Grids and the arithmetics of Jordan pairs
The author uses \textit{E. Neher}'s theory of grids [Jordan triple systems by the grid approach, Lect. Notes Math. 1280, Springer (1987; Zbl 0621.17001)]\ to derive several new results on orders in finite-dimensional Jordan pairs over local fields. In the first part, the concept of a norm on a Jordan pair \(V\) over a valued field \((K,\Gamma,\lambda)\) is introduced, as follows. Let \(\Delta\) be a totally ordered abelian group containing \(\Gamma\) as a totally ordered subgroup. A \(\lambda\)-norm on \(V\) with values in \(\Delta\) is a pair \(\rho=(\rho^+,\rho^-)\) of mappings \(\rho^\varepsilon:V^\varepsilon\to\Delta_\infty\) (\(\varepsilon=\pm\)) satisfying the following conditions: (1) \(\rho^\varepsilon(x)=\infty \Leftrightarrow x=0\), (2) \(\rho^\varepsilon(x+x')\geq\min(\rho^\varepsilon(x),\rho^\varepsilon(x'))\), (3) \(\rho^\varepsilon(Q(x)y)\geq 2\rho^\varepsilon(x)+\rho^{-\varepsilon}(y)\), and (4) \(\rho^\varepsilon(ax)=\lambda(a)+\rho^\varepsilon(x)\). The following ``Norm Theorem'' guarantees the existence of norms in a particular case: Let \(V\) be a Jordan pair over a Henselian field \((K,\Gamma,\lambda)\). Let \(G\) be a covering division grid in \(V\) whose Peirce components are \(2\)-finite-dimensional (i.e.\ every subpair generated by two elements \(x,y\) satisfying \(x^+=y^+\) or \(x^-=y^-\) is finite-dimensional). Then \(G\) canonically determines a \(\lambda\)-norm \(\rho\) of \(V\) with values in \(\Gamma\otimes_{\mathbb Z}\mathbb Q\). The main idea is to define \(\rho\) on the Peirce components of \(V\) relative to \(G\); since these Peirce components are Jordan division pairs, Niggemann's results on valuations of Jordan division rings [\textit{D. Niggemann}, Allgemeine Bewertungstheorie für Jordan-Divisionsringe, Thesis, Hagen (1995)] can be used for this task. The Norm Theorem is used to classify saturated maximal orders (i.e.\ maximal orders containing a covering division grid): Let \(V\) be a finite-dimensional Jordan pair over the local field \(K\) with complete, discrete, surjective valuation \(\lambda : K \rightarrow \mathbb Z_\infty\). Then \({\mathcal O} \subseteq V\) is a saturated maximal order if, and only if, there exists a covering standard division grid \(G\) in \(V\) such that \({\mathcal O} = \{ x \in V \mid \rho^+(x^+)\), \(\rho^-(x^-) \geq 0\}\), where \(\rho\) is the \(\lambda\)-norm defined by \(G\). By a result of Neher (whose proof is contained in the paper under review), every weakly separable order is a saturated maximal one. The existence of separable and weakly separable orders is linked to ramification properties: Let \(V\) be a finite-dimensional Jordan pair over the local field \(K\). If \(V\) is a division pair, then call \(V\) weakly unramified, if it contains weakly separable orders or, equivalently, if \(V\) is isomorphic to \((J,J)\) for some finite-dimensional weakly unramified Jordan division algebra \(J\) over \(K\). If \(V\) is arbitrary, then call \(V\) weakly unramified if it is semisimple and if certain division subpairs (the so-called anisotropic parts) of its simple summands are weakly unramified. For separable Jordan pairs, the term ``unramified'' is introduced in a similar way. It turns out that the following statements are equivalent: (i) \(V\) is weakly unramified (resp.\ unramified), (ii) \(V\) contains weakly separable (resp.\ separable) orders, and (iii) \(V\) is semisimple (resp.\ separable) and every saturated maximal order in \(V\) is weakly separable (resp.\ separable). Finally, a global version of the preceding result is given: Let \(X\) be a regular integral scheme of dimension \(1\) with function field \(k\). For \(p \in X\), let \(K_p\) denote the local field derived from the local ring of \(X\) at \(p\) by the process of completion. Consider a finite-dimensional Jordan pair \(V\) over \(k\). Then there exists a separable Jordan pair over \(X\) with generic fibre \(V\) if, and only if, \(V\otimes_k K_p\) is unramified over \(K_p\) for every \(p \in X\).
scientific article; zbMATH DE number 1270875
scientific article