Epimorphisms in Alg(R) are reflected from CRing

[catdat-issue-creation]

I claim that epimorphisms in Alg(R) are simply morphisms of R-algebras which happen to be epimorphisms of commutative rings.

Let f : A -> B be a morphism of R-algebras. It is clear that if f is an epimorphism in CRing, then it is an epimorphism in Alg(R). Conversely, suppose f is an epimorphism in Alg(R).

Now we notice that the two natural maps B -> B \otimes_A B are R-algebra homomorphisms.

Since the two composites A -> B -> B \otimes_A B are equal and f is an epimorphism, we conclude that the two natural maps B -> B \otimes_A B are equal.

This implies that f is an epimorphism of commutative rings (see https://stacks.math.columbia.edu/tag/04VN).

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This issue has been created by Ben Spitz via the submission form on https://catdat.app/category/Alg(R)

[ScriptRaccoon]

Thanks! I assume you mean commutative algebras. I have added the proof here: #58

[diracdeltafunk]

Yes I did mean commutative algebras! Thanks for catching that :)