User - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T02:56:04Z http://mathoverflow.net/feeds/user/18087 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112443/constructing-a-field-from-a-spherical-building Constructing a field from a spherical building unknown (google) 2012-11-15T01:28:38Z 2012-11-15T22:28:03Z <p>Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism $\Delta(G,F)\simeq\Delta(G',F')$ induces a field isomorphism $F\to F'$.</p> <p>This shows that the field is somehow coded up in the isomorphism type of the building. I'm wondering whether a construction is floating around anywhere that shows how to construct the field from the combinatorics of the building.</p> <p>To help clarify what I'm after: I've seen a construction of the real field from the incidence structure or geometry of the real projective plane (pick a pair of lines, prove they're bijective, define addition via some more lines and unique intersection points which must exist, etc.). In the end you have parametrized a projective line by matching up points on it with the underlying field. The construction requires a choice of a few points in general position and so is given sort of "up to collineation". I'm under the impression that I should view those results on spherical buildings as generalizations of results in projective geometry, and I'm looking for an analogous construction. That is, a construction of a field in terms of the apartments, relations between them, etc. In the end presumably some collection of objects is parametrized by the underlying field. Or is this not the case?</p> http://mathoverflow.net/questions/112443/constructing-a-field-from-a-spherical-building/112529#112529 Comment by 2012-11-15T23:43:00Z 2012-11-15T23:43:00Z Thanks for the Von Staudt and Mnev references. http://mathoverflow.net/questions/112443/constructing-a-field-from-a-spherical-building/112528#112528 Comment by 2012-11-15T23:33:42Z 2012-11-15T23:33:42Z @Koen This is helpful. Do you happen to know a good reference for getting at the Moufang projective plane mentioned your 3rd paragraph? If not, thanks for pointing out that path anyway. http://mathoverflow.net/questions/112443/constructing-a-field-from-a-spherical-building/112488#112488 Comment by 2012-11-15T18:49:55Z 2012-11-15T18:49:55Z Thanks for the response. Sorry if this is basic, but could you comment on what objects end up being parametrized when we construct the field in the building case? The usual proj. geom. argument parametrizes a line in a projective space; its points are field elements. Say we start with a high rank euclidean building, take its building at infinity, and construct the field, what are we parametrizing? We can't be parametrizing things inside a given apartment, that seems like it would always give $\mathbb R$. Is the correspondence that apartments are field elements?