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Stream: learning: questions

Topic: Pushouts of monomorphisms of noncommutative k-algebras

David Michael Roberts (Oct 12 2023 at 04:04):

The pushout of a monomorphism in an abelian category is a monomorphism, but what about in a category of non-commutative but associative algebras (over a field, say)? I have no intuition for this. I'm happy to assume unital algebras, but the non-unital case (which is of course is more subtle) would be of interest, too.

Todd Trimble (Oct 12 2023 at 10:59):

If you accept the red herring principle that noncommutative algebras can be commutative, then consider the pushout of the map $\mathbb{Z} \to \mathbb{Q}$ along $\mathbb{Z} \to \mathbb{Z}/(3)$.

David Michael Roberts (Oct 12 2023 at 11:39):

Ok, gotcha. I'll have to think to see if I should be using a stronger notion of monomorphism that rules out such a thing.

But.... Z is not an algebra over a field!

David Michael Roberts (Oct 12 2023 at 11:47):

In particular @Todd Trimble your example is mixing characteristics, but I want to fix a base field.

Todd Trimble (Oct 12 2023 at 12:46):

Yes, I was reading too quickly. But, okay, does something like pushing out $k[x] \hookrightarrow k(x)$ along $k[x] \to k[x]/(x^2)$ work? Same idea, really.

Todd Trimble (Oct 12 2023 at 12:51):

Basically, you can't have $x$ going to an element $r$ in a $k$-algebra $R$ that is both nilpotent and invertible at the same time.