Timeline for Tensor powers of an algebra all isomorphic
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 26, 2014 at 4:27 | comment | added | Jesse Elliott | I found this reference, which shows that $A \otimes_k A \longrightarrow A$ is an isomorphism if and only if $k \longrightarrow A$ is an epimorphism of rings. mathoverflow.net/questions/56591/… So the question reduces to that of epimorphisms. Also found this: mathoverflow.net/questions/109/… | |
| Jan 26, 2014 at 1:43 | vote | accept | Jesse Elliott | ||
| Jan 26, 2014 at 1:42 | comment | added | Jesse Elliott | Ok, thanks. Now I see why flatness might need to be assumed. | |
| Jan 26, 2014 at 1:14 | comment | added | Jason Starr | @JesseElliott: No, that does not get rid of those types of counterexamples -- please see my third edit. | |
| Jan 26, 2014 at 0:24 | comment | added | Jesse Elliott | I added the assumption that $k \longrightarrow A$ be injective, which gets rid of some easy counterexamples, including the $\mathbb{Z}/2\mathbb{Z}$ example. | |
| Jan 26, 2014 at 0:18 | comment | added | Jason Starr | @JesseElliot: "Please add $A\neq 0$ to my assumptions." Why don't you add that to your assumptions, yourself? In fact, I suggest that you just accept one of the two correct answers. If you want to ask another question, you can always do that. | |
| Jan 26, 2014 at 0:13 | comment | added | Jesse Elliott | Please add $A \neq 0$ to my assumptions. | |
| Jan 26, 2014 at 0:07 | comment | added | Laurent Moret-Bailly | "Which means $A\subset K$": yes assuming e.g. that $k$ is a domain and $A\neq0$. In fact, the zero algebra is a counterexample to many claims. Also, "finitely presented" is irrelevant, I think. | |
| Jan 25, 2014 at 23:23 | comment | added | Jesse Elliott | Flatness is sufficient for the given conclusion, but is it necessary? And what is a counterexample when $k \longrightarrow A$ is injective? | |
| Jan 25, 2014 at 20:30 | comment | added | Jason Starr | That makes sense to me. | |
| Jan 25, 2014 at 19:54 | history | answered | abx | CC BY-SA 3.0 |