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-INVARIANT IDEALS IN IWASAWA ALGEBRAS K. ARDAKOV, S. J. WADSLEY
 

Summary: -INVARIANT IDEALS IN IWASAWA ALGEBRAS
K. ARDAKOV, S. J. WADSLEY
Abstract. Let kG be the completed group algebra of a uniform pro-p group
G with coefficients in a field k of characteristic p. We study right ideals I in
kG that are invariant under the action of another uniform pro-p group . We
prove that if I is non-zero then an irreducible component of the characteristic
support of kG/I must be contained in a certain finite union of rational linear
subspaces of Spec gr kG. The minimal codimension of these subspaces gives a
lower bound on the homological height of I in terms of the action of a certain
Lie algebra on G/Gp. If we take to be G acting on itself by conjugation,
then -invariant right ideals of kG are precisely the two-sided ideals of kG, and
we obtain a non-trivial lower bound on the homological height of a possible
non-zero two-sided ideal. For example, when G is open in SLn(Zp) this lower
bound equals 2n - 2. This gives a significant improvement of the results of
Ardakov, Wei and Zhang [AWZ1] on reflexive ideals in Iwasawa algebras.
1. Introduction
1.1. Prime ideals in Iwasawa algebras. In recent years, several attempts have
been made to understand the structure of prime ideals in non-commutative Iwasawa
algebras. These are the completed group algebras G of compact p-adic analytic
groups G with coefficients in the finite field Fp; we refer the reader to the survey

  

Source: Ardakov, Konstantin - School of Mathematical Sciences, University of Nottingham

 

Collections: Mathematics