Post Closed as "no longer relevant" by Harry Gindi, Andy Putman, Ryan Budney, Yemon Choi, José Figueroa-O'Farrill
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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.

5 changed etale to étale

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, etaleétale, 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 etale étale topology and constructs algebraic stacks above that. (I assume that one gets Artin stacks if one replaces the etale 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.

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