Timeline for Cohomology of sheaves in different Grothendieck topologies
Current License: CC BY-SA 2.5
14 events
| when toggle format | what | by | license | comment | |
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| S Jun 13, 2017 at 6:59 | history | suggested | Topological |
Cohomology instead of etale cohomology
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| Jun 13, 2017 at 5:36 | review | Suggested edits | |||
| S Jun 13, 2017 at 6:59 | |||||
| S Apr 14, 2017 at 13:07 | history | suggested | Ascenso |
Added Grothendieck topology tag
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| Apr 14, 2017 at 12:37 | review | Suggested edits | |||
| S Apr 14, 2017 at 13:07 | |||||
| Sep 28, 2012 at 23:22 | vote | accept | Sam Derbyshire | ||
| Sep 26, 2012 at 11:01 | answer | added | Thomas Geisser | timeline score: 19 | |
| Mar 26, 2012 at 23:03 | comment | added | Keenan Kidwell | Regarding the vanishing of higher direct images, it tells you that the Leray spectral sequence degenerates into a family of isomorphisms $H^p(X_{Zar},f_*(\mathcal{F}))\rightarrow H^p(X_{et},F)$. | |
| Apr 3, 2010 at 23:42 | comment | added | BCnrd | A concrete way to see the vanishing for the qcoh case is Cartan's criterion for vanishing of higher sheaf cohomology (in terms of Cech-like cohomologies, applied in the etale topology) and the use of a cofinal system of affine etale covers of an affine scheme, coupled with the exactness of the long Cech-like complex built for a module and a faithfully flat ring extension (which in turn rests on the essential content of Grothendieck's brilliant trick with a section to prove fpqc descent for qcoh sheaves). | |
| Apr 3, 2010 at 22:00 | history | edited | Sam Derbyshire | CC BY-SA 2.5 |
Added a comment about the second question, following Scott's comment.
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| Apr 3, 2010 at 19:17 | history | edited | Sam Derbyshire | CC BY-SA 2.5 |
Changed wording for necessity of R^qf_* being 0
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| Apr 3, 2010 at 19:08 | history | edited | Sam Derbyshire | CC BY-SA 2.5 |
Added an example of failure
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| Apr 3, 2010 at 18:57 | history | edited | Sam Derbyshire | CC BY-SA 2.5 |
Explained the little understanding I have of the proof in SGA 4
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| Apr 3, 2010 at 18:50 | comment | added | S. Carnahan♦ |
Regarding your subquestion, the restriction functor $\epsilon: X_{\'et} \to X_{Zar}$ has a right adjoint $\epsilon^{-1}$, and it returns the sheaf associated to the presheaf $(U,f) \mapsto \mathcal{G}(f(U))$. You have to take a tensor product to get the $\mathcal{O}$-module functor $\epsilon^*$, and this turns out to be an equivalence on quasicoherent modules.
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| Apr 3, 2010 at 18:09 | history | asked | Sam Derbyshire | CC BY-SA 2.5 |