Timeline for equivalence of Grothendieck-style versus Cech-style sheaf cohomology
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| S Mar 14, 2018 at 4:55 | history | suggested | Alex Shpilkin | CC BY-SA 3.0 |
use MathJax to typeset ∞-things everywhere (not just ∞-groupoids)
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| Mar 13, 2018 at 22:04 | review | Suggested edits | |||
| S Mar 14, 2018 at 4:55 | |||||
| Nov 11, 2010 at 0:43 | comment | added | Matt | Wow. Thank you so much! Our student seminar meets again next Tues, so I have about a week to sort out what this means and bring them an answer. | |
| Nov 11, 2010 at 0:14 | comment | added | Urs Schreiber | ... This darn comment section here is clearly not the right place to discuss these issues. Come over to the nForum if you want me to discuss this more. Or have a look at this article of mine, where all this is spelled out in some detail over a site of smooth spaces: ncatlab.org/schreiber/show/… | |
| Nov 11, 2010 at 0:13 | comment | added | Urs Schreiber | ... the correct hom-space whose homotopy groups give you the desired cohomology groups is the simplicial hom-complex from that Cech nerve into your given coeefficient object. If you look at what such a morphism from a Cech nerve is, you see that these are precisely Cech cocycles with respect to the chose cover. So we find: Dugger's theorem tells us when Cech cohomology computes the correct hom-space: namely when it is a "good cover" in the sense that its Cech nerve is a simplicial presheaf that is degreewise a coproduct of representables. ... | |
| Nov 11, 2010 at 0:10 | comment | added | Urs Schreiber | ... If these patches are "small enough" it is easy to check for a given coefficient sheaf if it is locally fibrant. So all the work is then moved to finding cofibrant resolutions of the object that you want to compute the cohomology of. Now comes a theorem by Dan Dugger on cofibrant replacement in the projective model structure: it tells us that Cech nerves of good covers are cofibrant, where I call a cover good if all finite intersections of the covering patches (computed as presheaves) are again representable. So if you have that, general model category nonsense tells you that... | |
| Nov 11, 2010 at 0:07 | comment | added | Urs Schreiber | I can give you the fully general answer, which however may require a bit more work to unwind over a specific choice of site. You need to know about the "model structure on simplicial presheaves" for what I say now. If you don't check the nLab entry with that title. Cech cohomology is -- if done right (see below) -- a tool for computing hom spaces (aka derived global section functors) in the projective model structure. For that, you pick a site of definition for your ambient topos whose objects are "small patches" of the kind that you want to build covers from. For instance affine spaces. ... | |
| Nov 10, 2010 at 4:44 | comment | added | Matt | I glanced through these links and they are great! However, my AG club was stumped today when we tried to figure out if there was some condition on stacks (in particular we were considering moduli stacks of curves) that would guarantee Cech cohomology (for some cover) agree with sheaf cohomology. And we only really cared about $H^1$. Do you know any places that have such conditions? | |
| Nov 17, 2009 at 17:07 | history | edited | Urs Schreiber | CC BY-SA 2.5 |
added 4 characters in body; edited body
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| Nov 17, 2009 at 13:40 | history | answered | Urs Schreiber | CC BY-SA 2.5 |