Timeline for When are there enough projective sheaves on a space X?
Current License: CC BY-SA 3.0
17 events
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| Feb 11, 2018 at 3:50 | comment | added | Tim Campion | As far as non-locally-connected spaces go, see also Cor 1.9 of loc. cit.. It shows that if $X$ is a regular topological space, then the category $Ab(X)$ of abelian sheaves on $X$ satisfies AB4* (i.e. has projective effacements) iff the topology has a basis of sets of cohomological dimension 0. In particular, if $Ab(X)$ has enough projectives, then $X$ locally has cohomological dimension 0. | |
| Feb 11, 2018 at 3:38 | comment | added | Tim Campion | To run this argument, we need only assume there are enough projective effacements rather than enough projectives. Since this is a Grothendieck category, having projective effacements is equivalent to assuming the AB4* axiom (cf. Roos, Cor. 1.4). Results comparable but not identical to VA.'s answer -- and which go beyond the locally connected case -- appear in e.g. Thm 1.7 and Cor 1.13 of loc. cit.. Also, I think VA's argument works also for $\mathcal O_X$-modules when $(X,\mathcal O_X)$ is locally ringed. | |
| Nov 30, 2017 at 20:56 | history | edited | Will Sawin | CC BY-SA 3.0 |
Link was to the wrong kind of Alexandrov space
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| S Nov 30, 2017 at 17:35 | history | suggested | Mike Pierce | CC BY-SA 3.0 |
Mathjaxxed, added some links, and regrammared that last paragraph a tad
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| Nov 30, 2017 at 17:04 | review | Suggested edits | |||
| S Nov 30, 2017 at 17:35 | |||||
| Jul 18, 2013 at 4:27 | history | wiki removed | François G. Dorais | ||
| Nov 17, 2009 at 2:24 | vote | accept | Pete L. Clark | ||
| Nov 14, 2009 at 20:36 | history | edited | VA. | CC BY-SA 2.5 |
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| Nov 14, 2009 at 0:50 | history | answered | VA. | CC BY-SA 2.5 |