Timeline for Natural morphism appearing in Grothendieck spectral sequence
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7 events
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| Mar 7, 2014 at 12:37 | answer | added | user19475 | timeline score: 2 | |
| Jun 21, 2010 at 11:42 | comment | added | Boyarsky | @Martin: By the way, for the first edge map, the only "direct description" I ever found for it was in terms of injective/acyclic resolutions (certainly more direct than saying "edge map in spectral sequence", so not a tautology, and still useful for concrete identification of it in some practical situations). Not as nice as for the 2nd edge map. If you notice something better, please mention it here. | |
| Jun 21, 2010 at 11:34 | comment | added | Boyarsky | @Martin: Look at 12.2.5 for the relation of the 2nd edge map with "sheafified pullback" (you may want to provide some justification omitted in EGA), and 12.1.3 for the proof of functoriality in the morphism for sheaf-pullback. Combining these, the commutative square drops out. Concerning "non-exact $\delta$-functor" stuff, I wasn't assuming you knew about it already, which was why I thought pointing you in that direction was a good hint. (I noticed it myself for this application!) I say more: use erasability and look back at the proof of universality of erasable $\delta$-functors. | |
| Jun 21, 2010 at 9:27 | comment | added | Martin Brandenburg | Could you please elaborate this? In the section of EGA you mentioned I can't find any answer to my questions. Also, I don't know why univeral $\delta$-functors are also initial (or at least here) for non-exact $\delta$-functors. Otherwise I would not have asked ... | |
| Jun 20, 2010 at 16:28 | comment | added | Boyarsky | In the above I should have said at the end that it is the unique such map which is the evident equality for $q=0$. | |
| Jun 20, 2010 at 11:55 | comment | added | Boyarsky | The compatibility with natural transformations in $F$ and $G$ is an exercise in looking at the construction. The final commutative diagram is explained in section 12 of Ch. 0 of EGA III; best seen via the answer to the question on the 2nd edge map (for which your question is surely to give intrinsic characterizations of each so they can be identified/computed in specific situations). What's the answer? It's the unique such map from the universal $\delta$-functor ${\rm{R}}^q(FG)$ to the ``non-exact $\delta$-functor'' $F \circ {\rm{R}}^q(G)$ (exercise: make precise what such a statement means!) | |
| Jun 20, 2010 at 10:36 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |