## Left-Handed Commutative Algebra

We will construct the category of Algebraic spaces following the program outlined by James Borger, that is, relying only on commutative algebra, category theory, and the powerful theory of Descent. At the moment, we intend to follow classical descent theory as outlined in Vistoli’s notes, but it is possible that we may change lanes, so to speak, and pursue the theory of homotopical descent of Toen and Vessozi, to generalize our results.

We assume familiarity with category theory throughout

Results involving commutative algebra may be stated without proof, but the final goal is to have all relevant theorems proven.

The plan (more will eventually be added):

1.) Review of the Yoneda Lemma and a proof of the 2-Yoneda lemma for pseudofunctors
2.) Grothendieck Topologies
3.) Descent for sheaves and stacks – All definitions given using the 2-Yoneda descent condition
4.) Overview of Commutative Algebra Results necessary
– Zariski’s Main Theorem
– Definitions: Finitely presented, formally unramified, formally smooth, formally etale
– Lemmata: The above are stable under base change and composition.
5.) The Etale topology on $CRing^{op}$
6.) The topos of sheaves $\mathfrak{T}:= Sh( CRing^{op}_{\acute{e}tale} )$
7.) Definitions of important algebro-geometric morphisms between sheaves where possible.
8.) The category of Algebraic Spaces $\mathcal{A} \subset \mathfrak{T}$
9.) The etale, nisnevich, fppf, and fpqc topologies on $\mathcal{A}$.
10.) The category of Stacks on $\mathcal{A}$ in the fppf and fpqc topologies.
11.) Algebraic Stacks

The theory of schemes will be discussed in examples, but it is not the focus of our exposition. Upon the completion of the above agenda in sufficient rigour and generality, the theory of schemes may be investigated more thoroughly.

1. The advantage of purely fuctorial view point is that one can deal with presheaves other than sheaves. But it seems that you do not mention this

Dealing with sheaves will lose lots of functoriality

• I wasn’t aware we could do it any other way. This is for classical AG, at least. If you’d like to elaborate, I’d be interested in hearing what you have to say.

• John
• February 14th, 2010

Hi, I’m not sure this is the right place to be posting this but I saw you question on MO on Toen’s typo and can’t seem to find it anymore. Since I couldn’t find any mail address I’m posting here.
I think you’re right, it is a typo and what you suggested seems to be right. Although if Toen gives you an answer let us/me know please.
Anyways, I hope you keep up with this project, it’s something I also thought of trying to do, but didn’t have the courage for. I’d be more than happy to participate if you’re interested.
What I think would be really important to do, after all the formalities/technicalities have been sorted out, is to find a good dictionary between the objects you define (e.g. schemes) and the pictures you want to/can draw.

• I shot you an e-mail, but I’d be glad to have someone work on this with me.