Formula omitted]-invariant ideals in Iwasawa algebras

Abstract

Let k G 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 k G that are invariant under the action of another uniform pro- p group Γ . We prove that if I is non-zero then an irreducible component of the characteristic support of k G / I must be contained in a certain finite union of rational linear subspaces of Spec gr k G . 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 / G p . If we take Γ to be G acting on itself by conjugation, then Γ -invariant right ideals of k G are precisely the two-sided ideals of k G , 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 SL n ( Z p ) this lower bound equals 2 n − 2 . This gives a significant improvement of the results of [K. Ardakov, F. Wei, J.J. Zhang, Reflexive ideals in Iwasawa algebras, Adv. Math. 218 (2008) 865–901].