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We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X 1 (M , M N) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman. [ABSTRACT FROM AUTHOR]

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h (M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h (M) in the case when M is the canonical module ω R. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups. [ABSTRACT FROM AUTHOR]

We study a family of (possibly non topological) deformations of BF theory for the Lie algebra obtained by quadratic extension of so (3 , 1) by an orthogonal module. The resulting theory, called quadratically extended General Relativity (qeGR), is shown to be classically equivalent to certain models of gravity with dynamical torsion. The classical equivalence is shown to promote to a stronger notion of equivalence within the Batalin–Vilkovisky formalism. In particular, both Palatini–Cartan gravity and a deformation thereof by a dynamical torsion term, called (quadratic) generalised Holst theory, are recovered from the standard Batalin–Vilkovisky formulation of qeGR by elimination of generalised auxiliary fields. [ABSTRACT FROM AUTHOR]

Our main goal in this note is to use a version of a lemma by Hanes and Huneke to provide characterizations of when certain one-dimensional reduced local rings are regular. This is of interest in view of the long-standing Berger's Conjecture (the ring is predicted to be regular if its universally finite differential module is torsion-free), which in fact we show to hold under suitable additional conditions, mostly toward the G-regular case of the conjecture. Furthermore, applying the same lemma to a Cohen-Macaulay local ring which is locally Gorenstein on the punctured spectrum but of arbitrary dimension, we notice a numerical characterization of when an ideal is strongly non-obstructed and of when a given semidualizing module is free. [ABSTRACT FROM AUTHOR]

The three famous problems concerning units, zero-divisors and idempotents in group rings of torsion-free groups, commonly attributed to Kaplansky, have been around for more than 60 years and still remain open in characteristic zero. In this article, we introduce the corresponding problems in the considerably more general context of arbitrary rings graded by torsion-free groups. For natural reasons, we will restrict our attention to rings without non-trivial homogeneous zero-divisors with respect to the given grading. We provide a partial solution to the extended problems by solving them for rings graded by unique product groups. We also show that the extended problems exhibit the same (potential) hierarchy as the classical problems for group rings. Furthermore, a ring which is graded by an arbitrary torsion-free group is shown to be indecomposable, and to have no non-trivial central zero-divisor and no non-homogeneous central unit. We also present generalizations of the classical group ring conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we report on a computation demonstrating that if A and B are nonempty subsets of a torsion-free group such that A B has no unique product, then | A | + | B | ≥ 1 6. Moreover, this bound is sharp, as there are examples where | A | = | B | = 8 , and in fact A = B for at least two such examples. More generally, when | A | is small, we find lower bounds on | B |. One consequence of this work is that any counterexample to Kaplansky's zero-divisor conjecture must be quite complicated, if it exists at all. Another advance is that we give new examples of torsion-free groups that are not unique product groups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of NN-torsion-free modules (respectively, NN-injective modules) as a generalization of nonnil-torsion-free modules (respectively, injective modules). We show that they behave in a way similar to the classical ones. As an application, we characterize ϕ-rings over which every NN-torsion-free module is NN-injective. [ABSTRACT FROM AUTHOR]

We generalize the "facial weak order" of a finite Coxeter group to a partial order on a set of intervals in a complete lattice. We apply our construction to the lattice of torsion classes of a finite-dimensional algebra and consider its restriction to intervals coming from stability conditions. We give two additional interpretations of the resulting "facial semistable order": one using cover relations, and one using Bongartz completions of 2-term presilting objects. For |$\tau $| -tilting finite algebras, this allows us to prove that the facial semistable order is a semidistributive lattice. We then show that, in any abelian length category, our new partial order can be partitioned into a set of completely semidistributive lattices, one of which is the original lattice of torsion classes. [ABSTRACT FROM AUTHOR]

In this paper, we study a Drinfeld module analogue of the Uniform Boundedness Conjecture on the torsion of abelian varieties. As a result, we prove the -primary Uniform Boundedness Conjecture for one-dimensional families of Drinfeld modules of arbitrary rank, which extends a result of Poonen. This result can be regarded as a Drinfeld module analogue of the Cadoret–Tamagawa's result on the p -primary Uniform Boundedness Conjecture for one-dimensional families of abelian varieties. [ABSTRACT FROM AUTHOR]

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials. [ABSTRACT FROM AUTHOR]

Elliptic curves are abelian varieties of dimension one; the two-dimensional analogues are abelian surfaces. In this work we present an algorithm to compute (2 n , 2 n) -isogenies between abelian surfaces defined over finite fields. These isogenies are the natural generalization of 2 n -isogenies of elliptic curves. The efficient computation of such isogeny chains gained a lot of attention as the runtime of the attacks on SIDH (Castryck–Decru, Maino–Martindale, Robert) depends on this computation. Different results deduced in the development of our algorithm are also interesting beyond these applications. For instance, we derive a formula for the evaluation of (2, 2)-isogenies. Given an element in Mumford coordinates, this formula outputs the (unreduced) Mumford coordinates of its image under the (2, 2)-isogeny. Furthermore, we study 4-torsion points on Jacobians of hyperelliptic curves and explain how to extract square roots of coefficients of 2-torsion points from these points. [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

For a quadratic field K without rationally defined complex multiplication, we prove that there exists of a prime p K depending only on K such that if d is a positive integer whose minimal prime divisor is greater than p K , then for any extension L / K of degree d and any elliptic curve E / K , we have E (L) tors = E (K) tors. By not assuming the GRH, this is a generalization of the results by Genao, and Gonález-Jiménez and Najman. [ABSTRACT FROM AUTHOR]

Let p be a prime. We prove that a positive solution to Efrat's Elementary Type Conjecture implies a positive solution to a strengthened version of Minač–Tân's Massey Vanishing Conjecture in the case of finitely generated maximal pro- p Galois groups whose pro- p cyclotomic character has torsion-free image. Consequently, the maximal pro- p Galois group of a field K containing a root of 1 of order p (and also − 1 if p = 2) satisfies the strong n -Massey vanishing property for every n > 2 (which is equivalent to the cup-defining n -Massey product property for every n > 2 , as defined by Minač–Tân) in several relevant cases. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring. An R-module M is said to be w-split if Ext R 1 (M , N) is a GV-torsion R-module for all R-modules N. It is known that every projective module is w-split, but the converse is not true in general. In this paper, we study the w-split dimension of a flat module. To do so, we introduce and study the so-called w-copure (resp., strongly w-copure) projective modules, which is in some way a generalization of the notion of copure (resp., strongly copure) projective modules. An R-module M is said to be w-copure projective (resp., strongly w-copure projective) if Ext R 1 (M , N) (resp., Ext R n (M , N) ) is a GV-torsion R-module for all flat R-modules N and any n ≥ 1 . [ABSTRACT FROM AUTHOR]

Let d ≥ 1 be an integer and let p be a rational prime. Recall that p is a torsion prime of degree d if there exists an elliptic curve E over a degree d number field K such that E has a K-rational point of order p. Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) have computed the torsion primes of degrees 4, 5, 6 and 7. We verify that the techniques used in Derickx et al. (Algebra Number Theory 17(2):267–308, 2023) can be extended to determine the torsion primes of degree 8. [ABSTRACT FROM AUTHOR]

The displacement field and governing equation completely depend on the cross-section shape in the torsion of nano-bars. Due to the torsion of structure and warping of its cross-section, axial strain is created. A review of the literature demonstrates that the effect of the normal strain in the torsional analysis of nanostructures with non-circular cross-sections is ignored for the sake of simplicity and causes a computational error. While in the torsion of short nano-bars due to the great warping of the cross-section the normal strain appears and it shouldn't be ignored. Therefore, for the first time, the effect of normal strain based on the torsion of Timoshenko-Gere theory is considered in this research. In this theory, the twist rate of non-circular sections in the axial direction is not considered constant, unlike studies in the literature. The governing equation of torsional vibration is extracted using the Hamilton principle, and the nonlocal strain gradient theory is used to show the size-dependent effects. The fundamental frequency is calculated for the arbitrary cross-section shape by using Galerkin's method. The effect of parameters such as size effects, the thickness of nano-bar, mode number, and dimensions changes of cross-section versus natural frequency of nano-bar for both the Timoshenko-Gere and typical theories are investigated. Results indicate that neglecting normal strain due to warping the cross-section causes a significant error in the short nano-bars, especially at higher mode numbers. Also, a comparison is made between the obtained natural frequencies and those of the results reported in the literature. [ABSTRACT FROM AUTHOR]

For any number field, we prove that there exists an elliptic curve defined over this field such that its Shafarevich-Tate group has a nontrivial 2-torsion subgroup. [ABSTRACT FROM AUTHOR]

We give an algorithm to explicitly determine all elements of the q-torsion (for q an odd prime) of the Brauer group of an elliptic curve over any base field of characteristic different from q, containing a primitive q-th root of unity. These elements of the Brauer group are given as tensor products of symbol algebras over the function field of the elliptic curve. We give sufficient conditions to determine if the Brauer classes that arise are trivial. Using our algorithm, we derive an upper bound on the symbol length of the prime torsion of Br(E)/Br(k). [ABSTRACT FROM AUTHOR]

For a smooth projective variety X over an algebraic number field k a conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map of X is a torsion group. In this article we consider a product X = C 1 × ⋯ × C d of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true for X. For a product X = C 1 × C 2 of two curves over Q with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map J 1 (Q) ⊗ J 2 (Q) → ε CH 0 (C 1 × C 2) is finite, where J i is the Jacobian variety of C i . Our constructions include many new examples of non-isogenous pairs of elliptic curves E 1 , E 2 with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products X = C 1 × ⋯ × C d for which the analogous map ε has finite image. [ABSTRACT FROM AUTHOR]

We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no "torsion-free version" of P. Kropholler's theorem, which characterises solvable groups of infinite rank via their metabelian subquotients. [ABSTRACT FROM AUTHOR]

Poincaré Gauge Theories are a class of Metric-Affine Gravity theories with a metric-compatible (i.e. Lorentz) connection and with an action quadratic in curvature and torsion. We perform an explicit one-loop calculation starting with a single term of each type and show that not only are all other terms generated, but also many others. In our particular model all terms containing torsion are redundant and can be eliminated by field redefinitions, but there remains a new term quadratic in curvature, making the model non-renormalizable. We discuss the likely behavior of more general theories of this type. [ABSTRACT FROM AUTHOR]

The coordinate ring O q (K n) of quantum affine space is the K -algebra presented by generators x 1 , ... , x n and relations x i x j = q i j x j x i for all i , j. We construct simple O q (K n) -modules in a more general setting where the parameters q i j lie in a torsion subgroup of K ∗ and show that analogous results hold as in the uniparameter case. [ABSTRACT FROM AUTHOR]

It is shown that there is a finitely generated metabelian group of finite torsion-free rank associated with each non-constant integer polynomial. It is shown how many structural properties of the group can be detected by inspecting the polynomial. [ABSTRACT FROM AUTHOR]

We prove that the $p^\infty$ -torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a 'flat Tate conjecture' for divisors. We also study other geometric Galois-invariant $p^\infty$ -torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$ -divisible. We explain how the existence of these $p$ -divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$. [ABSTRACT FROM AUTHOR]

We prove that every finitely generated, residually finite group G$G$ embeds into a finitely generated perfect branch group Γ$\Gamma$ such that many properties of G$G$ are preserved under this embedding. Among those are the properties of being torsion, being amenable and not containing a non‐abelian free group. As an application, we construct a finitely generated, non‐amenable torsion branch group. [ABSTRACT FROM AUTHOR]

We develop silting theory of a Noetherian algebra |$\Lambda $| over a commutative Noetherian ring |$R$|⁠. We study mutation theory of |$2$| -term silting complexes of |$\Lambda $|⁠ , and as a consequence, we see that mutation exists. As in the case of finite-dimensional algebras, functorially finite torsion classes of |$\Lambda $| bijectively correspond to silting |$\Lambda $| -modules, if |$R$| is complete local. We show a reduction theorem of |$2$| -term silting complexes of |$\Lambda $|⁠ , and by using this theorem, we study torsion classes of the module category of |$\Lambda $|⁠. When |$R$| has Krull dimension one, we describe the set of torsion classes of |$\Lambda $| explicitly by using the set of torsion classes of finite-dimensional algebras. [ABSTRACT FROM AUTHOR]

Cogging torque can affect the performance of permanent magnet (PM) homopolar inductor machines (HIMs). In order to find the reduction methods of the PM HIM cogging torque, it is necessary to investigate its production mechanism and analytical model. In this paper, the production mechanism of the PM HIM cogging torque is revealed from the perspective of air‐gap field modulation principle. It is found that the air‐gap permeance of PM HIMs can modulate their air‐gap magneto‐motive force (MMF) and then a large number of modulated air‐gap magnetic fields are generated. These magnetic fields with identical production condition but different rotation speeds or rotation directions can interact with each other, hence the production of the PM HIM cogging torque. By the combination of energy method and air‐gap field modulation principle, the cogging torque analytical model of PM HIMs is derived. Based on the obtained analytical model, the methods for reducing the PM HIM cogging torque are further analyzed. Finally, a 48‐slot/4‐pole (48S4P) PM HIM is designed and exampled. The air‐gap flux density of the 48S4P PM HIM is simulated by three‐dimensional (3‐D) finite element analysis (FEA). Based on the harmonic analysis for the simulated results, the correctness of the proposed cogging torque production mechanism is validated. In addition, a prototype of the 48S4P PM HIM is manufactured. The cogging torque in the prototype is obtained by the analytical model, 3‐D FEA and experiments, respectively. The simulation and measurement results verify the correctness of the cogging torque analytical model. © 2023 Institute of Electrical Engineer of Japan and Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of ℚ . In particular, we prove that every elliptic curve E over ℚ has the weak Hilbert property of Corvaja and Zannier both over the maximal abelian extension ℚ ab of ℚ , and over the field ℚ ⁢ (A tor) obtained by adjoining to ℚ all torsion points of some abelian variety A over ℚ . [ABSTRACT FROM AUTHOR]

Let R be a commutative domain with 1 and Q (≠ R) its field of quotients. In this note an R -module M is called w ∞ -Warfield cotorsion if M ∈ W C ∩ P w ∞ ⊥ , where W C denotes the class of all Warfield cotorsion R -modules and P w ∞ the class of all w ∞ -projective R -modules. It is shown that R is a PVMD if and only if all w -cotorsion R -modules are w ∞ -Warfield cotorsion, and that R is a Krull domain if and only if every w -Matlis cotorsion strong w -module over R is a w ∞ -Warfield cotorsion w -module. [ABSTRACT FROM AUTHOR]

The graphical theory of direct decompositions of torsion-free Abelian groups of a certain type is considered as the basis for the algorithmic construction of the set of their direct decompositions satisfying some specified conditions. [ABSTRACT FROM AUTHOR]

For a smooth projective curve defined over an algebraic number field k, we investigate the finiteness of the set of generalized Jacobians of associated with modules defined over such that a fixed divisor representing a class of finite order in the Jacobian J of provides the torsion class in the generalized Jacobian . Various results on the finiteness and infiniteness of the set of generalized Jacobians with the above property are obtained depending on the geometric conditions on the support of , as well as on the conditions on the field k. These results are applied to the problem of the periodicity of a continued fraction expansion constructed in the field of formal power series for special elements of the field of functions of the hyperelliptic curve . [ABSTRACT FROM AUTHOR]

We investigate the maximal finite length submodule of the Breuil–Kisin prismatic cohomology of a smooth proper formal scheme over a $p$ -adic ring of integers. This submodule governs pathology phenomena in integral $p$ -adic cohomology theories. Geometric applications include a control, in low degrees and mild ramifications, of (1) the discrepancy between two naturally associated Albanese varieties in characteristic $p$ , and (2) the kernel of the specialization map in $p$ -adic étale cohomology. As an arithmetic application, we study the boundary case of the theory due to Fontaine and Laffaille, Fontaine and Messing, and Kato. Also included is an interesting example, generalized from a construction in Bhatt, Morrow and Scholze's work, which illustrates some of our theoretical results being sharp, and negates a question of Breuil. [ABSTRACT FROM AUTHOR]

It is well known from the results of Ferrero and Kida [2,7] that the Z 2 -torsion part of the unramified abelian Iwasawa module X ∞ of any imaginary quadratic number field is trivial or cyclic of order 2. We will determine an infinite family of CM number fields, in which the Z 2 -torsion of the Iwasawa module X ∞ is of arbitrary large rank, giving also the exact value of the rank of X ∞ . [ABSTRACT FROM AUTHOR]

For any positive integer |$N$|⁠ , let |$J_0(N)$| be the Jacobian variety of the modular curve |$X_0(N)$| over |${\mathbb {Q}}$| and |${\mathcal {C}}_N$| its cuspidal subgroup. Let |$F_N$| denote the splitting field of |${\mathcal {C}}_N$|⁠ , which is the smallest number field whose absolute Galois group acts trivially on |${\mathcal {C}}_N$|⁠. Let |${\mathcal {J}}_N=J_0(N)(F_{N})_{\textrm {tor}}$| be the torsion subgroup of the group of |$F_N$| -rational points on |$J_0(N)$|⁠. We prove that |${\mathcal {J}}_N$| coincides with |${\mathcal {C}}_N$| outside |$6N[F_N:{\mathbb {Q}}]$|⁠. [ABSTRACT FROM AUTHOR]

We use recent advances in the local evaluation of Brauer elements to study the role played by odd torsion elements of the Brauer group in the arithmetic of diagonal quartic surfaces over arbitrary number fields. We show that over a local field if the order of the Brauer element is odd and coprime to the residue characteristic then the evaluation map it induces on the local points is constant. Over number fields we give a sufficient condition on the coefficients of the equation, which is mild and easy to check, under which the odd torsion does not obstruct weak approximation. We also note a systematic way to produce K3 surfaces over \mathbb {Q}_2 with good reduction and a non-trivial 2-torsion element of the Brauer group with Swan conductor zero. [ABSTRACT FROM AUTHOR]

We compute the Reidemeister-Turaev torsion of homology cylinders which takes values in the K_1-group of the I-adic completion of the group ring \mathbb {Q}\pi _1\Sigma _{g,1}, and prove that its reduction to \widehat {\mathbb {Q}\pi _1\Sigma _{g,1}}/\hat {I}^{d+1} is a finite-type invariant of degree d. We also show that the 1-loop part of the LMO homomorphism and the Enomoto-Satoh trace can be recovered from the leading term of our torsion. [ABSTRACT FROM AUTHOR]

For any commutative ring A we introduce a generalization of Snoetherian rings using a hereditary torsion theory σ instead of a multiplicatively closed subset S ⊆ A. It is proved that totally noetherian w.r.t. σ is a local property, and if A is a totally noetherian ring w.r.t σ, then σ is of finite type. [ABSTRACT FROM AUTHOR]

In the context of f(T, B) modified teleparallel gravity, we investigate the influence of torsion scalar T and boundary term B on the confinement of both the gauge vector and Kalb–Ramond fields. Both fields require a suitable coupling in five-dimensional braneworld scenarios to yield a normalizable zero mode. We propose a Stueckelberg-like geometrical coupling that non-minimally couples the fields to the torsion scalar and boundary term. To set up our braneworld models, we use the first-order formalism in which two kinds of superpotential are taken: sine-Gordon and ϕ 4 -deformed. The geometrical coupling is used to produce a localized zero mode. Moreover, we analyze the massive spectrum for both fields and obtain possible resonant massive modes. Furthermore, we do not find tachyonic modes leading to a consistent thick brane. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set \mathsf {tors} A of torsion classes over a finite-dimensional algebra A. We show that \mathsf {tors} A is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of \mathsf {tors} A. In particular, we give a representation-theoretical interpretation of the so-called forcing order , and we prove that \mathsf {tors} A is completely congruence uniform. When I is a two-sided ideal of A, \mathsf {tors} (A/I) is a lattice quotient of \mathsf {tors} A which is called an algebraic quotient , and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of \mathsf {tors} A that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras \Pi, for which \mathsf {tors} \Pi is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between \mathsf {tors} k Q and the Cambrian lattice when Q is a Dynkin quiver. We also prove that, in type A, the algebraic quotients of \mathsf {tors} \Pi are exactly its Hasse-regular lattice quotients. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and S a given multiplicative closed subset of R. In this paper, we introduce the new concept of S -torsion exact sequences (respectively, S -torsion commutative diagrams) as a generalization of exact sequences (respectively, commutative diagrams). As an application, they can be used to characterize two classes of modules that are generalizations of projective modules. [ABSTRACT FROM AUTHOR]

In this paper, we introduced and studied δ-small submodule over prime module. Two concepts are very important namely strongly prime submodule and completely prime submodule. Multiple results led to obtaining a δ-small submodule of a singular, divisible and Bezout module with R is local. Important terms that appeared in this article, together with some terms, produced the submodule that we were interested in. [ABSTRACT FROM AUTHOR]

We introduce an algorithm to compute the structure of the rational torsion subgroup of the Jacobian of a hyperelliptic curve of genus 3 over the rationals. We apply a Magma implementation of our algorithm to a database of curves with low discriminant due to Sutherland as well as a list of curves with small coefficients. In the process, we find several torsion structures not previously described in the literature. The algorithm is a generalisation of an algorithm for genus 2 due to Stoll, which we extend to abelian varieties satisfying certain conditions. The idea is to compute p-adic torsion lifts of points over finite fields using the Kummer variety and to check whether they are rational using heights. Both have been made explicit for Jacobians of hyperelliptic curves of genus 3 by Stoll. This article is partially based on the second-named author's Master thesis. [ABSTRACT FROM AUTHOR]

For a prime number p, we characterize the groups that may arise as torsion subgroups of an elliptic curve with complex multiplication defined over a number field of degree 2p. In particular, our work shows that a classification in the strongest sense is tied to determining whether there exist infinitely many Sophie Germain primes. [ABSTRACT FROM AUTHOR]

Unlike the classical Brauer group of a field, the Brauer-Grothendieck group of a singular scheme need not be torsion. We show that there exist integral normal projective surfaces over a large field of positive characteristic with non-torsion Brauer group. In contrast, we demonstrate that such examples cannot exist over the algebraic closure of a finite field. [ABSTRACT FROM AUTHOR]

In this note (R , m) denotes a complete regular local ring and B mostly denotes its absolute integral closure. The four objectives of this paper are the following: i) to determine the highest non-vanishing local cohomology of Ω B / R in equicharacteristic 0, ii) to establish a connection between each of Ω B / R and Ω B / V and pull-back of Ω A / V via a short exact sequence together with new observations on corresponding local cohomologies in mixed characteristic where V is the coefficient ring of R and A is its absolute integral closure, iii) to demonstrate that Ω B / R can be mapped onto a cohomologically Cohen-Macaulay module and iv) to study torsion-free property for Ω C / V and Ω C / k along with their respective completions where C is an integral domain and a module finite extension of R. In this connection an extension of Suzuki's theorem on normality of complete intersections to the formal set-up in all characteristics is accomplished. [ABSTRACT FROM AUTHOR]

We generalize Bloch's map on torsion cycles from algebraically closed fields to arbitrary fields. While Bloch's map over algebraically closed fields is injective for zero-cycles and for cycles of codimension at most two, we show that the generalization to arbitrary fields is only injective for cycles of codimension at most two but, in general, not for zero-cycles. Our result implies that Jannsen's cycle class map in integral ℓ-adic continuous étale cohomology is, in general, not injective on torsion zero-cycles over finitely generated fields. This answers a question of Scavia and Suzuki. [ABSTRACT FROM AUTHOR]

Let R !A be a homomorphism of associative rings, and let .F ;C/be a hereditary complete cotorsion pair in R-Mod. Let .FA;CA/be the cotorsion pair in A-Mod in which FA is the class of all left A-modules whose underlying R-modules belong to F. Assuming that the F -resolution dimension of every left R-module is finite and the class F is preserved by the coinduction functor HomR.A;􀀀/, we show that CA is the class of all direct summands of left A-modules finitely (co)filtered by A-modules coinduced from R-modules from C. Assuming that the class F is closed under countable products and preserved by the functor HomR.A;􀀀/,we prove that CA is the class of all direct summands of leftA-modules cofiltered by A-modules coinduced from R-modules from C, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from F have finite F -resolution dimension bounded by k, involves cofiltrations indexed by the ordinal ! Ck. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra (2010). In addition, we discuss the n-cotilting and n-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules. [ABSTRACT FROM AUTHOR]

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity that is doubly exponential (and this is so for almost all data). [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, and let M be an R -module. In this paper, we focus on the ideals of R that are annihilators of torsion elements of M. In analogy with definitions and results on zero-divisor and annihilator graphs of rings, we define the annihilator graph of a module. We investigate the structure, the diameter, and the girth of this graph and the closely related torsion graph of a module introduced by Ghalandarzadeh and Malakooti Rad. Given the right definitions, the properties of the modules are reflected in the graph theoretic properties of the graphs. We thus modify and extend results on zero divisors of rings to the much more general setting of modules and their torsion elements. In addition, we significantly strengthen the known results on the torsion graphs of modules. Some of the results will be refined further for the cases when M is a multiplication module or a reduced module. [ABSTRACT FROM AUTHOR]