Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

Extending tamely ramified strict 1-motives into két log 1-motives

We define két abelian schemes, két 1-motives and két log 1-motives and formulate duality theory for these objects. Then we show that tamely ramified strict 1-motives over a discrete valuation field can be extended uniquely to két log 1-motives over the corresponding discrete valuation ring. As an application, we present a proof to a result of Kato stated in [12, §4.3] without proof. To a tamely ramified strict 1-motive over a discrete valuation field, we associate a monodromy pairing and compare it with Raynaud’s geometric monodromy.

D3-MODULES VERSUS D4-MODULES – APPLICATIONS TO QUIVERS

A module M is called a D4-module if, whenever A and B are submodules of M with M = A ⊕ B and f : A → B is a homomorphism with Imf a direct summand of B, then Kerf is a direct summand of A. The class of D4-modules contains the class of D3-modules, and hence the class of semi-projective modules, and so the class of Rickart modules. In this paper we prove that, over a commutative Dedekind domain R, for an R-module M which is a direct sum of cyclic submodules, M is direct projective (equivalently, it is semi-projective) iff M is D3 iff M is D4. Also we prove that, over a prime PI-ring, for a divisible R-module X, X is direct projective (equivalently, it is Rickart) iff X ⊕ X is D4. We determine some D3-modules and D4-modules over a discrete valuation ring, as well. We give some relevant examples. We also provide several examples on D3-modules and D4-modules via quivers.

On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

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.

Integer-valued skew polynomials

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

DEFORMATION THEORY OF THE CHOW GROUP OF ZERO CYCLES

We study the deformations of the Chow group of zerocycles of the special fibre of a smooth scheme over a Henselian discrete valuation ring. Our main tools are Bloch’s formula and differential forms. As a corollary we get an algebraization theorem for thickened zero cycles previously obtained using idelic techniques. In the course of the proof we develop moving lemmata and Lefschetz theorems for cohomology groups with coefficients in differential forms.

Barrelled Linearly Topologized Modules Over a Discrete Valuation Ring

In this paper barrelled linearly topologized modules over an arbitrary discrete valuation ring are introduced. A general form of the Banach-Steinhaus theorem for continuous linear mappings on barrelled linearly topologized modules is established and some consequences of it are derived.

Barrelled Linearly Topologized Modules Over a Discrete Valuation Ring

In this paper barrelled linearly topologized modules over an arbitrary discrete valuation ring are introduced. A general form of the Banach-Steinhaus theorem for continuous linear mappings on barrelled linearly topologized modules is established and some consequences of it are derived.

ARITHMETIC STRUCTURES FOR DIFFERENTIAL OPERATORS ON FORMAL SCHEMES

Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$ -scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$ , for every sufficiently large positive integer $k$ , generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$ . The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$ , over all admissible blow-ups $\mathfrak{X}$ , is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$ , which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ , $\mathscr{D}_{\mathfrak{X},\infty }$ , and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ .

Clifford theory of Weil representations of unitary groups

Let {\mathcal{O}} be an involutive discrete valuation ring with residue field of characteristic not 2. Let A be a quotient of {\mathcal{O}} by a nonzero power of its maximal ideal, and let {*} be the involution that A inherits from {\mathcal{O}} . We consider various unitary groups {\mathcal{U}_{m}(A)} of rank m over A, depending on the nature of {*} and the equivalence type of the underlying hermitian or skew hermitian form. Each group {\mathcal{U}_{m}(A)} gives rise to a Weil representation. In this paper, we give a Clifford theory description of all irreducible components of the Weil representation of {\mathcal{U}_{m}(A)} with respect to all of its abelian congruence subgroups and a third of its nonabelian congruence subgroups.