A morphism of finite presentation X —> S is a morphism which is (a) locally of finite presentation, (b) quasi-separated, and (c) quasi-compact.<\/p>\n
Let \u03ba be an infinite cardinal. What should be a morphism of \u03ba-presentation? By analogy with the above I think it should be a morphism f : X —> S such that<\/p>\n
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<\/p>\n
X —> S is of κ-presentation ⇔ each X_i —> S_i is of κ-presentation<\/p>\n
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).<\/p>\n
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<\/strong> contain a cofinal subset!<\/p>\n 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<\/em>\/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.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 \u03ba be an infinite cardinal. What should be a morphism of \u03ba-presentation? By analogy with the … Continue reading