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Stream: deprecated: mathematics

Topic: epimorphisms of rings

John Baez (Sep 17 2021 at 21:13):

Epimorphisms of rings turn out to be pretty interesting. On the n-Cafe I wrote about rings $R$ for which the unique morphism from $\mathbb{Z}$ to $R$ is an epimorphism:

• Solid rings.

It turns out they're all commutative, so we can also think of them as subterminal objects in the category of affine schemes.

The ensuing conversation was pretty interesting - well, to me anyway! - and I think a lot of the questions raised may get sorted out nicely if the category CommRing (and/or Ring) has a (epi, regular mono) factorization system. This may have already been settled by Isbell; I haven't checked yet. Anyway, maybe some of you can settle some of the questions we raised.

John Baez (Sep 17 2021 at 21:14):

I'd prefer comments there to comments here, since the conversation is going on there.

Johan Commelin (Sep 19 2021 at 18:57):

@John Baez Sorry for posting a comment here anyway. My first gut reaction is: won't this give a name clash with 'solid' as used in condensed mathematics nowadays? Solid abelian groups are everywhere. And monoid objects in that category are probably called solid rings.

Mike Shulman (Sep 19 2021 at 19:50):

Wouldn't it be more accurate to complain about the use of "solid" in condensed mathematics clashing with this? Presumably the notion of "solid ring" is much older.

John Baez (Sep 19 2021 at 21:23):

Yeah, the people who invented the term "solid ring" - two guys named Bousfeld and Kan - weren't smart enough to look into the future and avoid the clash with Scholze's terminology.

David Michael Roberts (Sep 19 2021 at 23:18):

I guess the problem is that B&K were not algebraic geometers, so their terminology never caught on there.

Mike Shulman (Sep 20 2021 at 00:45):

I feel lIke I've also seen the adjective "solid" applied in another mathematical context that isn't the same as either of these, but I'm blanking on exactly where.

Graham Manuell (Sep 20 2021 at 20:42):

Mike Shulman said:

I feel lIke I've also seen the adjective "solid" applied in another mathematical context that isn't the same as either of these, but I'm blanking on exactly where.

Well, there are solid functors at least.

Mike Shulman (Sep 20 2021 at 20:46):

Perhaps that was it.

John Baez (Sep 20 2021 at 21:02):

There's "solid geometry". :upside_down:

Tim Hosgood (Oct 07 2021 at 08:05):

i forgot to mention this when this thread was alive, but epimorphisms of rings are so interesting that there was an entire seminar on them in the late 60s! http://www.numdam.org/issues/SAC_1967-1968__2_/

Tim Hosgood (Oct 07 2021 at 08:07):

a particularly nasty example which arises is an epimorphism of rings which is an inclusion, sends non-units to non-units, and the codomain has no idempotents, but the epimorphism is not an isomorphism (see e.g. https://mathoverflow.net/a/139/73622)

Tim Hosgood (Oct 07 2021 at 08:12):

also (from the same mathoverflow thread) is an interesting characterisation of epimorphisms of (commutative) rings!

A homomorphism $f\colon A \to B$ is an epimorphism if and only if for all $b \in B$ there exist matrices $C$, $D$, $E$ of sizes $1 \times n$, $n \times n$, and $n \times 1$ respectively, where

i. $C$ and $E$ have entries in $B$,
ii. $D$ has entries in $f(A)$,
iii. the entries of $CD$ and of $DE$ are elements of $f(A)$, and
iv. $b = CDE$.

(Such a triple is called a zig-zag for b.)

Fawzi Hreiki (Oct 07 2021 at 08:17):

I’ve seen this characterisation lots of places, but is it actually useful for verifying epimorphicity in practice?

Morgan Rogers (he/him) (Oct 07 2021 at 08:22):

Tim Hosgood said:

A homomorphism $f\colon A \to B$ is an epimorphism if and only if for all $b \in B$ there exist matrices $C$, $D$, $E$ of sizes $1 \times n$, $n \times n$, and $n \times 1$ respectively, where...

what is $n$ here?

Fawzi Hreiki (Oct 07 2021 at 08:28):

It’s any natural number n

Fawzi Hreiki (Oct 07 2021 at 08:28):

It’s just so that you can take their product

John Baez (Oct 08 2021 at 19:05):

I don't know what this characterization is used for, but I know what it's called: "the Silver-Mazet-Isbell Zigzag Lemma".

John Baez (Oct 08 2021 at 19:07):

Isbell is a category theorist (as many of you know), and it seems he wrote a least four papers on epimorphisms. Here's one:

• John R. Isbell, Epimorphisms and dominions. IV.

Unfortunately I have not read it, and now I'm interested in something else!

John Baez (Oct 08 2021 at 19:10):

There's more about this here:

• B. Mitchell, The dominion of Isbell.

This generalizes some ideas to arbitrary categories. Nice title! It makes Isbell sound like a king. :crown: