Timeline for Exactness is often an open condition. How often?
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7 events
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| Feb 12, 2013 at 5:12 | comment | added | Theo Johnson-Freyd | ... For abelian groups, I think I can do the infinitesimal case from a spectral-sequence argument, using the fact that the if the associated-graded of a filter abelian group is zero, then the filtered abelian group was already zero; but there, I don't even know "infinitesimal upper semicontinuity" except near exact complexes, and I really don't know how to say non-infinitesimal statements. | |
| Feb 12, 2013 at 5:11 | comment | added | Theo Johnson-Freyd | @Mariano: Right, for finite-dimensional vector spaces I know how to do everything I've asked. Already for infinite-dimensional vector spaces, though, I know how to prove semicontinuity for "infinitesimal" open neighborhoods, but I don't know enough about infinite-dimensional vector spaces to say things finitely... | |
| Feb 10, 2013 at 15:13 | comment | added | Martin Brandenburg | "I expect that there is a natural way to give the hom spaces in an abelian category, and thus the space of differentials, the structure of affine algebraic varieties;" I am not sure if this works without any assumption on the abelian category. | |
| Feb 10, 2013 at 11:25 | comment | added | Allen Knutson | Mariano, I think you mean affine variety, with equations $d^2=0$. | |
| Feb 10, 2013 at 9:13 | comment | added | S. Carnahan♦ | Regarding your parting comment, I think it is reasonable to require that the zero element be closed in any topologized hom space. | |
| Feb 10, 2013 at 6:07 | comment | added | Mariano Suárez-Álvarez | The set of complexes of vector spaces of given finite dimension vector "are" an affine space, and the dimension of their cohomology groups is a lower (upper? I never know which) semi-continuous function. Since it is possitive, that gives your continuity in that case. | |
| Feb 10, 2013 at 5:54 | history | asked | Theo Johnson-Freyd | CC BY-SA 3.0 |