Timeline for Galois cohomology of finite fields
Current License: CC BY-SA 3.0
7 events
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| Mar 24, 2017 at 9:42 | history | edited | nfdc23 | CC BY-SA 3.0 |
fixed missing finiteness condition on N.
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| Mar 24, 2017 at 9:39 | comment | added | nfdc23 | I'm sure there is a more direct elementary argument; that is just what first came to mind via general cohomological principles and inspired by the usual proof of compatibility of module-Ext with localization when the first argument of module-Ext is finitely presented (for a much wider cohomological context of the same, way beyond just on Spec of fields, see Prop. 4.18 and the associated footnote that are both stated without proof in Chapter I of the book of Freitag & Kiehl on etale cohomology, which focuses there on $n$-torsion sheaves). | |
| Mar 24, 2017 at 9:31 | comment | added | nfdc23 | For any scheme $X$, $\mathscr{E}xt^{\bullet}_X(\mathscr{F}, \mathscr{G})$ is sheafification of $U\mapsto {\rm{Ext}}^{\bullet}_U(\mathscr{F}|_U,\mathscr{G}|_U)$ (univ. $\delta$-functor proof). Thus, for $X = {\rm{Spec}}(k)$, the $k_s$-stalk of such sheaf-Ext is direct limit of ${\rm{Ext}}_K^{\bullet}(\mathscr{F}_K,\mathscr{G}_K)$ ($K/k$ finite inside $k_s$), and the natural map from this to ${\rm{Ext}}^{\bullet}_{\rm{Ab}}(\mathscr{F}_{k_s}, \mathscr{G}_{k_s})$ is an isomorphism for lcc $\mathscr{F}$ by reduction to $\mathscr{F}=\mathbf{Z}$ and compatibility of cohomology with limits of schemes. | |
| Mar 24, 2017 at 6:23 | comment | added | user1225 | Could you explain more specifically your statement that the sheaf Ext is equal to $Ext_{Ab}(N, M)$ as discrete $G$-modules? | |
| Mar 24, 2017 at 6:15 | vote | accept | user1225 | ||
| S Mar 24, 2017 at 5:30 | history | answered | nfdc23 | CC BY-SA 3.0 | |
| S Mar 24, 2017 at 5:30 | history | made wiki | Post Made Community Wiki by nfdc23 |