Timeline for Geometric (or intuitive) interpretation of additional derivatives in characteristic p > 0
Current License: CC BY-SA 2.5
6 events
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| Jun 24, 2010 at 9:06 | answer | added | Bugs Bunny | timeline score: 5 | |
| Jun 24, 2010 at 8:21 | comment | added | Lars | Using the composition formula, one sees that in order to write a given operator of order n as product of derivations you have to be able to divide by n!. That's what's not possible in pos. char., but divided power structures restore this ability. For a geometric meaning: The modules over the subring gen'd by deriviations are precisely the integrable connections of p-curvature 0. Is that what you're looking for? I can recommend the first chapter of Berthelot-Ogus' "Notes on Crystalline Cohomology". | |
| Jun 24, 2010 at 7:44 | comment | added | CFZ | But why is the PD structure the correct modification (other than "it works" in making the perceived "extras" go away)? To put it differently, is there some canonical geometric meaning to the subring of differential operators generated by derivations? The same coefficients appear in the Leibniz rule for degree $n$ derivations and in the PD structure, so possibly it is just forced algebraically, but is that actually the case or it only makes the PD enlargement one of many possible options? | |
| Jun 24, 2010 at 7:41 | history | edited | Charles Matthews | CC BY-SA 2.5 |
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| Jun 24, 2010 at 6:08 | comment | added | Lars | Could you make the question a little bit more precise, I don't really know what you are looking for. Differential operators are defined in terms of thickenings: The sheaf of operators of order <= n on X is dual to the structure sheaf of the n-th infinitesimal neighborhood of the diagonal of X in X x X. Similarly, the sheaf of all diff ops is dual to the structure sheaf of the formal nbhd. of the diagonal. For crystalline theory you add divided power structures, which make the extra operators go away. Ring of PD-DiffOps=ring gen'd by derivations and O_X if I remember correctly. | |
| Jun 24, 2010 at 3:57 | history | asked | CFZ | CC BY-SA 2.5 |