Timeline for Invariant ring of the subvariety
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 27, 2023 at 2:39 | vote | accept | It'sMe | ||
| Jan 27, 2023 at 2:19 | answer | added | LSpice | timeline score: 5 | |
| Jan 27, 2023 at 2:13 | comment | added | It'sMe | @LSpice Thanks for your explanation. I use the same definition. It's just that I didn't observe it in terms of the exact sequence, but I see it now. | |
| Jan 27, 2023 at 2:03 | comment | added | LSpice | Re, this is, again, for me the definition of linear reductivity: that every $G$-representation is completely reducible; and the definition of complete reducibility is that every stable subspace has a stable complement. Again, if you use another definition, could you tell me what it is so that I can see if I can find a source for the equivalence? | |
| Jan 27, 2023 at 1:10 | comment | added | It'sMe | @LSpice I meant the fact about the "$G$-module splitting". Sorry, I should have been more precise. | |
| Jan 27, 2023 at 0:55 | comment | added | LSpice | Re, for me, the exact sequence I mentioned is the definition of a closed subvariety of an affine variety. Could you say what is your definition? | |
| Jan 27, 2023 at 0:49 | history | edited | It'sMe | CC BY-SA 4.0 |
added 25 characters in body
|
| Jan 27, 2023 at 0:48 | comment | added | It'sMe | @LSpice Thanks for your reply. Yes, I meant $Y$ is $G$-stable. For my situation, $X$ is an affine space. Also, can you give some kind of reference for the exact sequence that you've mentioned. | |
| Jan 27, 2023 at 0:33 | comment | added | LSpice | Re, assuming you meant $Y$ to be $G$-stable. Otherwise, I don't know what $\mathbb K[Y]^G$ means. | |
| Jan 27, 2023 at 0:27 | comment | added | LSpice | Under your hypotheses, isn't $\mathbb K[Y]^G$ the quotient of $\mathbb K[X]^G$ by the $G$-fixed vectors in the ideal $I_{Y \subseteq X}$ of functions in $\mathbb K[X]$ vanishing on $Y$? At least this seems so as vector spaces, since linear reductivity implies that there is a $G$-module splitting of $0 \to I_{Y \subseteq X} \to \mathbb K[X] \to \mathbb K[Y] \to 0$. | |
| Jan 26, 2023 at 22:56 | history | asked | It'sMe | CC BY-SA 4.0 |