Timeline for Nonabelian reciprocity law
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Mar 10, 2017 at 15:34 | comment | added | Somerville Scholar | The cokernel of $H^i(X,\mathbf{Z}) \rightarrow H^i(X,\mathbf{F}_p)$ is $H^{i+1}(X,\mathbf{Z})[p]$; the cokernel of $H_i(X,\mathbf{Z}) \rightarrow H_i(X,\mathbf{F}_p)$ is $H_{i-1}(X,\mathbf{Z})[p]$. Here $M[p]$ denotes the $p$-torsion of $M$. | |
| Mar 10, 2017 at 5:29 | comment | added | stupid_question_bot | Ah, of course! Here I assume by "$C$" you meant some chain group with coefficients in... $\mathbb{Z}$? Do you know what the cokernel is of the reduction mod $p$ map is measured by? (In the homology case it's a certain Tor group) | |
| Mar 10, 2017 at 4:56 | comment | added | Somerville Scholar | Dear @Amy, I am indeed talking about the map $H^i(X,\mathbf{Z}) \rightarrow H^i(X,\mathbf{F}_p)$ (but one can similarly consider the map $H_i(X,\mathbf{Z}) \rightarrow H_i(X,\mathbf{F}_p)$). Even though Cohomology is contravariant in spaces, it is covariant in the coefficients. The cohomology is computed from the homology of the co-chain complex; think of a cohomology class as represented by a class in $\mathrm{Hom}(C,\mathbf{Z})$ for some $C$; this maps naturally to $\mathrm{Hom}(C,\mathbf{F}_p)$ and this induces a map on cohomology. I'm not sure if this answers your question or not. | |
| Mar 10, 2017 at 4:40 | comment | added | stupid_question_bot | I really enjoyed reading this! I'm a bit of a novice at this stuff, so perhaps you'd excuse my naive question: When you said "mod $p$ cohomology lifts/does not lift to characteristic 0", I assume you're talking about singular cohomology? Ie, you're asking if every cohomology class in $H^i(X,\mathbb{F}_p)$ is the image of a class in $H^i(X,\mathbb{Z})$? (How does one get this map? - for homology the map comes from the universal coeff. theorem) Or, were you talking about a different type of cohomology? | |
| Mar 10, 2017 at 1:31 | review | First posts | |||
| Mar 10, 2017 at 1:39 | |||||
| Mar 10, 2017 at 1:31 | history | answered | Somerville Scholar | CC BY-SA 3.0 |