Timeline for Why are Gabriel categories closed under localization?
Current License: CC BY-SA 4.0
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| Jan 21 at 23:38 | comment | added | Jeremy Rickard | Henning Kause, in Chapter 14 of his book Homological Theory of Representations, covers the “small” version of all this (i.e., for Serre, rather than localizing, subcategories of small abelian categories) in some detail. I’m not sure if there are added difficulties in the “localizing” case. | |
| Jan 19 at 8:15 | comment | added | Jeremy Rickard | Sorry, I won't have time for a few days to work out the details, but I'd be happy for anybody else to do so in the meantime, Or to point out why this won't work! | |
| Jan 19 at 8:13 | comment | added | Jeremy Rickard | For the other implications and inequalities, I think you need to consider a looser kind of filtration than the Gabriel filtration, where $\mathcal{K}_{\alpha+1}/\mathcal{K}_\alpha$ is generated by simple objects as a localizing subcategory of $\mathcal{K}/\mathcal{K}_\alpha$, but not necessarily all the available simple objects. Such a looser filtration may grow strictly more slowly than the Gabriel filtration, but should give an upper bound for the Gabriel dimension. And I think that splicing, and intersecting the Gabriel flitration of $\mathcal{K}$ with $\mathcal{T}$ does work now. | |
| Jan 19 at 8:08 | comment | added | Jeremy Rickard | Isn't $\mathcal{K}$ Gabriel iff it has no proper quotient by a localizing subcategory that has no simple modules? In which case, it is clear that $\mathcal{K}$ Gabriel implies $\mathcal{K}/\mathcal{T}$ Gabriel (and hypoabelian groups are stable under subgroups!) But the same argument doesn't show that $\mathcal{K}$ Gabriel implies $\mathcal{T}$ Gabriel. | |
| Jan 18 at 21:02 | history | edited | Tim Campion | CC BY-SA 4.0 |
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| Jan 18 at 19:54 | history | asked | Tim Campion | CC BY-SA 4.0 |