Approximations and Endomorphism Algebras of Modules

This series aims to report new developments in mathematical research and teaching -quickly, informally and at a high level.The type of material considered for publication includes: 1.Preliminary drafts of original papers and monographs 2. Lectures on a new field, or presenting a new angle on a classical field 3. Seminar work-outs 4. Reports of meetings, provided they are a) of exceptional interest or b) devoted to a single topic.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In modules many ‘positive’ versions of model-theoretic concepts turn out to be equivalent to concepts known in classical module theory—by ‘positive’ we mean that instead of allowing arbitrary first-order formulas in the model-theoretic definitions only positive primitive formulas are taken into consideration. (This feature is due to Baur's quantifier elimination for modules, cf. [Pr], however we will not make explicit use of it here.) Often this allows one to combine model-theoretic methods with algebraic ones. One instance of this is the result proved in [Rot1] (see also [Rot2]) that the Mittag-Leffler modules are exactly the positively atomic modules. This paper is parallel to the one just mentioned in that it is proved here, Theorem 3.1, that the pure-projective modules are exactly the positively constructible modules. The following parallel facts from module theory and from model theory led us to this result: every pure-projective module is Mittag-Leffler and the converse is true for countable (in fact even countably generated) modules, cf. [RG]; every constructible model is atomic and the converse is true for countable models, cf. [Pi].

In this chapter we develop basic algebraic notions and facts to the extent needed for the applications in this book. Equally important are the examples for such structures from number theory. At the center of attention are basic facts from group theory, especially about cyclic groups, which are central in the analysis of the deterministic primality test. We discuss (commutative) rings (with 1), with the central example being ℤ m . Finally, we develop the basic facts about finite fields, in particular we establish that in finite fields the multiplicative group is cyclic.KeywordsGroup OperationPrime NumberCyclic GroupFinite FieldBinary OperationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The so called dense pairings were studied mainly by Radford in his work on coreflexive coalegbras over fields. They were generalized in a joint paper with Gómez-Torricillas and Lobillo to the so called rational pairings over a commutative ground ring R to study the interplay between the comodules of an R-coalgebra C and the modules of an R-algebra A that admits an R-algebra morphism κ : A → C*. Such pairings, satisfying the so called α-condition, were called in the author's dissertation measuring α-pairings and can be considered as the corner stone in his study of duality theorems for Hopf algebras over commutative rings. In this paper we lay the basis of the theory of rational modules of corings extending results on rational modules for coalgebras to the case of arbitrary ground rings. We apply these results mainly to categories of entwined modules (e.g., Doi-Koppinen modules, alternative Doi-Koppinen modules) generalizing results of Doi, Koppinen, Menini et al.

This paper studies two dual notions in module theory—namely, radicals and socles—from the standpoint of reverse mathematics. We first consider radicals of $\\mathbb{Z}$ -modules, where the radical of a $\\mathbb{Z}$ -module $M$ is defined as the intersection of $pM=\\{px:x\\in M\\}$ with $p$ taken from all primes. It shows that $\\mathrm{ACA}_{0}$ is equivalent to the existence of radicals of $\\mathbb{Z}$ -modules over $\\mathrm{RCA}_{0}$ . We then study socles of modules over commutative rings with identity. The socle of an $R$ -module $M$ is the largest semisimple submodule of $M$ . We show that the existence of socles of modules over a commutative ring with identity is equivalent to $\\mathrm{ACA}_{0}$ over $\\mathrm{RCA}_{0}$ . Vector spaces are semisimple modules over fields. In general, semisimple modules possess nice properties of vector spaces. Lastly, we study characterizations of semisimple modules over commutative rings using techniques of reverse mathematics.

Functional models in linear algebra

Resolutions of determinantal ideals: The submaximal minors

The workshop aimed to study discrete and Lie groups and their actions using measure theoretic methods and their asymptotic invariants, such as \ell^2 -invariants, the rank gradient, cost, torsion growth, entropy-type invariants and invariants coming from random walks and percolation theory. The participants came from a wide range of mathematics: asymptotic group theory, geometric group theory, ergodic theory, \ell^2 -theory, graph convergence, representation theory, probability theory, descriptive set theory and algebraic topology.

Quantum topology is the branch of mathematics that studies objects from the realm of low-dimensional topology, such as knots, surfaces or 3-manifolds, using algebraic structures that arose in the 1980s inspired by ideas from theoretical physics. This fascinating new area of topology combines various fields of mathematics, including monoidal category theory, Hopf algebras, representation theory, Lie algebras, topological quantum field theories, algebraic topology, etc., all revolving around the so-called quantum knot invariants.<br/><br/>This book focuses on two such invariants: the universal invariant subject to a ribbon Hopf algebra —that dominates the Reshetikhin-Turaev invariants coming from the representation theory of the Hopf algebra— and the Kontsevich invariant —which is universal among finte type invariants—. More precisely, Gaussian calculus techniques developed by Bar-Natan and van der Veen are exploited to study a family of knot polynomial invariants that determines the universal invariant subject to a specific algebra, and to investigate how the most elementary polynomial of this collection is closely related to the so-called 2-loop polynomial of the knot, an invariant that encodes a certain part of its Kontsevich invariant.<br/><br/>Although this book covers several topics of current research, it has been written entirely in a self-contained way, so that it can be accessible to a broad audience.

The cohomology of groups

This paper presents the way of integrating various mathematical structures into one system that will result in the improvement of academic knowledge and liberate more useful applications. The work establishes a single approach, which considers groups, rings, fields, modules, and lattices in an inclusive algebraic perspective by applying universal algebra and category theory. This can be seen as such that it is easier to map morphisms and representation theories that retain significant algebraic properties the same across structures. The proposed framework holds a great potential to improve such area as algebraic topology, cryptography, and quantum algebra through the provision of new ideas and simplification of the processes. The issues that may arise and how this unified theory can be extended in the future are also discussed.

The general subject of the conference was geometric group theory, that is the study of groups via suitable actions on spaces endowed with additional structures, for instance a distance, a measure (class), or both of them with natural compatibilities. The interactions with other mathematical fields such as algebraic topology (e.g. via homological methods), functional analysis (e.g. via the theory of operator algebras), probability theory (e.g. via random walks) or even representation theory and number theory, play an increasing role in the development of this part of group theory.

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211-3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes.

The computational complexity of module socles