Timeline for Why is the prime spectrum not useful in non-archimedean analytic geometry?
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| Apr 20, 2010 at 2:22 | comment | added | BCnrd | Dear Matt: Yep, I should have made clearer that my comment to your answer was mainly aimed at Artie. To illustrate the consistency of your answer with the stuff in my comment, the non-functoriality of generic points as you point out meshes well with the analogous map on "reductions" being non-dominant and so also not respecting generic points: constant map of affine line over residue field to origin of another affine line over the residue field. | |
| Apr 20, 2010 at 2:00 | comment | added | Emerton | Dear Brian, What I had in mind is that if we, say, include the disk of radius $|p|$ into the unit disk, this does not induce a corresponding map of generic points (although the map on algebras is an injection of one domain in another, and so maps the zero ideal of one to the zero ideal of another). | |
| Apr 19, 2010 at 21:59 | comment | added | BCnrd | By taking norm to be sup-norm one gets the "Gauss point", which is "generic" in many ways. First, upon removing it the Berkovich disc disconnects into disjoint open union of residue discs (of radius 1). Thus, path-connectedness of Berkovich disc rests on that "non-classical" point; paths linking classical points on "boundary" must pass through the Gauss point. Second, under "reduction map" onto affine line over residue field its unique over that generic point (corresponds to Shilov boundary, another "genericity" property). Third, local ring at this point is a field, similar to varieties. | |
| Apr 19, 2010 at 19:58 | history | answered | Emerton | CC BY-SA 2.5 |