35 research outputs found
Prime ideals in nilpotent Iwasawa algebras
Let G be a nilpotent complete p-valued group of finite rank and let k be a
field of characteristic p. We prove that every faithful prime ideal of the
Iwasawa algebra kG is controlled by the centre of G, and use this to show that
the prime spectrum of kG is a disjoint union of commutative strata. We also
show that every prime ideal of kG is completely prime. The key ingredient in
the proof is the construction of a non-commutative valuation on certain
filtered simple Artinian rings
Gamma-invariant ideals in Iwasawa algebras
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
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D-modules on rigid analytic spaces I
We introduce a sheaf of infinite order differential operators D-cap on smooth
rigid analytic spaces that is a rigid analytic quantisation of the cotangent
bundle. We show that the sections of this sheaf over sufficiently small
affinoid varieties are Fr\'echet-Stein algebras, and use this to define
co-admissible sheaves of D-cap-modules. We prove analogues of Cartan's Theorems
A and B for co-admissible D-cap-modules.Award identifier / Grant number: EP/L005190/1
The first author was supported by EPSRC grant EP/L005190/1
D-modules on rigid analytic spaces II: Kashiwara’s equivalence
We prove that the category of coadmissible D-cap-modules on a smooth rigid
analytic space supported on a closed smooth subvariety is naturally equivalent
to the category of coadmissible D-cap-modules on the subvariety, and use this
result to construct a large family of pairwise non-isomorphic simple
coadmissible D-cap-modules.The first author was supported by EPSRC grant EP/L005190/1
Equivariant D-modules on rigid analytic spaces
We define coadmissible equivariant -modules on smooth rigid analytic spaces and relate them to admissible locally analytic representations of semisimple -adic Lie groups
Primeness, semiprimeness and localisation in Iwasawa algebras
Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions are found for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al
The centre of completed group algebras of pro-p groups
We compute the centre of the completed group algebra of an arbitrary countably based pro-p group with coefficients in double-struck F sign p or ℤp. Some other results are obtained