Maximal prime homomorphic images of mod-p Iwasawa algebras

• Published in: Mathematical proceedings of the Cambridge Philosophical Society Vol. 171; no. 2; pp. 387 - 419
• Main Author: WOODS, WILLIAM
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2021
• Subjects: Fields (mathematics); Isomorphism; Ring structures; Subgroups; 16S35; 16D25; 16S34
• ISSN: 0305-0041; 1469-8064
• DOI: 10.1017/S0305004120000262

Let k be a finite field of characteristic p, and G a compact p-adic analytic group. Write kG for the completed group ring of G over k. In this paper, we describe the structure of the ring kG/P, where P is a minimal prime ideal of kG. We give an explicit isomorphism between kG/P and a matrix ring with coefficients in the ring ${(k'G')_\alpha }$ , where $k'/k$ is a finite field extension, $G'$ is a large subquotient of G with no finite normal subgroups, and (–) α is a “twisting” operation that preserves many desirable properties of the ring structure. We demonstrate the usefulness of this isomorphism by studying the correspondence induced between certain ideals of kG and those of ${(k'G')_\alpha }$ , and showing that this preserves many useful “group-theoretic” properties of ideals, in particular almost-faithfulness and control by a closed normal subgroup.