Reflexive ideals in Iwasawa algebras

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 Γ. 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 Γ to be G acting on itself by conjugation, then Γ-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.

Centres of Skewfields and Completely Faithful Iwasawa Modules

“…It is shown in [6,Theorem 7.3] that KH has no non-trivial reflexive two-sided ideals. As I is non-zero, it follows that I = KH.…”

Homological properties of noncommutative Iwasawa algebras

“…Ring-theoretic and homological properties of the Iwasawa algebras are useful for understanding the structure of the Pontryagin dual of the Selmer groups [14,17] and other modules over the Iwasawa algebras [6,13]. Several recent papers [2][3][4]15,16] are devoted to ring-theoretic properties of the Iwasawa algebras. Certain homological aspects of the Iwasawa algebras have been studied in [2,6,13,15,16].…”

“…They are complete noetherian semilocal algebras, which are in general noncommutative. Theorem C of [3] states that every prime ideal of the Iwasawa algebra Ω G over any open torsion-free subgroup of SL 2 (Z p ) is either zero or maximal. In this case, Ω G is local and extremely noncommutative since the only non-zero prime ideal is the maximal ideal.…”

“…Motivated by the main conjecture of Iwasawa theory, and more generally by the roles of Λ G and Ω G in the arithmetic geometry of elliptic curves and their Selmer groups, there has been considerable ring-theoretic activity concerning Λ G and Ω G (see [2][3][4]7,[16][17][18]). Many Iwasawa algebras are nice examples of "just-infinite algebras" which both satisfy the Auslander-Gorenstein condition and are thus amenable to Lie theoretic analysis.…”

Prime ideals in nilpotent Iwasawa algebras

“…A longrunning project aims to understand the prime spectrum Spec(kG) of kG, guided in part by the list of open questions in §6 of this survey paper. Progress so far has been rather limited: the strongest known result to date, [3,Theorem 4.8] asserts that (under mild restrictions on the prime p) when the Lie algebra g of G is split semisimple, the homological height of a non-zero prime ideal in kG is bounded below by an integer u depending only g; for example if g = sl n (Q p ) then u = 2n − 2. …”