Timeline for Are the rings $\mathbb{C}[X]/\langle X^2- c \mathrm{Tr}(X) X \rangle$ isomorphic when $c$ ranges over a neighborhood of 0?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Mar 18, 2020 at 23:04 | answer | added | Libli | timeline score: 1 | |
| Mar 10, 2020 at 8:00 | comment | added | YCor | For $N=2$ I prove (in a comment below) that the (non-reduced) $\mathbf{C}$-algebras $R_{2,c}$, for $c\notin\{1,1/2,0\}$ are all isomorphic. | |
| Mar 10, 2020 at 0:31 | history | edited | Libli | CC BY-SA 4.0 |
this is not a vriety but a scheme, it is not reduced
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| Mar 9, 2020 at 17:02 | comment | added | YCor | For $N=1$ the $J_c$ ($c\neq 1$)are clearly all equal to $J_0$. On the other hand, for $N=2$ (hence $N\ge 2$) it's also true that the $J_c$ are pairwise distinct. Indeed, consider the (local artinian) ring $S=\mathbf{C}[u,v]/(u^3,v^3,uv,v^2-(2c-1)u^2)$. Let $M$ be the diagonal matrix $\mathrm{diag}(u+v,u-v)$. Then $M^2-c'\mathrm{Tr}(M)M=(u^2+v^2)I_2-2c'u^2I_2=2(c-c')u^2I_2$. Since $u^2\neq 0$, we see that $M$ belongs to $V_{N,c'}(S)$ iff $c=c'$. | |
| Mar 9, 2020 at 16:22 | comment | added | YCor | OP, you shouldn't accept an answer which doesn't answer the question. | |
| Mar 9, 2020 at 16:21 | comment | added | YCor | To summarize context, Libli's answer shows (for given $N$) that for $c\notin\{1,1/2,\dots,1/N\}$, we have $\sqrt{J_c}=\sqrt{J_0}$, where $\sqrt{\;}$ denotes the radical of an ideal. And the answer by "guest" asserts that (for $N\ge 3$) $J_0$ is not contained in $J_c$ for any $c\neq 0$ (so an isomorphism can't be induced by the identity map). | |
| Mar 9, 2020 at 16:08 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo (due to me). added practical notation to be used
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| Mar 9, 2020 at 14:36 | answer | added | guest | timeline score: 3 | |
| Mar 8, 2020 at 2:45 | vote | accept | Yuji Tachikawa | ||
| Mar 10, 2020 at 15:54 | |||||
| Mar 6, 2020 at 9:46 | history | edited | YCor | CC BY-SA 4.0 |
fixed clumsy notation
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| Mar 6, 2020 at 9:42 | comment | added | YCor | Could you locate where it occurs in this 60-page paper? | |
| Mar 6, 2020 at 9:40 | history | edited | YCor | CC BY-SA 4.0 |
fixed clumsy notation
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| Mar 6, 2020 at 9:34 | history | edited | YCor | CC BY-SA 4.0 |
fixed clumsy notation
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| Mar 5, 2020 at 14:22 | history | edited | LSpice | CC BY-SA 4.0 |
Name of paper; deleted 'irresponsible'
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| Mar 5, 2020 at 13:01 | answer | added | Libli | timeline score: 7 | |
| Mar 5, 2020 at 2:12 | history | asked | Yuji Tachikawa | CC BY-SA 4.0 |