Just an update on what’s been going on since the last update. The following list is roughly in chronological order.

- Jonathan Wang send us a bunch of lemmas which help determine whether a given stack in groupoids is an algebraic stack.
- We added enough material on finite Hilbert stacks so we can use them. These results are mainly contained in the chapter entitled “Criteria for representability”. It came as a big relief to me that these results are painless to prove given the results on algebraic spaces at our disposal.
- Removed the ridiculous term “distilled” and replaced it by “quasi-DM” as suggested by Brian Conrad.
- Started a chapter entitled “Quot and Hilbert Spaces” where we will eventually put results on existence (as algebraic spaces) of Quot spaces and Hilbert spaces. So far it only contains a discussion of “the locus where a morphism has property P”.
- Added an example of a module which is a direct sum of countably many locally free modules of rank 1 but is not itself locally free.
- Added a bunch more basic results on modules on algebraic spaces, and on morphisms of algebraic spaces.
- The pullback of a flat module along a morphism of ringed topoi is flat. We only proved this in case the topoi have enough points. The general case (due to Deligne) is a bit harder to prove, and we’ll likely never use it.
- The fppf topology is the topology generated by open coverings and finite locally free morphisms. Discussed previously on the blog.
- Basics of flatness and morphisms of algebraic spaces (openness, criterion par fibre, etc).
- Added an example of a formally etale nonflat ring map due to Brian Conrad.
- Infinitesimal thickenings of algebraic spaces. We study these using the earlier results on algebraic spaces as locally ringed topoi discussed earlier on this blog. A key technical ingredient is that a first order thickening of an affine scheme in the category of algebraic spaces is an affine scheme. This can be tremendously generalized (see work by David Rydh), but that would require a _lot_ more work.
- Universal first order thickenings for formally unramified morphisms of algebraic spaces.
- Fixed section on formally etale morphisms of algebraic spaces.
- Section on infinitesimal deformations of maps of algebraic spaces. This is now very slick, due to the work on thickenings above.
- Fixed proof of relationship formally smooth morphisms of algebraic spaces and smooth morphisms of algebraic space.
- Formal smoothness for algebraic spaces is etale local on the source.
- Relative effective Cartier divisors.
- Lots of material on regular sequences, regular immersions, relative regular immersions, all intended to be used eventually to define local complete intersection morphisms.
- Introduced the following algebra notions:
- Pseudo-coherent complexes
- Tor amplitude and complexes of finite tor dimension
- Perfect complexes
- Relatively pseudo-coherent complexes
- Pseudo-coherent ring maps
- Perfect ring maps

- Introduced the following types of morphisms of schemes:
- Pseudo-coherent morphisms of schemes
- Perfect morphisms of schemes
- Local complete intersection morphisms

Among some of the properties of these we proved that local complete intersection morphisms are fppf local on the target and syntomic local on the source. Hence it makes sense to say that a morphism of algebraic spaces is a local complete intersection morphism. We should now be in a good position to define the “lci-locus” in the Hilbert stack, which is our next goal.