Bad Categorical Quotients - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T01:42:15Z http://mathoverflow.net/feeds/question/801 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/801/bad-categorical-quotients Bad Categorical Quotients Harold Williams 2009-10-16T21:04:32Z 2009-10-19T15:40:00Z <p>Let G be an algebraic group acting on a scheme X. Then f: X --> Y is a categorical quotient if it is constant on G-orbits and any other G-invariant morphism factors through it in a unique fashion. We say f is a 'good' categorical quotient if:</p> <p>1) f is a surjective open submersion (i.e. the topology on Y is induced from X).</p> <p>2) for any open U &sub; Y, the induced map from functions on U to G-invariant functions on f^-1(U) is an isomorphism.</p> <p>Does anyone know an example of a 'bad' categorical quotient (by which I mean...well...a not good one). </p> http://mathoverflow.net/questions/801/bad-categorical-quotients/1069#1069 Answer by Jarod Alper for Bad Categorical Quotients Jarod Alper 2009-10-18T20:37:52Z 2009-10-19T15:40:00Z <p>Note that if f: X&rarr;Y is a categorical quotient in the category of schemes which is stable under base change by open immersions, then the second condition (ie. O<sub>Y</sub>&rarr;(f<sub>*</sub> O<sub>X</sub>)<sup>G</sup> is an isomorphism) is automatically satisfied. </p> <p>In the paper "<a href="http://arxiv.org/abs/math/0002096" rel="nofollow">Examples and counterexamples for existence of categorical quotients</a>" by A'Campo-Neuen and Hausen, there is an example of a categorical quotient f: X&rarr;<b>A</b><sup>1</sup> such that f<sup>-1</sup>(<b>A</b><sup>1</sup> - 0)&rarr;<b>A</b><sup>1</sup> - 0 is not a categorical quotient. I haven't checked but I believe this should also give an example where condition (2) fails.</p> <p>I don't know of example of a categorical quotient where condition (1) fails. </p>