Random matrices over a DVR and LU factorization

Let D be an integral domain with identity and let K denote the quotient field of D. If P is a prime ideal of D denote by $P[[X]]$ that prime ideal of $D[[X]]$ consisting of all those formal power series each of whose coefficients belongs to P. In this paper the following question is considered: When is ${(D[[X]])_{P[[X]]}}$ a valuation ring? Our main theorem states that if ${(D[[X]])_{P[[X]]}}$ is a valuation ring, then ${D_P}$ must be a rank one discrete valuation ring. Moreover, we show that if ${D_P}$ is a rank one discrete valuation ring and if $PD[[X]] = P[[X]]$, then ${(D[[X]])_{P[[X]]}}$ is a valuation ring. We also give an example to show that ${(D[[X]])_{P[[X]]}}$ need not be a valuation ring when ${D_P}$ is rank one discrete.

The fundamental result of linear network coding asserts the existence of optimal codes over acyclic networks when the symbol field is sufficiently large. The restriction to just acyclic networks turns out to stem from the customary algebraic structure of data symbols as a finite field. Adopting data units that belong to a discrete valuation ring (DVR), that is, a PID with a unique maximal ideal, much of the network coding theory extends to cyclic networks. Being a PID with the maximal ideal 0, a field can be regarded as a degenerated DVR. Thus the field-based theory becomes a degenerated version of the DVR-based theory. Meanwhile, convolutional network coding becomes the instance when the DVR consists of rational power series over a field. Besides the treatise in commutative algebra, the present paper also delves into the efficiency issue of code construction. Given a cyclic network, a quadratically large acyclic network is constructed so that every optimal code on the acyclic network subject to some straightforward restriction induces an optimal code on the given network. In this way, existing construction algorithms over acyclic networks can be adapted for cyclic networks as well.

In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

Normal toric varieties over a field or a discrete valuation ring are classified by rational polyhedral fans. We generalize this classification to normal toric varieties over an arbitrary valuation ring of rank one. The proof is based on a generalization of Sumihiro's theorem to this non-noetherian setting. These toric varieties play an important role for tropicalizations.

In this paper, we mainly study on the valuation and Dedekind modules and we also give some equal conditions to Dedekind modules. Moreover, by using these classes, we give some characterizations for a Dedekind ring, a valuation ring and a Discrete valuation ring.

We give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real field corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying $\operatorname{GL}_2$-representation can be chosen to take values in a discrete valuation ring. A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works. Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements. To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a $p$-adic interpolation series for central derivative values when the root number is generically equal to $-1$, in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves.

The relative form of Gersten's conjecture over a discrete valuation ring: The smooth case

In this paper we investigate the ring Ar(R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on Ar(R). We show that a certain extension of Ar(R) is a discrete valuation ring. We also investigate the metric structure of the ring Ar(R).

Let R be a normal Noetherian local domain of Krull dimension two. We examine intersections of rank one discrete valuation rings that birationally dominate R . We restrict to the class of prime divisors that dominate R and show that if a collection of such prime divisors is taken below a certain “level,” then the intersection is an almost Dedekind domain having the property that every nonzero ideal can be represented uniquely as an irredundant intersection of powers of maximal ideals.

Essential dimension of group schemes over a local scheme

Let G \mathbf {G} be the General Linear or Special Linear group with entries from the finite quotients of the ring of integers of a non-archimedean local field and U \mathbf {U} be the subgroup of G \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) of G \mathbf {G} obtained from a non-degenerate character θ \theta of U \mathbf {U} is multiplicity free for all ℓ ≥ 2. \ell \geq 2. This is analogous to the multiplicity one theorem regarding Gelfand-Graev representation for the finite Chevalley groups. We prove that for many cases the regular representations of G \mathbf {G} are characterized by the property that these are the constituents of the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) for some non-degenerate character θ \theta of U \mathbf {U} . We use this to prove that the restriction of a regular representation of General Linear groups to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases.

Models of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mi>μ</mml:mi><mml:mrow><mml:msup><mml:mi>p</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>,</mml:mo><mml:mi>K</mml:mi></mml:mrow></mml:msub></mml:math> over a discrete valuation ring

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

We prove, among other results, that the one-dimensional local domain A is Henselian if and only if for every maximal ideal M in the Laurent polynomial ring A[T, T~l], either M n A[T] or M C\ A[T~^\ is a maximal ideal.The discrete valuation ring A is Henselian if and only if every pseudo-Weierstrass polynomial in A[T] is Weierstrass.We apply our results to the complete intersection problem for maximal ideals in regular Laurent polynomial rings.

When is a field, the classical Jacobian criterion computes the singular locus of an equidimensional, finitely generated ‐algebra as the closed subset of an ideal generated by appropriate minors of the so‐called Jacobian matrix. Recently, Hochster‐Jeffries and Saito have extended this result for algebras over any unramified discrete valuation ring of mixed characteristic via the use of ‐derivations. Motivated by these results, in this paper, we state and prove an analogous Jacobian criterion for algebras over ramified discrete valuation rings of mixed characteristic.