Timeline for Are schematic fixed-points of a Cohen-Macaulay scheme Cohen-Macaulay?
Current License: CC BY-SA 2.5
11 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Oct 26, 2010 at 5:48 | history | edited | Hailong Dao | CC BY-SA 2.5 |
added 222 characters in body
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| Oct 26, 2010 at 3:40 | comment | added | Ben Webster♦ | Hailong- You might want to add a note to the answer explaining this. | |
| Oct 26, 2010 at 2:07 | comment | added | Hailong Dao | @ Allen and BCnrd: thank you! I will leave my answer, so people might benefit from your explanations and avoid my mistakes! | |
| Oct 26, 2010 at 1:28 | comment | added | BCnrd | Dear Hailong: No ad hoc constructions/definitions are required. For any scheme $X$ over a ring $k$ and action on $X$ by a $k$-gp scheme $G$, define functor $X^G$ on $k$-algebras as follows: for $k$-algebra $R$, $X^G(R)$ is set of $x \in X(R)$ fixed by the $G_R$-action on $X_R$ (i.e., for any $R$-algebra $R'$, $x$ viewed in $X(R')$ is $G(R')$-invariant). Is this represented by a closed subscheme of $X$? Yes, provided $X$ is locally of finite type and separated over $k$ and $G$ is affine and fppf over $k$ with connected fibers. (See Prop. A.8.10(1)ff. in "Pseudo-reductive groups" for details.) | |
| Oct 26, 2010 at 0:43 | comment | added | Allen Knutson | @Hailong: you applied the fixed-point functor to the algebra rather than to the space. That is, the notation itself has a consistent meaning. (I'm not surely how to reasonable define schematic fixed points; in characteristic zero, with a connected group, one could look where the generating vector fields vanish.) | |
| Oct 26, 2010 at 0:10 | comment | added | Hailong Dao | Hmm, sorry, I always thought of this notation as the invariants. What do you mean by schematic fixed pts? Are they just literally the pts of $X$ fixed by the group action? | |
| Oct 25, 2010 at 23:12 | comment | added | Ben Webster♦ | Hailong- this is the categorical quotient, not schematic fixed points. The schematic fixed points here are a single point with reduced structure and thus Cohen-Macaulay. | |
| Oct 25, 2010 at 23:05 | vote | accept | Ben Webster♦ | ||
| Oct 25, 2010 at 23:10 | |||||
| Oct 25, 2010 at 22:41 | history | edited | Hailong Dao | CC BY-SA 2.5 |
edited body
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| Oct 25, 2010 at 22:32 | history | answered | Hailong Dao | CC BY-SA 2.5 |