Timeline for Prove that $\overline{a}_{11}$ is a prime element in $R$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| May 13, 2022 at 16:36 | vote | accept | It'sMe | ||
| May 13, 2022 at 13:55 | history | edited | It'sMe | CC BY-SA 4.0 |
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| May 12, 2022 at 17:31 | answer | added | Jason Starr | timeline score: 1 | |
| May 11, 2022 at 10:33 | comment | added | Jason Starr | @FreePawn. The ring in your case is a $\mathbb{Z}_{\geq 0}$-graded ring, and the element $a_{1,1}$ is a homogeneous element. Thus, the element $a_{1,1}$ is prime if and only if the image in the associated local ring is prime (localization at the maximal ideal generated by all positive degree homogeneous elements). | |
| May 11, 2022 at 9:23 | history | edited | It'sMe | CC BY-SA 4.0 |
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| May 11, 2022 at 6:06 | comment | added | It'sMe | Dear sir @JasonStarr isn't the Samuel's conjecture and Grothendieck's proof of Samuel's conjecture involve Local rings? The ring $R$ in my case is not a local ring(unless I'm missing something) | |
| May 10, 2022 at 21:16 | history | edited | LSpice | CC BY-SA 4.0 |
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| May 10, 2022 at 20:49 | comment | added | It'sMe | Dear Sir @JasonStarr, My main aim is to prove $R$ is a UFD. So, I was trying to use Nagata's criterion for UFD (first prove that $R$ is an integral domain and then $\overline{a}_{11}$ and $\overline{c}_{22}$ are prime elements and then, the localization of $R$ by multiplicative set generated by these two prime elements is a UFD and hence, $R$ is UFD). But, it seems you have better approach. | |
| May 10, 2022 at 20:47 | comment | added | It'sMe | Dear Sir @JasonStarr , I don't have advanced knowledge in algebraic geometry. But it'll be nice if you can elaborate your approach little bit. I would like to know. | |
| S May 10, 2022 at 20:42 | history | suggested | CommunityBot | CC BY-SA 4.0 |
some MathJax copy-editing improvements
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| May 10, 2022 at 20:39 | comment | added | Jason Starr | What background are you supposed to know? That ring is a complete intersection ring whose singular locus appears to be of codimension $4$. Thus, by Grothendieck's proof of the Samuel conjecture, the ring is factorial. Thus, the given element is prime if and only if it is irreducible. | |
| May 10, 2022 at 20:15 | review | Suggested edits | |||
| S May 10, 2022 at 20:42 | |||||
| May 10, 2022 at 19:08 | history | asked | It'sMe | CC BY-SA 4.0 |