Value functions and Dubrovin valuation rings on simple algebras (Q2787955)

In commutative valuation theory, value functions and valuation rings are two sides of the same coin. While these notions become separate when the theory extends to (finite dimensional) simple algebras, there are tight connections, which the current paper elegantly outlines and strengthens.NEWLINENEWLINEThroughout, let \(F\) be a field and \(\nu \,{:}\, F \rightarrow \Gamma\) a valuation, where \(\Gamma\) is a totally ordered abelian group. The valuation induces a graded field \(\mathsf{gr}_{\nu}(F)\). Let \(A\) be a finite dimensional semisimple algebra over \(F\). A {\textit{\(\nu\)-gauge}} is a function \(\alpha \,{:}\, A \rightarrow \Gamma\cup\{\infty\}\) inducing a graded algebra \(\mathsf{gr}(A)\) which is semisimple over \(\mathsf{gr}(F)\), and of the same dimension as \(A\) over \(F\). The theory of gauges and their induced graded algebras is developed in the recent book [\textit{J.-P. Tignol} and \textit{A. R. Wadsworth}, Value functions on simple algebras, and associated graded rings. Cham: Springer (2015; Zbl 1357.16002)].NEWLINENEWLINEGauges for a simple \(F\)-algebra \(A\) are constructed from gauges of central simple algebras as follows. Let \(\nu_1,\dots,\nu_r\) be the extensions of \(\nu\) to the center \(K = \text{Cent}(A)\). Every \(\nu\)-gauge has the form \(\alpha = \min\{\alpha_1,\dots,\alpha_r\}\) where \(\alpha_i\) is a \(\nu_i\)-gauges. Conversely, assuming \(\nu\) is defectless (which is guaranteed, e.g., in characteristic zero), \(\alpha = \min\{\alpha_1,\dots,\alpha_r\}\) is a \(\nu\)-gauge if and only if the \(\nu_i\)-gauges \(\alpha_i\) are compatible (Theorem 2.8).NEWLINENEWLINEA {\textit{Dubrovin valuation ring}} is a subring \(B\) of a central simple \(F\)-algebra \(A\) with a (consequently unique) maximal ideal \(J\), such that \(B/J\) is simple Artinian, and for every \(s \in A\) outside of \(B\), \(sB \cap B\) and \(Bs \cap B\) have elements outside of \(J\). For example, Azumaya algebras over commutative valuation rings are Dubrovin. An overring \(S\) of a Dubrovin valuation ring \(B\) is Dubrovin, and the radical \(J(S)\) is a prime ideal of \(B\); every prime ideal of \(B\) has this form. On the other hand, for a subring \(R \subseteq B\) containing \(J\), being Dubrovin is equivalent to \(R/J\) being a Dubrovin subring of \(B/J\).NEWLINENEWLINELet \(V\) be a valuation ring of \(F\). There is a Dubrovin valuation ring \(B \subseteq A\) such that \(B \cap F = V\), unique up to conjugation in \(A\). Dubrovin valuation rings \(B_1,\dots,B_n\) of \(A\) satisfy the ``intersection property'' (IP) if the intersection \(J(S) \cap R\) is a prime ideal of \(R = B_1 \cap \cdots \cap B_n\) for every overring \(S\) of any of the \(B_i\). The IP for \(B_1,\dots,B_n\) is equivalent to IP for each pair \(B_i,B_j\). A subring \(R \subseteq A\) which is integral over its center is a {\textit{Gräter ring}} if it is the intersection of finitely many Dubrovin valuation rings of \(A\) satisfying the IP. The (noncomparable) Dubrovin rings participating in the intersection are the localizations of \(R\) with respect to its maximal ideals. Every finite intersection of valuation rings of \(F\) is the center of a Gräter subring of \(A\), unique up to conjugation.NEWLINENEWLINEThe {\textit{extension number}} \(\zeta_{V,A}\) of a valuation ring \(V \subseteq F\) in \(A\) is the number of maximal ideals in a Gräter ring of \(A\) with center \(V\). This number only depends on \(V\) and the Brauer class of \(A\). The extension number equals \(1\) if and only if every Gräter ring of \(A\) whose intersection with \(F\) is \(V\) is a Dubrovin valuation ring.NEWLINENEWLINEFor a \(\nu\)-gauge \(\alpha\) of \(A\), let \(\omega(\alpha)\) denote the number of simple components of the zero component \(A_0 \subset \mathsf{gr}(A)\). Assume \(A\) is central simple over \(F\). Then \(\omega(\alpha) \geq \zeta_{V,A}\) (Theorem 3.5). The \(\nu\)-gauge \(\alpha\) is {\textit{minimal}} if \(\omega(\alpha) = \zeta_{V,A}\). When \(\alpha\) is minimal, the gauge ring \(\{a \in A\,|\,\alpha(a) \geq 0\}\) is Gräter and its center is \(V\). When \(\alpha\) is defectless in \(A\), every Gräter ring of \(A\) with center \(V\) has this form for some minimal gauge \(\alpha\). The notion of minimality can be extended to gauges of semisimple \(F\)-algebras; in the most intricate part of the paper, the authors prove that if \(\nu\) is defectless, minimal \(\nu\)-gauges always exist.NEWLINENEWLINEIn the final section, the quaternion algebra \((1+x,y)\) over \(F = k(x)(\!(y)\!)\) is analyzed in details, demonstrating that nonisomorphic minimal gauges can have the same gauge ring.