Since the last update we have added the following material:
- results on limits of algebraic spaces (including results on quasi-coherent modules),
- results on coherent modules on locally Noetherian algebraic spaces, see
Section Tag 07U9 and Section Tag 07UI,
- devissage of coherent modules on Noetherian algebraic spaces, see Section Tag 07UN
- a decent singleton algebraic space is a scheme (Lemma Tag 047Z),
- a qc + qs algebraic space such that H^1 is zero on any quasi-coherent module is an affine scheme (Proposition Tag 07V6),
- if X —> Y is a surjective integral morphism, X is an affine scheme, and Y an algebraic space, then Y is an affine scheme (as far as I know this result is due to David Rydh), see Proposition Tag 07VT,
- the previous result in particular implies that if an algebraic space has a reduction which is a scheme then it is a scheme (you can find this in a paper by Conrad, Lieblich, and Olsson). This allowed us to significantly improve the exposition on thickenings of algebraic spaces which leads into the next item,
- pushouts of algebraic spaces, see Section Tag 07SW,
- this is applied to get a very general version of the Rim-Schlessinger condition for algebraic stacks, see Section Tag 07WM,
- a section about what happens with deformation theory when you have a finite extension of residue fields (possibly inseparable), see Section Tag 07WW,
- a (partial) solution to question 04PZ thanks to Philipp Hartwig, see Lemma Tag 07VM,
- a bunch more stuff in the chapter on Artin’s Axioms including an approach to checking openness of versality which works exactly as explained here and here.
A short term goal is now to apply the results of the chapter on Artin’s Axioms to show that some natural moduli problems (in restricted generality) are representable by algebraic stacks or algebraic spaces. For example: Picard stacks, moduli of curves, moduli of canonically polarized smooth projective varieties, Hilbert schemes/spaces, Quot schemes/spaces, etc.
A longer term goal would be to get the most general results of this type, for example the stack (of flat families) of finite covers of P^n (this is a made up example). For the longer term goal I see no way around working with the full cotangent complex (and not the naive one). Do you?
[Edit July 5, 2012: Jason Starr points out that in his preprint on “Artin’s Axioms” in Remark 4.5 he proves the stack mentioned above is an algebraic stack without using the full cotangent complex.]