Timeline for Tensor product and homomorphism
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 7, 2023 at 23:04 | history | became hot network question | |||
| Feb 7, 2023 at 22:04 | vote | accept | marco2013 | ||
| Feb 7, 2023 at 20:53 | answer | added | Martin Brandenburg | timeline score: 8 | |
| Feb 7, 2023 at 20:50 | comment | added | Benjamin Steinberg | @MartinBrandenburg, maybe it isn't. I'm used to finite dimensional algebras | |
| Feb 7, 2023 at 20:47 | comment | added | Martin Brandenburg | I also highly doubt that this is sufficient. | |
| Feb 7, 2023 at 20:45 | comment | added | Benjamin Steinberg | Maybe one should ask M to be finitely presented to be safe | |
| Feb 7, 2023 at 20:36 | comment | added | Martin Brandenburg | @BenjaminSteinberg Maybe I misunderstood your comment, but you are saying that $f$ is always injective? I doubt that. | |
| Feb 7, 2023 at 20:07 | comment | added | Benjamin Steinberg | I think it is injective and the image is all endomorphisms M that factor through a finitely generated projective. I don't have time to check details but the inverse map should involve choosing a factorization through some $A^m$ and then use the components to build something in the tensor product. One would need to show well defined | |
| Feb 7, 2023 at 17:19 | history | edited | marco2013 |
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| Feb 7, 2023 at 16:11 | comment | added | marco2013 | Yes, if $M$ is projective, $f$ is a monomorphism, but the reverse is not true. For example, if $A=\mathbb{Z}$ and $M=\mathbb{Z}/n \mathbb{Z}$, then $M^*=0$, so $M \otimes M^*=0$, so $f$ is injective, but $M$ is not a projective module. | |
| Feb 7, 2023 at 15:50 | comment | added | Sampah | It is a classical result that the mentioned map is a bijection if and only if $M$ is finitely generated and projective (fgp), so being fgp is a sufficient condition. But I believe we can weaken it to the module $M$ being merely projective. | |
| Feb 7, 2023 at 15:41 | review | Close votes | |||
| Feb 7, 2023 at 21:57 | |||||
| Feb 7, 2023 at 15:04 | history | asked | marco2013 | CC BY-SA 4.0 |