Timeline for Top local cohomology - recommendations
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8 events
| when toggle format | what | by | license | comment | |
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| Oct 17, 2022 at 4:40 | comment | added | Karl Schwede | A common structure in commutative algebra is to consider $H^d_{\mathfrak{m}}(R)$ where $R$ is local Noetherian. That local cohomology module is the Matlis dual of $\omega_R$, the canonical module of $R$ (assuming one exists, for instance if $R$ is complete or finite type over a field or $Z$). There are lots of other things one might say too. In particular, one way to study anything about the canonical module is to instead study the local cohomology module. | |
| S Oct 17, 2022 at 4:34 | history | suggested | CommunityBot |
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| Oct 17, 2022 at 2:26 | history | became hot network question | |||
| Oct 17, 2022 at 0:22 | comment | added | roy smith | I presume you have looked at Grothedieck's Local Cohomology, pages 30 and 64, where these groups are related to Ext groups and dualizing modules. Apologies if this is obvious, as I have only consulted this one source. | |
| Oct 16, 2022 at 21:30 | comment | added | Pulcinella | You can define local cohomology for topological spaces (and singular cohomology), or for algebraic varieties (and $\ell$-adic/de Rham/... cohomlogy). In all cases, the local cohomology of closed subspace $i:Z\hookrightarrow X$ defined in terms of the six functors to be $i^!k_X$, where $k_X$ is the constant sheaf on $X$. When both spaces are smooth or $i$ is a regular embedding, this is just (a shift of) the constant sheaf. Thus, the local cohomology (and presumably the top degree part too) measures the singular behaviour "in the normal direction". | |
| Oct 16, 2022 at 21:21 | review | Suggested edits | |||
| S Oct 17, 2022 at 4:34 | |||||
| Oct 16, 2022 at 21:21 | answer | added | Neil Strickland | timeline score: 7 | |
| Oct 16, 2022 at 18:24 | history | asked | user340793 | CC BY-SA 4.0 |