Timeline for Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?
Current License: CC BY-SA 3.0
7 events
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| May 28, 2011 at 9:05 | history | edited | Chris Brav | CC BY-SA 3.0 |
Fixed typo and added pointer to comment of 3tsuji.; edited body
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| May 27, 2011 at 18:45 | comment | added | t3suji | I am not sure how you expect codimension 2 to help in part 3. Basically, the only advantage is that for a locally free sheaf, the local cohomology will be coherent in cohomological degrees zero and one. However, even for locally free sheaves, you gen non-coherent higher local cohomology. And of course, if your sheaf is not locally free (for instance, torsion), you may have non-coherent cohomology in lower degrees, too. So it would seem there is no recollement for coherent categories, only for quasicoherent ones. | |
| May 27, 2011 at 17:01 | comment | added | Chris Brav | But perhaps an expert will show up and explain this to us. | |
| May 27, 2011 at 17:01 | comment | added | Chris Brav | I mean that usually one considers the various kinds of derived categories as triangulated categories and that these triangulated categories don't satisfy descent with respect to any reasonable topology on the base since a morphism in the triangulated category can be globally non-zero but locally zero. This problem can be fixed by working with natural pretriangulated dg categories or stable $\infty$-categories whose associated triangulated categories are the usual derived categories. This question is discussed in lectures notes of Toen on dg categories and in a paper of Hirschowitz-Simpson. | |
| May 27, 2011 at 15:08 | comment | added | Mikhail Bondarko | Thank you! Unfortunately, I didn't quite understand your part 4; could you explain this in more detail? | |
| May 27, 2011 at 15:07 | vote | accept | Mikhail Bondarko | ||
| May 27, 2011 at 14:14 | history | answered | Chris Brav | CC BY-SA 3.0 |