Gamma-invariant ideals in Iwasawa algebras

TL;DR

This paper investigates Gamma-invariant ideals in Iwasawa algebras of uniform pro-p groups, establishing bounds on their homological height and advancing understanding of their structure, especially for groups like SL_n(Zp).

Contribution

It introduces new bounds on the homological height of invariant ideals, linking algebraic invariants to geometric properties of the characteristic support.

Findings

Non-zero Gamma-invariant ideals have characteristic support contained in finite unions of rational linear subspaces.

The minimal codimension of these subspaces bounds the homological height of the ideals.

For G open in SL_n(Zp), the lower bound on height is 2n - 2.

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 Gamma. 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/G^p. If we take Gamma to be G acting on itself by conjugation, then Gamma-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 SL_n(\Zp) this lower bound equals 2n - 2. This gives a significant improvement of the results of Ardakov, Wei and Zhang on reflexive ideals in Iwasawa algebras.