Timeline for De Rham Cohomology in positive characteristic
Current License: CC BY-SA 3.0
8 events
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| Jul 20, 2016 at 16:41 | comment | added | SashaP | @ChrisDodd Probably, there is. But I doubt that there will be a complete non-conjectural solution as long as there are no resolution of singularities in positive characteristic. | |
| Jul 19, 2016 at 19:57 | comment | added | Chris Dodd | Hi Sahsa; thanks for the comment. This was one motivation for my question. As far as I can see, no one addresses this problem one way or the other in the literature; it seems like a strange gap. My feeling was that there must be some "folklore" thinking on the problem; we'll see if the MO community offers some. | |
| Jul 19, 2016 at 19:05 | comment | added | SashaP | Also, as you suggest rigid cohomology can be computed via the log-de Rham-Witt complex given a smooth compactification. Every such compactification gives an integral structure but it seems unclear to me how to identify these integral structures for different compactifications. | |
| Jul 19, 2016 at 18:42 | comment | added | SashaP | There is a well-defined theory called "rigid cohomology" due to Berthelot which gives finitely generated answer for arbitrary varieties. A drawback of it is that it is defined only rationally while crystalline cohomology has a structure of $W(k)$-module. As far as I know it is still an open problem to give rigid cohomology an integral structure. | |
| Jul 19, 2016 at 1:18 | comment | added | Chris Dodd | Well, if $X$ is a proper scheme then the De Rham cohomology groups are finite dimensional; since they are computed as the hypercohomology groups of a finite complex of vector bundles. So in that sense it is "due to the non-properness." | |
| Jul 19, 2016 at 0:53 | comment | added | Mariano Suárez-Álvarez | Infinite dimensionality is not due, in positive characteristic, to non-properness but to the actual characteristic, no? | |
| Jul 18, 2016 at 15:15 | review | First posts | |||
| Jul 18, 2016 at 15:36 | |||||
| Jul 18, 2016 at 15:12 | history | asked | Chris Dodd | CC BY-SA 3.0 |