A morphism of finite presentation X —> S is a morphism which is (a) locally of finite presentation, (b) quasi-separated, and (c) quasi-compact.

Let κ be an infinite cardinal. What should be a morphism of κ-presentation? By analogy with the above I think it should be a morphism f : X —> S such that

- for any affine opens U, V of X, S with f(U) ⊂ V the algebra O(U) is of the form O(V)[x_i; i ∈ I]/(f_j; j ∈ J) with |I|, |J| ≤ κ,
- for any U, U’ affine open in X over an affine V of S the intersection U ∩ U’ can be covered by κ affine opens, and
- for any affine V in S the inverse image f^{-1}(V) can be covered by κ affine opens.

It is my guess that all the usual things we prove for morphisms of finite presentation also hold for morphisms of κ-presentation. Namely, it should be enough to check the conditions over the members of an affine open covering of Y, the base change of a morphism of κ-presentation is a morphism of κ-presentation, etc. In particular, if should also be true that if {S_i —> S} is an fpqc covering and X_i —> S_i is the base change of f : X —> S, then

X —> S is of κ-presentation ⇔ each X_i —> S_i is of κ-presentation

Of course this is completely orthogonal to most of algebraic geometry and I hope you’ve already stopped reading several lines above (maybe when I used the key word “cardinal”). For those of you still reading let me indicate what prompted me to write this post. Namely, suppose that X, Y are schemes over a base S which are fpqc locally isomorphic. Then the above says that X and Y have roughly the same “size” (this is defined precisely in the chapter on sets in the stacks project).

As an application this tells us for example that given a group scheme G over S there is a set worth of isomorphism classes of principal homogeneous G-spaces over S! A principal homogeneous G-space is defined in the stacks project, as in SGA3, to be a pseudo G-torsor which is fpqc locally trivial — and note that the collection of fpqc coverings of S forms a proper class, which does **not** contain a cofinal subset!

Another potential application, internal to the stacks project and with notation and assumptions as in the stacks project, is that, given a group algebraic space G over S, it guarantees that the stack of principal homogeneous G-spaces form a stack in groupoids over (*Sch*/S)_{fppf}. Instead of working this out in detail in the stacks project I will for now put in a link to this blog post.