Timeline for When does a subspace of the affine space form a regular sequence in a ring of regular functions?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 27, 2018 at 3:11 | answer | added | benblumsmith | timeline score: 1 | |
| Sep 26, 2018 at 8:20 | history | edited | BrianT | CC BY-SA 4.0 |
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| Sep 26, 2018 at 8:20 | comment | added | BrianT | Oh ok, thank you. I erased this sentence, which is not necessary anyway. Suppose then that $J$ is not necessarily radical. Is there a way to relate the fact that $I$ is generated by a regular sequence in $\mathbb{C}[u] / J$ and the zero sets $\mathbb{C}^k$ and $V(J)$ ? | |
| Sep 26, 2018 at 8:12 | comment | added | red_trumpet | For $\mathbb{C}[u]/J$ to be isomorphic to the ring of regular functions on the zero zet $V(J)$ you need $J$ to be radical. For a counterexample consider $n = 1$. Then $V(u_1) = V(u_1^2)$, but obviously $\mathbb{C}[u_1]/(u_1) \ncong \mathbb{C}[u_1]/(u_1^2)$. | |
| Sep 26, 2018 at 7:51 | history | edited | BrianT | CC BY-SA 4.0 |
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| Sep 26, 2018 at 5:59 | comment | added | BrianT | I also don’t understand your statement “and implied by vanishing after any faithlully flat base change”. | |
| Sep 25, 2018 at 19:38 | comment | added | BrianT | Thanks @Jason Starr. I don’t understand your statement about $J$ being radical. Seems that the flatness of the ring homomorphism is enough (does this depend on radicality of $J$ ?). | |
| Sep 25, 2018 at 16:37 | comment | added | Jason Starr | Your claim in the second sentence is only valid if $J$ is a radical ideal. Regularity of a sequence is equivalent to the vanishing of the higher homology modules of the Koszul complex associated to the sequence. This vanishing is preserved by every flat base change (and implied by vanishing after any faithfully flat base change). The ring homomorphism $\mathbb{C}[u]/J \to \mathbb{C}[u,u^{-1}]/J\mathbb{C}[u,u^{-1}]$ is flat (usually not faithfully flat). Thus, the image in $\mathbb{C}[u,u^{-1}]/J\mathbb{C}[u,u^{-1}]$ of a regular sequence in $\mathbb{C}[u]/J$ is still a regular sequence. | |
| Sep 25, 2018 at 15:53 | history | asked | BrianT | CC BY-SA 4.0 |