Epimorphism of rings

We added a section on epimorphisms of rings to the algebra chapter. Everything is completely straightforward except for the following fact: If A —> B is an epimorphism of rings then |B| ≤ |A|. Since an epimorphism of rings is not necessarily surjective this is not a triviality. We learned this from the exposee by Mazet in the Seminaire Samuel.

You can use this to show that if X —>Y is a monomorphism of schemes then size(X) ≤ size(Y), which is a technical condition on the cardinalities of some sets associated to X and Y. See the chapter on sets.

Here is a consequence: Given a scheme Y there is a set worth of isomorphism classes of monomorphisms X —> Y. I don’t think this is formal since I vaguely remember reading somewhere about a category (maybe spaces up to homotopy?) where such a thing is not true. Leave a comment if you know the correct statement.

You can be an author too!

Take a look at the following commit. Observe how the author of this commit is Hendrik! But if you look at the summary of all commits, then you find that most commits have been authored by me.

What happened is that Hendrik sent me a patch formatted in such a way that git was able to understand who authored this patch. Maybe he used the instructions here, or he figured it out himself. It doesn’t matter.

But, if you want your name to show as an author in the logs of the stacks project, then you can do this too!


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

  1. 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,
  2. 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
  3. 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.