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Actions of Algebraic Groups on the Spectrum of Rational Ideals, II

โœ Scribed by Nikolaus Vonessen

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
461 KB
Volume
208
Category
Article
ISSN
0021-8693

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โœฆ Synopsis

We study rational actions of a linear algebraic group G on an algebra V, and the ลฝ . ลฝ induced actions on Rat V , the spectrum of rational ideals of V a subset of ลฝ . . 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, we relate properties of a rational ideal J and its ลฝ . ลฝ . orb, the ideal J : G s F โฅ J . The rational ideals of V containing the orb of โฅ g G ลฝ . J are precisely those in the Zariski-closure X of the orbit of J in Rat V . The ลฝ 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 . We show that the G-stratum 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 Goodearl and Letzter. We further show that ลฝ . the orbit of J is open in its closure in Rat V , provided that J is locally closed.

ลฝ . Among other results, we show that the semiprime ideal J : G is Goldie, and we ลฝ . relate the uniform and GelfandแŽKirillov dimensions of VrJ and Vr J : G .

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