Actions of algebraic groups on the spectrum of rational ideals. II
[For part I cf. ibid. 182, No. 2, 383-400 (1996; Zbl 0867.16020).] The author studies rational actions of a linear algebraic group \(G\) on an algebra \(V\), and the induced actions on \(\text{Rat}(V)\), the spectrum of rational ideals of \(V\) (a subset of \(\text{Spec}(V)\) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions on \(V\). In particular, some properties of the rational ideal \(J\) are related with its orb, the ideal \((J:G)=\bigcap_{\gamma\in G}\gamma(J)\). The rational ideals of \(V\) containing the orb of \(J\) are precisely those in the Zariski-closure \(X\) of the orbit of \(J\) in \(\text{Rat}(V)\). The \(G\)-stratum of \(J\) consists of all rational ideals in \(X\) whose orbit is dense in \(X\) (i.e. whose orb is equal to the orb of \(J\)). The author shows that the \(G\)-spectrum of a rational ideal consists of exactly one \(G\)-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by \textit{K. R. Goodearl} and \textit{E. S. Letzter} [Trans. Am. Math. Soc. (to appear)]. It is shown further that the orbit of \(J\) is open in its closure in \(\text{Rat}(V)\), provided that \(J\) is locally closed. Among other results, the author proves that the semiprime ideal \((J:G)\) is Goldie, and relates the uniform and Gelfand-Kirillov dimensions of \(V/J\) and \(V/(J,G)\).
scientific article; zbMATH DE number 1221662
scientific article