Timeline for Can $\mathcal O_X$ be recognized abstract-nonsensically?
Current License: CC BY-SA 3.0
15 events
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| Dec 15, 2016 at 16:36 | comment | added | მამუკა ჯიბლაძე | @WillSawin Only now I managed to understand it, thanks! The sheaf of endomorphisms is large but has very few global sections, yes of course that may happen. | |
| Dec 12, 2016 at 17:05 | comment | added | Will Sawin | @მამუკაჯიბლაძე A stable rank $2$ vector bundle on a projective curve of positive genus should do the trick. On any affine subset its endomorphisms ring is large, but on the curve there are no nonscalar homomorphisms. | |
| Dec 12, 2016 at 16:49 | comment | added | მამუკა ჯიბლაძე | @WillSawin Wow that's interesting! Now that you said it I somehow see this might happen but cannot really come up with an example... | |
| Dec 12, 2016 at 13:09 | comment | added | Will Sawin | @მამუკაჯიბლაძე I think the problem is that a sheaf may have commutative endomorphism ring but noncommutative endomorphism rings on an open cover. | |
| Dec 10, 2016 at 9:20 | comment | added | მამუკა ჯიბლაძე | Also, it seems one more question still remains for me unclarified. Can it happen that an object with noncommutative endomorphism ring restricts to projective generators with commutative endomorphism rings on every piece of some cover? | |
| Dec 10, 2016 at 6:18 | comment | added | მამუკა ჯიბლაძე | @HeinrichD My previous question was stupid - of course there are projective generators with noncommutative endomorphism rings. But, regardless of projectivity, does not having commutative endomorphism ring descend to any quotient? | |
| Dec 9, 2016 at 23:16 | comment | added | HeinrichD | I've made an edit which clarifies this. | |
| Dec 9, 2016 at 23:15 | history | edited | HeinrichD | CC BY-SA 3.0 |
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| Dec 9, 2016 at 22:41 | comment | added | მამუკა ჯიბლაძე | I see. Regarding (2), do you mean it can happen that a sheaf has noncommutative endomorphism ring but restricts to projective generators for some open cover? | |
| Dec 9, 2016 at 21:59 | comment | added | HeinrichD | @მამუკაჯიბლაძე: Yes, it is equivalent (if $X$ is quasi-separated and we also include 2), as mentioned), and I also wondered about a first-order condition, but couldn't find one so far. Actually I am not sure if there is such a condition. The problem is really that we are not allowed to use the tensor product, and hence neither internal homs. | |
| Dec 9, 2016 at 20:54 | comment | added | მამუკა ჯიბლაძე | Very interesting, thank you! I have some questions though, if you don't mind. Does not your (5) alone suffice for the characterization (in the case of quasicoherent sheaves)? Can it be further formulated without the use of subcategories, just in terms of objects and morphisms? That is, is this second order condition equivalent to some first order one? | |
| Dec 9, 2016 at 20:00 | comment | added | HeinrichD | Yes, hence it is also true for quasi-coherent modules of finite type. | |
| Dec 9, 2016 at 18:29 | comment | added | Denis Nardin | (1) is true for every finitely generated module over a commutative ring | |
| Dec 9, 2016 at 16:52 | history | edited | HeinrichD | CC BY-SA 3.0 |
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| Dec 9, 2016 at 16:29 | history | answered | HeinrichD | CC BY-SA 3.0 |