10 at 21:45
You win. Thank you. Meanwhile, is this definition of descent using holims more straighforward than the really awesome two-line definition using sieves? That is, a fibered category F is called a stack provided that the induced inclusion $Hom(h_X,F) \hookedrightarrow Hom(S,F)$ is an isomorphism for all objects X of the base category and all covering sieves of S of X. That is, where $S\subset h_X$ is distinguished subfunctor of the representable functor $h_X$. That's the "best" definition according to Vistoli's notes. Is the definition using holims even better? –  Harry Gindi Jan 26 '10 at 10:38
The unknown control sequence there is too hard to fix, so here is what it should have said, except it now has an unhooked arrow. $Hom(h_X,F) \rightarrow Hom(S,F)$ –  Harry Gindi Jan 26 '10 at 10:40
Well, maybe it's not the best definition to work with, but for sure, it's the slickest. –  Harry Gindi Jan 26 '10 at 10:46
No, Vistoli's definition is much simpler. But the definition using holims paves the way for higher stacks. Just replace groupoids by simplicial sets and start the same machine. –  Timo Schürg Jan 26 '10 at 10:48

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.


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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… "

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Metaphorically speaking, the approach I'd like to see takes an elevator to the top of the building, and investigates each room going down one level at a time. The standard approach investigates each floor, then walks up the stairs after the investigation is done. I'm not saying either way would be better. I'm just saying that with my specific background, the approach I'm looking for is more motivated. –  Harry Gindi Jan 23 '10 at 22:51
@Harry. You could carry it out in your mind. Read a book like Demazure and Gabriel for the basic machinery, and read an article at Romagny's, with the constant aim in mind to see whether each statement can be proved without the crutches of schemes. Maybe you can write down your conclusions; or maybe it is sufficient to carry it out in your mind. –  Anweshi Jan 23 '10 at 23:25
I could do that, but if there is a book about it, I'd rather read the book. –  Harry Gindi Jan 24 '10 at 0:39

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

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I've read through some of Demazure-Gabriel, and while it's quite good, I'm looking for a generalization of their approach directly to stacks. This course that you're describing by Olsson seems to make the same "pause" to describe schemes, rather than just "taking it all the way." –  Harry Gindi Jan 23 '10 at 21:40

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.


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


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This does not seem to be relevant. The book does not deal with algebraic spaces. –  Martin Brandenburg May 24 '11 at 16:17

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