Timeline for How to introduce notions of flat, projective and free modules?
Current License: CC BY-SA 2.5
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| Nov 20, 2010 at 19:24 | history | edited | roy smith | CC BY-SA 2.5 |
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| Nov 19, 2010 at 5:24 | comment | added | roy smith | Gee, this is nice. In line with the idea that flat maps are a slight weakening of smooth ones, one can just read the titles of the first few sections of SGA 1: i.e. etale maps, smooth maps, flat maps,..... Thanks for prompting me to look at Grothendieck. | |
| Nov 19, 2010 at 5:22 | comment | added | BCnrd | Dear Roy: My impression is that Serre viewed local analytic dimension as part of algebra. Usefulness of flatness for geometric problems (Hilbert schemes, deformation theory, etc.) is what I think surprised him. There are too many amazing geometric uses of flatness by Grothendieck, so no "favorite": fpqc descent (to work intuitively with $G/H$ and etale topology, etc.), fibral flatness criteria, openness results for loci defined by fibral properties relative to proper fppf maps, flatness of formally etale lfp maps,... Some of your examples are really about excellence (another miracle!), btw. | |
| Nov 19, 2010 at 5:02 | comment | added | roy smith | BC, I agree there is no visible occurrence in GAGA of the geometry of a general flat map. I was just to imagine how the idea extends. So that was Grothendieck's contribution, that's nice to know. | |
| Nov 19, 2010 at 4:53 | comment | added | roy smith | Thanks. Do you have a favorite reference for where Grothendieck exploits it, analogous to this one I just perused by Serre? It seems to me that even in GAGA Serre shows how it relates local geometric concepts like dimension, but maybe I am stretching, if he says otherwise! | |
| Nov 19, 2010 at 4:49 | history | edited | roy smith | CC BY-SA 2.5 |
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| Nov 19, 2010 at 4:43 | comment | added | BCnrd | Dear Roy: Last spring I asked Serre about the history. He viewed it as a purely algebraic notion, inspired by usefulness for the back-and-forth passage between analytic and algebraic local rings through their common completion (noetherian property of the former being pretty serious, of course) and clarifying localization. It came as a total surprise to him later that flatness is an important concept in algebraic geometry, and he said that all credit for its geometric significance belongs to Grothendieck; he (Serre) did not anticipate it would be of interest beyond pure algebra. | |
| Nov 19, 2010 at 3:42 | history | answered | roy smith | CC BY-SA 2.5 |