On the uniqueness of topologies for some constructions of rings and modules

Arnautov, V. I.; Ursul, Mihail

doi:10.1007/bf02107321

Cited by 5 publications

(7 citation statements)

References 3 publications

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“…Case 1: d > 2 or F = Q p or D is (absolutely) irreducible. These three conditions correspond to parts (1), (2), and (3) of Proposition 3.52 respectively, and indeed Proposition 3.52 implies that the complement of V Kirr,χ in X gen,χ has dimension at most dim X gen,χ − 2. Hence, we may take V 0,χ = V Kirr,χ .…”

Section: Proposition 413mentioning

confidence: 73%

“…In this case, µ = {1} so χ = 1 and thus X gen,1 = X gen . It follows from Proposition 3.25, Lemma 3.29, Lemma 3.51 that (2). We may also write V 0,χ = V Kirr ∪ V ′ Pmax , where we use the notation introduced in the proof of Proposition 3.26.…”

Section: Proposition 413mentioning

confidence: 99%

“…By considering this modulo (̟, c), we would conclude that (d 3 − d 6 , π) is the maximal ideal in k b, d . Since (d 3 − d 6 , π) → (b, d)/(b, d)2 is not surjective, we obtain a contradiction. The same argument shows that R ,χ ρ /̟ is also not factorial.…”

mentioning

confidence: 93%

“…In all cases, the functor D ρx is pro-represented by a complete local Noetherian Λ-algebra R ρx with residue field κ(x). The arguments of Mazur and Kisin carry over to the case when κ(x) is a local field of characteristic p and yield a presentation (2) R ρx ∼ = Λ x 1 , . .…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On local Galois deformation rings

Böckle¹,

Iyengar²,

Paškūnas³

2021

Preprint

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Section: Proposition 413mentioning

confidence: 73%

Section: Proposition 413mentioning

confidence: 99%

mentioning

confidence: 93%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On local Galois deformation rings

Böckle¹,

Iyengar²,

Paškūnas³

2021

Preprint

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“…Perhaps the present argument is more direct. Warner's theorem also emerges as a consequence of this recent very general result of Arnautov and Ursul [4,Theorem 4.1]: If the Jacobson radical of a ring R admits exactly one compact ring topology, then R itself admits at most one compact ring topology.…”

Section: Corollary a Semisimple Ring Has At Most One Compact Ring Tomentioning

confidence: 90%

Extending Ring Topologies

Comfort¹,

Remus²,

Szambien³

2000

Journal of Algebra

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Throughout this abstract, R denotes a compact (Hausdorff) topological ring. The authors extend to ring topologies on R which are totally bounded or even pseudocompact; a principal tool is the Bohr compactification of a topological ring. They show inter alia: If the Jacobson radical J R of R satisfies w R/J R > ω then there is a pseudocompact ring topology on R strictly finer than ; if in addition w R = w R/J R = α with cf α > ω then there are exactly 2 2 R -many such topologies. The ring R is said to be a van der Waerden ring if is the only totally bounded ring topology on R. Theorem 4.13 asserts that if R is semisimple, then R is a van der Waerden ring if and only if in the Kaplansky representation R = n<ω R n α n of R as a product of full matrix rings over finite fields each α n is finite. Other classes of van der Waerden rings are constructed, and it is shown that there are non-compact totally bounded rings S such that is the only totally bounded ring topology on S.

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Banach space representations and Iwasawa theory

2002

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The lack of a p-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact p-adic Lie group G in Banach spaces over a given p-adic field K. For example, the abelian group G = Z Z p has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations, as may be seen in the paper by Diarra [Dia]. We therefore address the problem of finding an additional "finiteness" condition on such representations that will lead to a reasonable theory. We introduce such a condition that we call "admissibility". We show that the category of all admissible G-representations is reasonablein fact, it is abelian and of a purely algebraic nature -by showing that it is anti-equivalent to the category of all finitely generated modules over a certain kind of completed group ringIn the first part of our paper we deal with the general functional-analytic aspects of the problem. We first consider the relationship between K-Banach spaces and compact, linearly topologized o-modules where o is the ring of integers in K. As a special case of ideas of Schikhof [Sch], we recall that there is an anti-equivalence between the category of K-Banach spaces and the category of torsionfree, linearly compact o-modules, provided one tensors the Hom-spaces in the latter category with Q. In addition we have to investigate how this functor relates certain locally convex topologies on the Hom-spaces in the two categories. This will enable us then to derive a version of this anti-equivalence with an action of a profinite group G on both sides relating K-Banach space representations of G and certain topological modules for the ringHaving established these topological results we assume that G is a compact padic Lie group and focus our attention on the Banach representations of G that correspond under the anti-equivalence to finitely generated modules over the ring. We characterize such Banach space representations intrinsically. We then show that the theory of such "admissible" representations is purely algebraic -one may "forget" about topology and instead study finitely generated modules over the noetherian ringAs an application of our methods we determine the topological irreducibility as well as the intertwining maps for representations of GL 2 (Z Z p ) obtained by induction of a continuous character from the subgroup of lower triangular matrices. Let us stress the fact that topological irreducibility for an admissible Banach space representation corresponds to the algebraic simplicity of the dual K[[G]]-module. It is indeed the latter which we will analyze. These results are a complement to the treatment of the locally analytic principal series representations studied in [ST1].

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On the uniqueness of topologies for some constructions of rings and modules

Cited by 5 publications

References 3 publications

On local Galois deformation rings

On local Galois deformation rings

Extending Ring Topologies

Banach space representations and Iwasawa theory

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