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.

2 thoughts on “Epimorphism of rings

  1. If Sub(X) is small for every X in a category C, then C is said to be _wellpowered_ in the categorical literature. Here by Sub(X) I mean the collection of equivalence classes of monomorphisms in to X. It is certainly _not_ the case that every locally small category is wellpowered. A simple example of a non-wellpowered category (from _The Joy of Cats_, by Ad├ímek, Herrlich, and Strecker) is this: Let C^op be a large well-ordered set. Then C is not wellpowered. Other examples come when derived or homotopy categories are embedded into e.g. abelian categories — see e.g. [this MathOverflow question](http://mathoverflow.net/questions/93853/abelian-category-which-is-not-well-powered).

    Anyway, it sounds like you’re claiming that wellpoweredness of the category schemes follows from wellpoweredness of the category of affine schemes, which would provide a positive answer to [this MathOverflow question](http://mathoverflow.net/q/160681/2362). However, on inspection of the relevant Stacks Project theorem ([Lemma 9.8 of “Set Theory”](http://stacks.math.columbia.edu/tag/04VA)), it appears that there are extra technical conditions that prevent the argument from actually applying to _arbitrary_ monomorphisms of schemes. I wonder if these technical conditions can be lifted?

  2. It seems the second paragraph of my first comment was spoken too hastily. Laurent Moret-Bailly has supplied the (easy) argument from wellpoweredness of affine schemes to wellpoweredness of schemes at [the MathOverflow question I linked to above](http://mathoverflow.net/a/160871/2362). It’s so simple, I suppose it went without saying! The extra technical conditions are only needed to get specific cardinal bounds on |Sub(X)|.

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