Structure of central torsion Iwasawa modules

Susan Howson

Displaying similar documents to “Structure of central torsion Iwasawa modules”

On the structure theory of the Iwasawa algebra of a p-adic Lie group

Otmar Venjakob (2002)

Journal of the European Mathematical Society

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This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, $Λ$ of a $p$ -adic analytic group $G$ . For $G$ without any $p$ -torsion element we prove that $Λ$ is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null $Λ$ -module. This is classical when $G = ℤ_{p}^{k}$ for some integer $k \geq 1$ , but was previously unknown in the non-commutative case. Then the category...

On $τ$ -extending modules

Y. Talebi, R. Mohammadi (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we introduce the concept of $τ$ -extending modules by $τ$ -rational submodules and study some properties of such modules. It is shown that the set of all $τ$ -rational left ideals of $_{R} R$ is a Gabriel filter. An $R$ -module $M$ is called $τ$ -extending if every submodule of $M$ is $τ$ -rational in a direct summand of $M$ . It is proved that $M$ is $τ$ -extending if and only if $M = R e j_{M} E (R / τ (R)) \oplus N$ , such that $N$ is a $τ$ -extending submodule of $M$ . An example is given to show that the direct sum of $τ$ -extending modules need not...

Some results on $(n, d)$ -injective modules, $(n, d)$ -flat modules and $n$ -coherent rings

Zhanmin Zhu (2015)

Commentationes Mathematicae Universitatis Carolinae

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Let $n, d$ be two non-negative integers. A left $R$ -module $M$ is called $(n, d)$ -injective, if ${Ext}^{d + 1} (N, M) = 0$ for every $n$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called $(n, d)$ -flat, if ${Tor}_{d + 1} (V, N) = 0$ for every $n$ -presented left $R$ -module $N$ . A left $R$ -module $M$ is called weakly $n$ - $F P$ -injective, if ${Ext}^{n} (N, M) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . A right $R$ -module $V$ is called weakly $n$ -flat, if ${Tor}_{n} (V, N) = 0$ for every $(n + 1)$ -presented left $R$ -module $N$ . In this paper, we give some characterizations and properties of $(n, d)$ -injective modules and $(n, d)$ -flat modules in...

$ℒ$ -invariants and Darmon cycles attached to modular forms

Victor Rotger, Marco Adamo Seveso (2012)

Journal of the European Mathematical Society

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Let $f$ be a modular eigenform of even weight $k \geq 2$ and new at a prime $p$ dividing exactly the level with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to $f$ a monodromy module $D_{f}^{F M}$ and an $ℒ$ -invariant $ℒ_{f}^{F M}$ . The first goal of this paper is building a suitable $p$ -adic integration theory that allows us to construct a new monodromy module $D_{f}$ and $ℒ$ -invariant $ℒ_{f}$ , in the spirit of Darmon. The two monodromy modules are isomorphic, and in particular the two $ℒ$ -invariants...

On the minimaxness and coatomicness of local cohomology modules

Marzieh Hatamkhani, Hajar Roshan-Shekalgourabi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$ -module. We wish to investigate the relation between vanishing, finiteness, Artinianness, minimaxness and $𝒞$ -minimaxness of local cohomology modules. We show that if $M$ is a minimax $R$ -module, then the local-global principle is valid for minimaxness of local cohomology modules. This implies that if $n$ is a nonnegative integer such that ${(H_{I}^{i} (M))}_{𝔪}$ is a minimax $R_{𝔪}$ -module for all $𝔪 \in Max (R)$ and for all $i < n$ , then the set ${Ass}_{R} (H_{I}^{n} (M))$ is finite. Also, if $H_{I}^{i} (M)$ is...

A note on Frobenius divided modules in mixed characteristics

Pierre Berthelot (2012)

Bulletin de la Société Mathématique de France

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If $X$ is a smooth scheme over a perfect field of characteristic $p$ , and if $𝒟_{X}^{(\infty)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of $𝒟_{X}^{(\infty)}$ on an $𝒪_{X}$ -module $ℰ$ is equivalent to giving an infinite sequence of $𝒪_{X}$ -modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f : X \to S$ of a smooth morphism in characteristic $p$ , endowed with Frobenius liftings. We also show that it...

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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The rational completion $\bar{M}$ of an $R$ -module $M$ can be characterized as a $\tau_{M}$ -injective hull of $M$ with respect to a (hereditary) torsion functor $\tau_{M}$ depending on $M$ . Properties of a torsion functor depending on an $R$ -module $M$ are studied.

On twisted group algebras of OTP representation type over the ring of p-adic integers

Leonid F. Barannyk, Dariusz Klein (2016)

Colloquium Mathematicae

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Let $ℤ ̂_{p}$ be the ring of p-adic integers, $U (ℤ ̂_{p})$ the unit group of $ℤ ̂_{p}$ and $G = G_{p} \times B$ a finite group, where $G_{p}$ is a p-group and B is a p’-group. Denote by $ℤ ̂_{p}^{λ} G$ the twisted group algebra of G over $ℤ ̂_{p}$ with a 2-cocycle $λ \in Z ² (G, U (ℤ ̂_{p}))$ . We give necessary and sufficient conditions for $ℤ ̂_{p}^{λ} G$ to be of OTP representation type, in the sense that every indecomposable $ℤ ̂_{p}^{λ} G$ -module is isomorphic to the outer tensor product V W of an indecomposable $ℤ ̂_{p}^{λ} G_{p}$ -module V and an irreducible $ℤ ̂_{p}^{λ} B$ -module W.

On the de Rham and $p$ -adic realizations of the elliptic polylogarithm for CM elliptic curves

Kenichi Bannai, Shinichi Kobayashi, Takeshi Tsuji (2010)

Annales scientifiques de l'École Normale Supérieure

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In this paper, we give an explicit description of the de Rham and $p$ -adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve $E$ defined over an imaginary quadratic field $𝕂$ with complex multiplication by the full ring of integers $𝒪_{𝕂}$ of $𝕂$ . Note that our condition implies that $𝕂$ has class number one. Assume in addition that $E$ has good reduction above a prime $p \geq 5$ unramified in $𝒪_{𝕂}$ . In this case, we prove that the specializations of the...

Relative Gorenstein injective covers with respect to a semidualizing module

Elham Tavasoli, Maryam Salimi (2017)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$ -module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$ -injective module $G$ , the character module $G^{+}$ is $G_{C}$ -flat, then the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${𝒢ℐ}_{C} (R) \cap 𝒜_{C} (R)$ ...

Separable $ℵ_{k}$ -free modules with almost trivial dual

Daniel Herden, Héctor Gabriel Salazar Pedroza (2016)

Commentationes Mathematicae Universitatis Carolinae

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An $R$ -module $M$ has an almost trivial dual if there are no epimorphisms from $M$ to the free $R$ -module of countable infinite rank $R^{(ω)}$ . For every natural number $k > 1$ , we construct arbitrarily large separable $ℵ_{k}$ -free $R$ -modules with almost trivial dual by means of Shelah’s Easy Black Box, which is a combinatorial principle provable in ZFC.

Some bounds for the annihilators of local cohomology and Ext modules

Ali Fathi (2022)

Czechoslovak Mathematical Journal

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Let $𝔞$ be an ideal of a commutative Noetherian ring $R$ and $t$ be a nonnegative integer. Let $M$ and $N$ be two finitely generated $R$ -modules. In certain cases, we give some bounds under inclusion for the annihilators of ${Ext}_{R}^{t} (M, N)$ and $H_{𝔞}^{t} (M)$ in terms of minimal primary decomposition of the zero submodule of $M$ , which are independent of the choice of minimal primary decomposition. Then, by using those bounds, we compute the annihilators of local cohomology and Ext modules in certain cases.

Lifting the field of norms

Laurent Berger (2014)

Journal de l’École polytechnique — Mathématiques

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Let $K$ be a finite extension of $Q_{p}$ . The field of norms of a $p$ -adic Lie extension $K_{\infty} / K$ is a local field of characteristic $p$ which comes equipped with an action of $Gal (K_{\infty} / K)$ . When can we lift this action to characteristic $0$ , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(ϕ, Γ)$ -modules, and give a condition for the existence of certain types of lifts.

Irregularity of an analogue of the Gauss-Manin systems

Céline Roucairol (2006)

Bulletin de la Société Mathématique de France

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In $𝒟$ -modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+} (𝒪 e^{g})$ of a $𝒟$ -module twisted by the exponential of a polynomial $g$ by another polynomial $f$ , where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can...

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

Tomasz Jędrzejak (2013)

Acta Arithmetica

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Consider the families of curves $C^{n, A} : y ² = x ⁿ + A x$ and $C_{n, A} : y ² = x ⁿ + A$ where A is a nonzero rational. Let $J^{n, A}$ and $J_{n, A}$ denote their respective Jacobian varieties. The torsion points of $C^{3, A} (ℚ)$ and $C_{3, A} (ℚ)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7, A} (ℚ)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7, A} (ℚ) [2]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3, A}$ (A ≠ 4) and $J^{5, A}$ . We...

Prescribing endomorphism algebras of $ℵ_{n}$ -free modules

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2014)

Journal of the European Mathematical Society

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It is a well-known fact that modules over a commutative ring in general cannot be classified, and it is also well-known that we have to impose severe restrictions on either the ring or on the class of modules to solve this problem. One of the restrictions on the modules comes from freeness assumptions which have been intensively studied in recent decades. Two interesting, distinct but typical examples are the papers by Blass [1] and Eklof [8], both jointly with Shelah. In the first case...