Download e-book for iPad: A Compactification of the Bruhat-Tits Building by Erasmus Landvogt

By Erasmus Landvogt

ISBN-10: 3540604278

ISBN-13: 9783540604273

The objective of this paintings is the definition of the polyhedral compactification of the Bruhat-Tits construction of a reductive workforce over an area box. moreover, an specific description of the boundary is given. so one can make this paintings as self-contained as attainable and likewise available to non-experts in Bruhat-Tits idea, the development of the Bruhat-Tits construction itself is given completely.

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Then there exists a unique smooth, affine OKgroup scheme ~l~,,a of finite type with generic fibre U~ and the property that for every good order of qj~d the K-isomorphism 11 U~ ~ Us can be extended aE~I/~ed to an oK-isomorphism [I [l~,a --+ t2,~,~. P r o o f . 5). 7. Let q~' C_ q~ C ~ be two positively closed subsets. l~,,n onto a closed oK-subgroup scheme of tl,~,a. P r o o f . One can show immediately by induction on card(qd) that every good ordering of ~ ' is induced by a good ordering of 9. 6).

So we will identify ( H - T H + ) / a n d (H+TL[-)/, in the following. In order to construct a smooth oK-group scheme with generic fibre G from ~A-TH+, we will define an og-birational group law on H-~:H + first. 5). We will recall briefly the definitions: Let R be a ring and let X be a smooth R-scheme. An open subscheme ~3 of :~ is called R - d e n s e , if all fibres of ~3 (over R) are Zariski-dense in the corresponding fibres of :~. If 2] is a further, not necessarily smooth R-scheme, then an R r a t i o n a l m a p 3~ - -+ ~ is an equivalence class of R-morphisms ~3 --+ ~ where ~3 is an open, R-dense subscheme of ~.

P r o o f . By [SGA 3] Exp. 13 the torus T is a product of tori of the form RL(~m/L) where L / K is a finite field extension. So we can restrict ourselves to the case T = TiL(~,,~/L). First of all, we will show that 7r176(~m/oc) has the universal property of the canonical oK-group scheme ~ associated with T. Let ~r be a smooth, affine oK-group scheme of finite type with generic fibre T and ~r = ~ o . Hence we have to show that there is exactly one extension ~' -'+ T~~ (Cm/On) of the identity on T.

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A Compactification of the Bruhat-Tits Building by Erasmus Landvogt


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