Timeline for What is the definition of algebro-gemetric quotient?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Feb 6, 2011 at 21:37 | comment | added | JBorger | Harry, a stack is a 2-categorical object, whereas an algebraic space is a 1-categorical concept. In any situation where your action has nontrivial stabilizers (such as $G$ acting on a point, as jlk points out), the two will disagree, because the algebraic space quotient doesn't have enough n-categorical headroom to carry stabilizer information. The quotients in the categories of sheaves and algebraic spaces also typically disagree. (See Dan Peterson's example of $\mathbf{C}^*$ acting on $\mathbf{C}^2$.) They agree precisely when the algebraic space quotient is universally effective. | |
| Feb 6, 2011 at 18:08 | comment | added | jlk | In this case, the categorical quotient $X/G$ that James Borger describes is just the point. The stack-theoretic quotient is the classifying stack of $G$. In other words, an object over a scheme $T$ is a principal $G$-bundle $P \to T$. This is not equivalent to a $1$-category (as I am sure you know). | |
| Feb 6, 2011 at 18:04 | comment | added | jlk | @Harry Gindi: I believe the two notions are distinct in the case of the quotient of a point by the trivial action of a (non-trivial) group $G$. | |
| Feb 6, 2011 at 12:21 | comment | added | Harry Gindi | Dear James, how does the stack-theoretic quotient differ from the quotient of algebraic spaces or the sheaf-theoretic quotient (and do these two generally agree)? | |
| Feb 6, 2011 at 7:03 | history | answered | JBorger | CC BY-SA 2.5 |