Reflexive ideals in Iwasawa algebras

K. Ardakov; F. Wei; J. J. Zhang

doi:10.1016/j.aim.2008.02.004

Reflexive ideals in Iwasawa algebras

K. Ardakov^*, F. Wei, J. J. Zhang

^*Corresponding author for this work

School of Mathematics and Statistics

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

Let G be a torsionfree compact p-adic analytic group. We give sufficient conditions on p and G which ensure that the Iwasawa algebra Ω_G of G has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every non-zero normal element in Ω_G is a unit. We show that these conditions hold in the case when G is an open subgroup of SL₂ (Z_p) and p is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in Ω_G when G is a congruence subgroup of SL₂ (Z_p): the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.

Original language	English
Pages (from-to)	865-901
Number of pages	37
Journal	Advances in Mathematics
Volume	218
Issue number	3
DOIs	https://doi.org/10.1016/j.aim.2008.02.004
Publication status	Published - 20 Jun 2008

Keywords

Iwasawa algebra
Microlocalisation
Normal element
Reflexive ideal
Uniform pro-p group

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10.1016/j.aim.2008.02.004

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AU - Ardakov, K.

AU - Wei, F.

AU - Zhang, J. J.

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N2 - Let G be a torsionfree compact p-adic analytic group. We give sufficient conditions on p and G which ensure that the Iwasawa algebra ΩG of G has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every non-zero normal element in ΩG is a unit. We show that these conditions hold in the case when G is an open subgroup of SL2 (Zp) and p is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in ΩG when G is a congruence subgroup of SL2 (Zp): the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.

AB - Let G be a torsionfree compact p-adic analytic group. We give sufficient conditions on p and G which ensure that the Iwasawa algebra ΩG of G has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every non-zero normal element in ΩG is a unit. We show that these conditions hold in the case when G is an open subgroup of SL2 (Zp) and p is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in ΩG when G is a congruence subgroup of SL2 (Zp): the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.

KW - Iwasawa algebra

KW - Microlocalisation

KW - Normal element

KW - Reflexive ideal

KW - Uniform pro-p group

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JO - Advances in Mathematics

JF - Advances in Mathematics

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Reflexive ideals in Iwasawa algebras

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Keywords

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