A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders

Simson, Daniel

doi:10.1090/S0002-9947-00-02575-7

A reduced Tits quadratic form and tameness of three-partite subamalgams of tiled orders
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Trans. Amer. Math. Soc. 352 (2000), 4843-4875 Request permission

Abstract:

Let $D$ be a complete discrete valuation domain with the unique maximal ideal ${\mathfrak {p}}$. We suppose that $D$ is an algebra over an algebraically closed field $K$ and $D/{\mathfrak {p}} \cong K$. Subamalgam $D$-suborders $\Lambda ^{\bullet }$ of a tiled $D$-order $\Lambda$ are studied in the paper by means of the integral Tits quadratic form $q_{\Lambda ^{\bullet }}: {\mathbb {Z}}^{n_{1}+2n_{3}+2 } \longrightarrow {\mathbb {Z}}$. A criterion for a subamalgam $D$-order $\Lambda ^{\bullet }$ to be of tame lattice type is given in terms of the Tits quadratic form $q_{{\Lambda ^{\bullet }}}$ and a forbidden list $\Omega _{1},\ldots ,\Omega _{17}$ of minor $D$-suborders of $\Lambda ^{\bullet }$ presented in the tables.

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