New PDF release: A Compactification of the Bruhat-Tits Building

By Erasmus Landvogt

ISBN-10: 3540604278

ISBN-13: 9783540604273

The target of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits construction of a reductive workforce over a neighborhood box. additionally, an particular description of the boundary is given. with the intention to make this paintings as self-contained as attainable and likewise obtainable to non-experts in Bruhat-Tits thought, the development of the Bruhat-Tits development itself is given completely.

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Now we want to lift /~s to the level of oK-schemes. distinguish two cases as in w In order to do this, we 1. Case: 2a ~ ~; let the notations be as in the first case of w Then (10 1)" 1 (-u' 01)= ( uu ~ -u' 1) 1 - and (_1721~)" (t0 tO1)"(~ 1)= I--IV' t--1~--~tVVt)~ from which follows that Ws is the standard open subset of Us • U - s with respect to the function ds: (xo(u), x - s ( u ' ) ) ~* N ~ ~ (1 - uu') and that ~s(Xs(U), X_s(U')) = (X_s((1 - uu')-lu'), 5(1 - u u ' ) , X s ( ( 1 - uul)-lu)) .

4) If 9 C_ ~ is positively closed, then the following product maps are bijections: (i) | a~red Dist H~,n -4 Dist il~,,n (for every good ordering of ~ d ) ; (ii) Dist ~ | Dist s -4 Dist ~H~,,n and Dist L[v,n | Dist ~ -4 Dist ~g~,,~; (iii) Dist H~ | Dist ~ | Disti[ + -4 Dist(/)). (iv) The canonical map K | Dist(lP) -4 Dist G is a bijection, too. P r o o f . 8) the assertions (i) and (ii) are clear. Ad (iii): The injectivity is clear because of the injectivity of U - • T • U + ~ G and the previous remark.

6. Let C E ~ be a chamber. 4) and the topology induced by V ~ coincide, (ii) V C is an open subset of V ~, (iii) the closures of C in V C and V ~ coincide and (iv) the closure of C in V ~ is compact. P r o o f . In order to prove (i) and (ii), it suffices to show that the canonical inclusion V c --+ V ~ is an open embedding. Let C1,C2 E ~ be two chambers, let F C }2 be a face of CI and let U C_ V be 26 an open subset. T h e n c~ n v c~ = c F - n v c~ U+F = (U c vF' +~)n vC~ = F' EE lJ C vF'_ +F F' E~ F'C~ F'C~ F' C C 2 E I where Cu+-f is an open subset of V C1 and V C2 for F ' C F M C1 N C2- Hence Cff N V C2 is an open subset of V C2.

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A Compactification of the Bruhat-Tits Building by Erasmus Landvogt


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