RECENT TRENDS IN RINGS AND ALGEBRAS. 2013 separados/Abstracts... · Group rings of ﬁnite strongly...

RECENT TRENDS IN RINGS AND ALGEBRAS. 2013June 3th-7th, 2013. Murcia, Spain.

Scientific Committee

Eric Jespers (Vrije Universiteit Brussel)

Manuel Saorín (Universidad de Murcia)

Jan Trlifaj (Charles University in Prague)

Organizing Committee

Sergio Estrada

José Luis García

Pedro A. Guil

Pedro Nicolás

Ángel del Río

Manuel Ruiz Marín

Juan Jacobo Simón

Web: http://www.um.es/rtra2013/

e-mail: [email protected]

Abstracts

Contents

1 Plenary talks 8Gradings on associative algebras and polynomial identities asymptotics 8

Eli Aljadeff, Ofir David

Large tilting sheaves on exceptional curves 9Lidia Angeleri Hügel, Dirk Kussin

Atiyah’s Conjecture and von Neumann regular rings 10Pere Ara

Group cohomology over partial actions 11M. Dokuchaev, M. Khrypchenko

Ideal Approximation Theory 13Ivo Herzog

SL2-tilings and triangulations of the strip 14Thorsten Holm, Peter Jørgensen

Cotorsion pairs and Gorenstein homological algebra 15Mark Hovey

Recent advances on torsion subgroups of integral group rings 16Wolfgang Kimmerle

Cluster structures on quantum coordinate rings 17Bernard Leclerc

Good colimits, weak factorization systems and model categories 18J. Rosický

2 Special session 19Computational tools for rings and algebras 19

Alexander Konovalov

3 Communications 20Second modules over noncommutative rings 20

M. Alkan, S.Ceken, P.F. Smith

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The symmetry, period and Calabi-Yau dimension of finite dimensional mesh alge-bras 21

Estefanía Andreu Juan, Manuel Saorín

Gluing of derived equivalences by Grothendieck constructions 23Hideto Asashiba

On artinian rings with restricted class of injectivity domains 25Pınar Aydogdu, Bülent Saraç

Rational conjugacy of torsion units in integral group rings of non-solvable groups 26Andreas Bächle, Leo Margolis

Equivalences to being a rigid ring 27Muhittin Baser, Fatma Kaynarca, Tai Keun Kwak

Partial representations of Hopf algebras 29Eliezer Batista, Marcelo M.S. Alves, Stefaan Caenepeel, Joost Vercruysse

The congruence subgroup problem in group rings 31Mauricio Caicedo, Ángel del Río

A note on split Malcev algebras 32Antonio J. Calderón, Manuel Forero, José M. Sánchez-Delgado

On the feedback classification problem over commutative rings 33Miguel V. Carriegos

Drazin invertibility for rings and its generalizations 34Jianlong Chen

Gorenstein conditions over triangular matrix rings 35Manuel Cortés-Izurdiaga, Blas Torrecillas

Mitchell lemma and Gabriel-Popescu theorem 36Septimiu Crivei, Constantin Nastasescu, Laura Nastasescu

Generic objects for G-simple algebras 37Ofir David

Modules, bimodules and complexes with a rigid structure 38Gabriella D’Este

Rings over which every simple module has an injective cover 39Yılmaz Durgun, Engin Büyükasık

On the involutive Yang-Baxter property in finite groups 41Florian Eisele

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Gorenstein injective envelopes 42Edgar Enochs, Alina Iacob

Purity in categories of sheaves 43Sergio Estrada, Sinem Odabası

On semidualizing and tilting adjoint pairs. Applications 44J. R. García Rozas

Cleft extensions, integrals and crossed products in a weak setting 45Ramón González Rodríguez

Rings whose pure-injective modules are direct sums of lifting modules 47Pedro A. Guil Asensio, Derya Keskin Tütüncü

Gorenstein right derived functors of −⊗− with respect to semidualizing modules 48Jiangsheng Hu, Dongdong Zhang, Nanqing Ding

Group rings of finite strongly monomial groups: central units and primitive idem-potents 49

Eric Jespers, Gabriela Olteanu, Ángel del Río, Inneke Van Gelder

The co-dimension sequence of matrices 51Yakov Karasik

On coretractable modules 52Derya Keskin Tütüncü, Berke Kalebogaz

Absolute co-supplement and absolute co-coclosed modules 54Derya Keskin Tütüncü and Sultan Eylem Toksoy

Discontinuous actions on H2 ×H2 56Ann Kiefer, Ángel del Río, Eric Jespers

Varieties of algebras of polynomial growth 57Daniela La Mattina

Dynamical method in constructive algebra 58Henri Lombardi

On the heart associated with a torsion pair 59Francesca Mantese

Torsion pairs through Giraud and co-Giraud subcategories 60Francesco Mattiello, Riccardo Colpi, Luisa Fiorot

Cleanness of group algebras 62Paula Murgel Veloso, Álvaro Pérez Raposo

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Construction of minimal non-abelian left group codes 64Gabriela Olteanu, Inneke Van Gelder

Precovering and preenveloping ideals 65Furuzan Ozbek

Hearts of t-structures which are Grothendieck categories 66Carlos Parra, Manuel Saorín

Recollements of module categories 68Chrysostomos Psaroudakis, Jorge Vitória

The codimension sequence of finite dimension G-simple algebras 69Yuval Shpigelman

Deconstructibility, approximations, and locally free modules 70Alexander Slávik

T-structures on hereditary categories 71Maria José Souto Salorio

The role of big cotilting modules in derived equivalences 72Jan Stovicek

Derived dualities induced by a 1-cotilting bimodule 73Alberto Tonolo

A∞-algebras, E∞-algebras, and Bousfield Localization 74David White

Forms of path algebras 75Adi Wolf

On classical rings of quotients of duo rings 76Michal Ziembowski

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1 Plenary talks

Gradings on associative algebras and polynomial identitiesasymptotics

Eli Aljadeff, Ofir David

Regev and Seeman introduced regular G-gradings on associative algebra where G is a fi-nite abelian group. Roughly, it consists of a decomposition of an algebra A into homogeneoussubspaces which commute up to nonzero scalars. Then, Bahturin and Regev conjectured thatif the grading is regular and minimal (i.e. the grading induced by a proper homomorphicimage of G is not longer regular) then the order of the group G is an invariant of the algebraA. We prove the conjecture and show that in fact the order of the group G coincides with aninvariant which arises in asymptotics PI theory. In the lecture I’ll recall the two topics in-volved (namely graded algebras and polynomial identities) and explain how to connect themvia graded polynomial identities. This result can be extended to nonabelian groups. Jointwork with Ofir David

Department of MathematicsTechnionHaifa, Israel [email protected], [email protected]@gmail.com

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Recent Trends in Rings and Algebras, Murcia, 3–7 June 2013

Large tilting sheaves on exceptional curvesLidia Angeleri Hügel, Dirk Kussin

We consider the category QcohX of quasi-coherent sheaves on a weighted projective line,or more generally (when the ground field is not algebraically closed), on an exceptional curveX . The localizations of QcohX at sets of finite length objects are related to tilting sheaves onX that are not coherent. We give a complete classification of the non-coherent tilting sheaveson X when X is of domestic or of tubular type. Moreover, we discuss the connection withthe infinite dimensional tilting modules over the derived equivalent canonical algebra.

Keywords: quasi-coherent sheaves, weighted projective line, tilting theory

Mathematics Subject Classification 2010: 14H45,16G10,16E35

Dipartimento di Informatica - Settore MatematicaUniversità degli Studi di VeronaStrada Le Grazie 15 - Ca’ Vignal 2 37134 [email protected]

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Atiyah’s Conjecture and von Neumann regular ringsPere Ara,

Atiyah’s Conjecture on the L2-Betti numbers of manifolds can be formulated in purelyalgebraic terms, involving only the values of the canonical trace on certain projections onmatrices over the von Neumann algebra of the fundamental groupN (Γ) [2]. We will state anequivalent formulation, which uses a canonical rank function on the ∗-regular algebra U(Γ)of Γ, which is the classical ring of quotients of N (Γ).

A version of Atiyah’s Conjecture asserted that the values of this rank function on matricesover QΓ are contained in the subgroup of Q generated by the inverses of the orders of thefinite subgroups of Γ. This version of the conjecture was disproved in [1], the counterexamplebeing the so-called lamplighter group G. It remains an open problem to determine, given agroup Γ, what are the possible values of the canonical rank function on matrices over QΓ.This problem is closely related to the determination of the ∗-regular closure ofQΓ inside UΓ.

In joint work-in-progress with Ken Goodearl, we have determined the structure of the∗-regular closure of certain natural subalgebra of the group algebra KG, where G is thelamplighter group and K is any subfield of the complex numbers closed under conjugation.The set of values of the canonical rank function on matrices over this ∗-regular subalgebra ofU(G) turns out to be exactly the set of all non-negative rational numbers.

Keywords: Atiyah’s Conjecture, regular ring, rank function

Mathematics Subject Classification 2010: 16D70

References[1] R. I. GRIGORCHUK, P. LINNELL, T. SCHICK, A. ZUK, On a question of Atiyah, C. R.

Acad. Sci. Paris Sér. I Math. 331 663–668 (2000).

[2] W. LÜCK, L2-invariants: theory and applications to geometry and K-theory. Ergebnisseder Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Math-ematics, 44. Springer-Verlag, Berlin, 2002.

Department de MatemàtiquesUniversitat Autònoma de Barcelona08193 Bellaterra (Barcelona), [email protected]

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Group cohomology over partial actions∗

M. Dokuchaev, M. Khrypchenko

The concept of a partial group action was gradually worked out in the theory of operatoralgebras which permitted one to endow important classes of C∗-algebras, such as the ap-proximately finite dimensional algebras and the Cuntz-Krieger algebras, with the structure ofa crossed product by a partial action (see [1]). The new construction gave a method to studyC∗-algebras generated by partial isometries with respect to their internal structure, K-theoryand representations. The notions of a (continuous) twisted partial action of a locally compactgroup on a C∗-algebra and of the corresponding crossed products were given in [5]. Thenew concept permitted one to show that any second countable C∗-algebraic bundle†, whichsatisfies a certain regularity condition (automatically verified if the unit fiber algebra is sta-ble), is a C∗-crossed product of the unit fiber algebra by a continuous partial action of thebase group. A purely ring theoretic treatment of twisted partial group actions including ananalogue of the above mentioned fact was given in [2].

A twisted partial action of a group G on a commutative ring A falls into two parts: apartial action α of G on A and its twisting. With an additional assumption that α is unital(which means that the domains involved in α are generated by central idempotents), one canderive the concept of a 2-cocycle (the twisting) whose values belong to groups of invertibleelements of appropriate ideals of A. The concept of a partial 2-coboundary then followsfrom that of an equivalence of twisted partial actions introduced in [3]. Replacing A by acommutative multiplicative monoid, one comes to the definition of the second cohomologygroup H2(G,A). Thus instead of a usual G-module we deal with a partial G-module, whichis a commutative monoid A with a unital partial action α of G on A. Replacing A by anappropriate submonoid one may actually assume that A is inverse. The groups Hn(G,A)with arbitrary n are defined in a similar way. Next one asks how to obtain these groups using,say, projective resolutions. It turns out that the category of partial G-modules is not abeliansince some sets of morphisms may be empty. Fortunately, our cohomology can be relatedto Lausch’s cohomology of inverse semigroups [7] via the inverse monoid S(G) introducedby R. Exel in [6] to deal with partial actions and partial representations of G. From a partialaction of G one comes to an action of S(G) and then to an “almost” Lausch’s S(G)-module.The latter can be seen as a module in the sense of H. Lausch over an epimorphic imageof S(G). Thus our category is made up of abelian “components” which are categories ofLausch’s modules over epimorphic images of S(G). This way we are able to define freeobjects and free resolutions which lead to Hn(G,A). We also show that the partial Schurmultiplier pM(G), introduced in [4] with respect to partial projective representations, is aunion of 2-cohomology groups with values in some, in general non-trivial, partialG-modules.∗The first author was partially supported by CNPq of Brazil and the second author was supported by FAPESP of

Brazil.†A C∗-algebraic bundle is roughly a C∗-algebra “graded” by a locally compact group.

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References[1] M. DOKUCHAEV, Partial actions: a survey, Contemp. Math., 537 (2011), 173-184.

[2] M. DOKUCHAEV, R. EXEL, J. J. SIMÓN, Crossed products by twisted partial actionsand graded algebras, J. Algebra, 320 (2008), (8), 3278–3310.

[3] M. DOKUCHAEV, R. EXEL, J. J. SIMÓN, Globalization of twisted partial actions, Trans.Am. Math. Soc., 362 (2010), (8), 4137–4160.

[4] M. DOKUCHAEV, B. NOVIKOV, Partial projective representations and partial actions,J. Pure Appl. Algebra, 214 (2010), 251–268.

[5] R. EXEL, Twisted partial actions: a classification of regularC∗-algebraic bundles, Proc.London Math. Soc., 74 (1997), (3), 417–443.

[6] R. EXEL, Partial actions of groups and actions of inverse semigroups, Proc. Am. Math.Soc. 126 (1998), (12), 3481–3494.

[7] H. LAUSCH, Cohomology of inverse semigroups, J. Algebra, 35 (1975), 273–303.

Departmento de Matemática, Instituto de Matemática e Estatística,Universidade de São Paulo,Rua do Matão, 1010, CEP 05508–090, São Paulo, [email protected]@gmail.com

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Ideal Approximation TheoryIvo Herzog,

Ideal Approximation Theory (developed jointly with X. Fu,. P.A. Guil Asensio and B.J.Torrecillas) is the study of complete ideal cotorsion pairs (I, J) in an exact category (A;E).We will show that Salce’s Lemma holds for ideal cotorsion pairs, and explain how the mor-phisms of the ideal I may be characterized as the phantom morphisms with respect to thesubfunctor of Ext associated to (I, J). Three examples, in various contexts, of completeideal cotorsion pairs will be described, but we will concentrate on the complete ideal co-torsion pair (in the category R-Mod of left modules over a ring) that is cogenerated by thepure-injective modules. The phantom morphisms in this case are the morphisms f : M → Nof left R-modules that satisfy Tor(−, f) = 0. The filtration of R-Mod given by the projec-tive powers Iα of the phantom ideal yields for every ordinal α a complete ideal cotorsion pair(Iα, (Iα)⊥). A generalization of Eklof’s Lemma is used to prove this last result (joint withX.Fu). The existance of this filtration permits us to ask certain concrete questions regardingthe relationship between cotorsion and pure-injective left R-modules.

Department of MathematicsOhio State UniversityLima - USA [email protected]

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SL2-tilings and triangulations of the stripThorsten Holm1, Peter Jørgensen1

SL2-tilings were introduced by Assem, Reutenauer, and Smith in connection with friezesand their applications to cluster algebras.

An SL2-tiling is a bi-infinite matrix of positive integers such that each adjacent 2-submatrixhas determinant 1.

We construct a large class of new SL2-tilings which contains the previously known ones.More precisely, we show that there is a bijection between our class of SL2-tilings and certaincombinatorial objects, namely triangulations of the strip.

Keywords: Arc, cluster algebra, cluster category, Conway–Coxeter friese, Ptolemy for-mula, tiling, triangulation

Mathematics Subject Classification 2010: 05E15, 13F60

1Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Fakultät für Mathe-matik und Physik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover,[email protected]

2School of Mathematics and Statistics, Newcastle University, Newcastle upon TyneNE1 7RU, United Kingdom [email protected]

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Cotorsion pairs and Gorenstein homological algebraMark Hovey

A (possibly noncommutative) Gorenstein ring R has a stable module category; this is thequotient category of the Gorenstein projectives obtained by sending the projective modulesto zero. This stable module category comes from the linked cotorsion pairs (Gorensteinprojectives, modules of finite projective dimension) and (projectives, all modules). One canalso recover this triangulated stable module category from cotorsion pairs on the categoryof chain complexes of R-modules. In joint work with Daniel Bravo and Jim Gillespie, weshow that this chain complex approach works more generally, allowing us to define and studyGorenstein homological algebra over non-Gorenstein rings.

Department of Mathematics & Computer ScienceWesleyan UniversityMiddletown, CT 06459 [email protected]

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Recent advances on torsion subgroups of integral grouprings

Wolfgang Kimmerle,

The structure of the unit groupU(ZG) of an integral group ringZG reflects many proper-ties of the group basisG. In other words the question is which properties ofG are determinedby its integral group ring ZG. This fundamental problem has been in the focus of researchsince G.Higmans thesis 1940. The talk reports on recent advances concerning the followingquestions mainly in the situation when G is supposed to be finite.

• The Zassenhaus conjecture concerning conjugacy of units of finite order within ZG.

• The prime graph question. Does the prime graph of U(ZG) coincide with that one ofG.

• Which torsion subgroups of U(ZG) are isomorphic to subgroups of G ?

Keywords: Zassenhaus Conjecture, Prime graph question, Normalizer problem

Mathematics Subject Classification 2010: 20C10, 16U60

References[1] A.BÄCHLE AND W.KIMMERLE, On torsion subgroups of integral group rings of finite

groups. J. of Algebra Vol.326,Issue 1, 34 – 46 (2011)

[2] W.KIMMERLE, On the Prime Graph of the Unit Group of Integral Group Rings of FiniteGroups. Contemporary Mathematics] Vol.420, 215 – 228 (2006)

[3] W.KIMMERLE AND A.KONOVALOV, On the Prime Graph of the Unit Group of IntegralGroup Rings of Finite Groups II. Stuttgarter Mathematische Berichte 2012-018, 1 –12(2012).

IGT, Fachbereich MathematikUniversity of StuttgartD 70550 - [email protected]

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Cluster structures on quantum coordinate ringsBernard Leclerc,

Many coordinate rings of varieties arising in algebraic Lie theory have non commutativequantum deformations coming from quantum groups. In this talk I will explain how one canendow a large family of quantum coordinate rings with the structure of a quantum clusteralgebra, in the sense of Berenstein and Zelevinsky, in particular quantum coordinate rings ofmatrices. This yields a quite different description of these algebras, which might be usefulfor studying their ring-theoretical properties. This is a joint work with Christof Geiss and JanSchröer [1].

References[1] C. GEISS; B. LECLERC; J. SCHRÖER, Cluster structures on quantum coordinate rings,

Selecta Mathematica 19 (2013), 337-397.

L.M.N.O.Université de CaenF-14032 Caen cedex, [email protected]

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Good colimits, weak factorization systems and modelcategories

J. Rosický

Good colimits introduced by J. Lurie [1] generalize transfinite compositions and providean important tool for understanding cofibrant generation in locally presentable categories.We will explain the relation of good colimits to transfinite compositions further with the em-phasis on good colimits which are at the same time directed. In particular, we show howthey eliminate the use of large objects in the usual small object argument. Another appli-cation is that combinatorial categories, i.e., locally presentable categories equipped with acofibrantly generated weak factorization system, are closed under appropriate 2-limits (in-cluding pseudopullbacks). This result can be used in the study of deconstructible classes inGrothendieck categories – good colimits replace the generalized Hill lemma. The existenceof pseudopullbacks of combinatorial model categories is open.

The talk reports the joint work with M. Makkai and L. Vokrínek (see [3] and [2]).

Keywords: good colimit, cofibrant generation, model category

Mathematics Subject Classification 2010: 18C35, 55U35

References[1] J. LURIE, Higher Topos Theory. Princeton Univ. Press, 2009.

[2] M. MAKKAI; J. ROSICKÝ, Cellular categories. arXiv:1304.7572.

[3] M. MAKKAI; J. ROSICKÝ; L. VOKRÍNEK, On a fat small object argument.arXiv:1304.6974.

Department of Mathematics and StatisticsMasaryk UniversityKotlárská 2, 61137 Brno, Czech [email protected]

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2 Special session

Computational tools for rings and algebrasAlexander Konovalov

In my talk, I would like to give some examples of using mathematical software in re-search in rings and algebras, with a particular emphasis on the GAP system [1] and some ofits specialised packages. Not limiting the scope of the talk by existing GAP capabilities, Iwill show how GAP may be used as a platform for rapid prototyping and more robust im-plementation of new algebraic objects and operations. The talk will be interleaved with shortsoftware demonstrations.

Keywords: Computational algebra, Algorithms, Algebraic structures

Mathematics Subject Classification 2010: 17–08, 97N80

References[1] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.6.4; 2013,

(http://www.gap-system.org).

School of Computer Science &Centre for Interdisciplinary Research in Computational AlgebraUniversity of St AndrewsJack Cole Building, North HaughSt Andrews, Fife, KY16 [email protected]

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3 Communications

Second Modules over Noncommutative RingsM. Alkan1, S.Ceken1

P.F. Smith2

Let R be an arbitrary ring. A non-zero unital right R-module M is called a second mod-ule if M and all its non-zero homomorphic images have the same annihilator in R. It isproved that if R is a ring such that R/P is a left bounded left Goldie ring for every primeideal P of R then a right R-module M is a second module if and only if Q = annR(M)is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies theascending chain condition on two-sided ideals then every non-zero R-module has a non-zerohomomorphic image which is a second module. Every non-zero Artinian module containssecond submodules and there are only a finite number of maximal members in the collectionof second submodules. If R is a ring and M is a non-zero right R-module such that M con-tains a proper submodule N with M/N a second module and M has finite hollow dimensionn, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals Pi(1 ≤ i ≤ k) such that if L is a proper submodule of M with M/L a second module thenM/L has annihilator Pi for some 1 ≤ i ≤ k. Every second submodule of an Artinian moduleis a finite sum of hollow second submodules.

Keywords: Second Module; Attached Prime ideal; Hollow Dimension; Semilocal Rings.

Mathematics Subject Classification 2010: 16D10,16N60,16L30,16L60,16L99.

1Department of Mathematics,Akdeniz University,Antalya, [email protected], [email protected]

2Department of Mathematics,University of Glasgow,Glasgow, [email protected]

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The symmetry, period and Calabi-Yau dimension of finitedimensional mesh algebras

Estefanía Andreu Juan and Manuel Saorín

The stable module category Λmod of a finite dimensional self-injective algebra Λ is atriangulated category with the syzygy operator Ω as its suspension functor ([6]). Under thisassumption, the algebra Λ is said to be stably Calabi-Yau of dimension m (CY −dimensionfor short) if, and only if, m is the smallest natural number such that Ωm+1

Λ and η(−) arenaturally isomorphic as triangulated functors Λmod −→ Λmod where η is the Nakayamaautomorphism of Λ. In addition, as defined in [4], Λ is Calabi-Yau Frobenius of dimension rif, and only if, r is the smallest natural number such that Ωr+1

Λe (Λ) ∼= ηΛ1 as Λ-bimodules.These two concepts are related and we always have CY − dim(Λ) ≤ CY F − dim(Λ).Nevertheless, in general, it is not known if the equality holds.

The problem of deciding when Λ is Calabi-Yau Frobenius is part of a more general pro-blem, namely, to determine under which conditions ΩrΛe(Λ) is isomorphic to a twisted bimo-dule 1Λϕ, for some automorphism ϕ of Λ, which is then determined up to inner automor-phism. By a result of Green-Snashall-Solberg ([5]), this condition on a finite dimensionalalgebra forces it to be self-injective. When ϕ is the identity (or an inner automorphism), thealgebra Λ is called periodic and the problem of determining the self-injective algebras whichare periodic is widely open. Even when an algebra Λ is known to be periodic, it is usuallyhard to calculate explicitly its period, that is, the smallest of the integers r > 0 such thatΩrΛe(Λ) is isomorphic to Λ as Λ-bimodules.

Another interesting problem in the context of finite dimensional self-injective algebras isthat of determining when such an algebra is weakly symmetric or symmetric. An algebra issymmetric when D(Λ) is isomorphic to Λ as a Λ-bimodule. This is equivalent to saying thatthe Nakayama functor DHomΛ(−,Λ) ∼= D(Λ)⊗Λ− : Λ−Mod −→ Λ−Mod is naturallyisomorphic to the identity functor. The algebra is weakly symmetric when this functor justpreserves the iso-classes of simple modules.

We tackle the problems mentioned above for a special class of finite dimensional self-injective algebras, which has deserved a lot of attention in recent times. Following [3], if ∆is one of the Dynkin quivers Ar, Dr of En (n = 6, 7, 8), an m-fold mesh algebra of type ∆is the mesh algebra of the stable translation quiver Z∆/G, where G is a weakly admissiblegroup of automorphisms of Z∆. In fact, it is shown in [2] that they are periodic.

Within the class of finite dimensional mesh algebras, we identify those which are sym-metric and those which are stably Calabi-Yau. We also give, in combinatorial terms, explicitformulas for the Ω-period of any such algebras, and for the Calabi-Yau Frobenius and thestable Calabi-Yau dimensions, when they are defined.

References[1] E. ANDREU; M. SAORÍN, The symmetry, period and Calabi-Yau dimension of finite

dimensional mesh algebras, preprint http://arxiv.org/abs/1304.0586.

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[2] S. BRENNER; M.C.R. BUTLER; A.D. KING, Periodic algebras which are almostKoszul, Algebras and Representation Theory 5 (2002), 331-367.

[3] K. ERDMANN; A. SKOWRONSKI, Periodic algebras, ’Trends in Representation Theoryof Algebras and related topics’. EMS Congress Reports (2008).

[4] C. EU; T.SCHEDLER, Calabi-Yau Frobenius algebras, J. Algebra 321 (2009), 774-815.

[5] E.L. GREEN; N. SNASHALL; O. SOLBERG, The Hochschild cohomology ring of a self-injective algebra of finite representation type, Proceedings AMS 131 (2003), 3387-3393.

[6] D. HAPPEL , Triangulated categories in the representation theory of finite dimensionalalgebras, London Mathematical Society Lecture Note Series 119. Cambridge Universitypress, Cambridge (1988).

Department of MathematicsUniversidad de MurciaCampus de Espinardo, 30100 Murcia, [email protected]; [email protected]

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Gluing of derived equivalences by Grothendieckconstructions

Hideto Asashiba

We fix a commutative ring k and a small category I , and denote by k-Cat (k-Ab,k-Tri)the 2-category of small (resp. small abelian, small triangulated) k-categories. By I0 and I1 wedenote the object set and the morphism set of I . Here we use the terminology “2-category”in a strict sense.

Let G be a group, which we regard as a category with a single object ∗. Then a G-action X on a k-category C is nothing but a functor X : G → k-Cat with X(∗) = C.C-modules are defined as functors from Cop to the category Mod k of k-modules, and thecategory Mod C = k-Cat(Cop,Mod k) of C-modules also has a G-action ModX : G →k-Ab defined by (ModX)(∗) := Mod C and by ((ModX)(a))(M) := M X(a−1)op

for all a ∈ G and M ∈ Mod C. This G-action ModX is naturally extended to a G-actionD(ModX) on D(Mod C). On the other hand, the orbit category C/G is nothing but (thek-linear version of) the Grothendieck construction Gr(X) of X [3, Exposé VI §8]. Themost general objects to which the Grothendieck construction is defined are lax functors orcolax functors. Since skew group algebras are Grothendieck constructions of colax functors,we take colax setting here. We generalize the above story from a group G to an arbitrarycategory I , and consider derived equivalences of colax functors X : I → k-Cat.

We define k-ModCat to be the 2-subcategory of k-Ab the objects of which are Mod Cwith C in k-Cat, the 1-morphisms of which are the functors between objects having ex-act right adjoints, the 2-morphisms of which are the natural transformations between 1-morphisms. Then we can define pseudofunctors

Mod : k-Cat→ k-ModCat and D : k-ModCat→ k-Tri

by extending the correspondence sending a functor F : C → C′ in k-Cat (resp. a functorE : Mod C → Mod C′ with an exact right adjoint) to

-⊗C C′(?, F (-)) : Mod C → Mod C′ (resp. LE : D(Mod C)→ D(Mod C′)).

To define the “module category” and the “derived module category” of a colax functor weuse the following.

Theorem 1 ([2]). Let B,C and D be 2-categories. Then a pseudofunctor V : C → Dinduces a pseudofunctor

←−−−Colax(B, V ) :

←−−−Colax(B,C)→

←−−−Colax(B,D), X 7→ V X.

In the above←−−−Colax(B,C) is a suitably defined 2-category consisting of the colax functors

B→ C. Apply this to colax functors I X // k-Cat Mod // k-ModCat D // k-Tri, whereboth Mod and D are pseudofunctors, to have colax functors

ModX := Mod X and D(ModX) := D Mod X,

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which are the desired “module category” and “derived module category” of X . Now twocolax functors X and X ′ are said to be derived equivalent if their derived module colaxfunctorsD(ModX) andD(ModX ′) are equivalent in the 2-category

←−−−Colax(I, k-Tri). Then

we have the following, whose proof uses the correspondence on 1-morphisms in Theorem 1.

Theorem 2 ([2]). Assume that X or X ′ is k-flat (e.g., this holds if k is a field). If X and X ′

are derived equivalent, then so are their Grothendieck constructions Gr(X) and Gr(X ′).

Recall that Gr(X) is a category obtained from the categories X(i) (i ∈ I0) by “gluing”them together with the functors X(a) (a ∈ I1) (see [1] for quiver presentations of Gr(X)).By Theorem 2 we can “glue” derived equivalences F (i) (i ∈ I0) together to have a derivedequivalence between Gr(X) and Gr(X ′). We will present an example of gluing of derivedequivalences F (i) between Brauer tree algebras X(i) and X ′(i) (2 ≤ i ∈ N) with quivers

1 2 3 · · · i

α1((

α2((

α3 **αi−1

''

β1

hhβ2

hhβ3

hhβi−1

jj and 1 2 3 · · · iγ1 // γ2 // γ3 // γi−1 //

γi

ii .

References[1] H. ASASHIBA; M. KIMURA: Presentations of Grothendieck constructions, to appear in

Comm. in Algebra (preprint arXiv:1111.3845).

[2] H. ASASHIBA: Gluing derived equivalences together, Adv. Math. 235 (2013) 134–160.

[3] A. GROTHENDIECK: SGA 1, Springer-Verlag, Lecture Notes in Math., 224.

Department of MathematicsGraduate School of ScienceShizuoka University836 Ohya, Suruga-kuShizuoka, 422-8529, [email protected]

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On Artinian Rings with Restricted Class of InjectivityDomains

Pınar Aydogdu, Bülent Saraç

In a recent paper of Alahmadi, Alkan and López–Permouth, a ring R is defined to haveno (simple) middle class if the injectivity domain of any (simple) R–module is the smallestor largest possible. Er, López–Permouth and Sökmez use this idea of restricting the classof injectivity domains to classify rings, and give a partial characterization of rings with nomiddle class. In this work, we continue the study of the property of having no (simple)middle class. We give a structural description of right Artinian right nonsingular rings withno right middle class. We also give a characterization of right Artinian rings that are not SIto have no middle class, which gives rise to a full characterization of rings with no middleclass. Furthermore, we show that commutative rings with no middle class are those Artinianrings which decompose into a sum of a semisimple ring and a ring of composition lengthtwo. Also, Artinian rings with no simple middle class are characterized. We demonstrate ourresults with several examples.

References[1] A. N. ALAHMADI; M. ALKAN; S. R. LÓPEZ–PERMOUTH, Poor Modules: The oppo-

site of injectivity, Glasgow Math. J. 52A (2010) 7–17.

[2] N. ER; S. R. LÓPEZ–PERMOUTH; N. SÖKMEZ, Rings whose modules have maximal orminimal injectivity domains, J. Algebra 330(1) (2011) 404–417.

Department of MathematicsHacettepe UniversityBeytepe 06800 Ankara [email protected]@hacettepe.edu.tr

25


Rational conjugacy of torsion units in integral group ringsof non-solvable groupsAndreas Bächle1, Leo Margolis2

We introduce a new method to study rational conjugacy of torsion units in integral grouprings of finite groups. This method involves modular and integral representation theory and isespecially interesting when combined with the standard HeLP-method. We use our methodto prove the Zassenhaus Conjecture for the groups PSL(2, 19) and PSL(2, 23), i.e. everynormalized torsion unit in the integral group ring over such a group is rationally conjugate toa group element. (The proof for PSL(2, 23) uses only the known HeLP-method.) We thenapply our method to show, that there are no units of order 6 in the integral group rings of thegroups PGL(2, 9) and M10. This completes the proof of a theorem of W. Kimmerle and A.Konovalov [1] stating that the prime graph of a group G coincides with the prime graph ofthe corresponding group of normalized units of the integral group ring, if the order of G isdivisible by at most three primes.

References[1] W, KIMMERLE; A. KONOVALOV, On the Prime Graph of the Unit Group

of Integral Group Rings of Finite Groups II, preprint, www.mathematik.uni-stuttgart.de/preprints/downloads/2012/2012-018.pdf.

1Vakgroep WiskundeVrije Universiteit BrusselPleinlaan 2B-1050 Brussels, [email protected]

2Universität Stuttgart, IGTPfaffanwaldring 5770569 Stuttgart, [email protected]

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Rigidness and Extended Armendariz PropertyMuhittin Baser1, Fatma Kaynarca1, Tai Keun Kwak2

A ring endomorphism α of a ring R is named by J. Krempa a rigid endomorphism ifaα(a) = 0 implies a = 0 for a ∈ R, and Hong et al. named R an α-rigid ring if there existsa rigid endomorphism α. A ring R is defined to be Armendariz by Rege and Chhawchhariaif whenever the product of any two polynomials in R[x] over R is zero, then so is the productof any pair of coefficients form the polynomials. The Armendariz property of polynomialswas extended to the skew polynomials (i.e., α-Armendariz and α-skew Armendariz rings)by Hong et al. In this paper, we study the connection between α-rigid rings and extendedArmendariz rings, and so we get various conditions on the rings which are equivalent to thecondition of being an α-rigid ring. Several known results on extended Armendariz rings canbe obtained as corollaries of our results.

References[1] E.P. ARMENDARIZ, A note on extensions of Baer and P.P.-rings, J. Austral. Math. Soc.

18(1974) 470–473.

[2] W. CHEN AND W. TONG, A note on Skew Armendariz Rings, Comm.Alg. 33 (2005)1137–1140.

[3] W. CHEN AND W. TONG, On Skew Armendariz Rings and Rigid Rings, Houston J.Math. 33 (2)(2007) 341–353.

[4] C.Y. HONG, N.K. KIM AND T.K. KWAK, Ore extensions of Baer and p.p.-rings, J.Pure and Appl. Algebra 151 (3)(2000)215–226.

[5] C.Y. HONG, N.K. KIM AND T.K. KWAK, On Skew Armendariz Rings, Comm. Algebra31 (1)(2003) 103-122.

[6] C.Y. HONG, T.K. KWAK AND S.T. RIZVI, Extensions of generalized Armendarizrings, Comm. Algebra 31 (1)(2003) 103–122.

[7] A.A.M. KAMAL, Some remarks on Ore extension rings, Comm. Algebra 22(1994)3637–3667.

[8] N.K. KIM AND Y. LEE, Armendariz rings and reduced rings, J. Algebra 223 (2000)477–488.

[9] J. KREMPA, Some examples of reduced rings, Algebra Colloq.3 (4)(1996) 289–300.

[10] T.K. LEE AND T.L. WONG, On Armendariz Rings, Houston J. Math.3(2005) 583–593.

27

[11] T.K. LEE AND Y.Q. ZHOU, Armendariz and reduced rings, Comm. Algebra 32(2004)2287–2299.

[12] M.B. REGE AND S. CHHAWCHHARIA, Armendariz rings, Proc. Japan Acad. Ser. AMath. Sci. 73(1997) 14–17.

1Department of MathematicsAfyon Kocatepe UniversityAfyonkarahisar 03200, [email protected]@aku.edu.tr

2Department of MathematicsDaejin UniversityPocheon 487-711, [email protected]

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Partial representations of Hopf algebrasEliezer Batista1, Marcelo M.S. Alves2 Stefaan Caenepeel3 Joost Vercruysse4

The notion of a partial actions of groups was resultant from the effort to endow impor-tant classes of C∗-algebras generated by partial isometries with a structure of a more generalcrossed product. Partial actions of groups are in correspondence with partial representationsof groups on algebras. In fact, to each partial action of a group on an algebra one can as-sociate a partial representation of the same group and each partial representation of a groupgives rise to an universal algebra which is isomorphic to a partial skew group algebra relatedto a given partial action of the same group. In fact, there are deeper connections betweenpartial representations of groups and inverse semigroups [7]. The algebraic study of partialactions and partial representations of groups was initiated in [5] and [6], in particular, this lastreference gave important structural results concerning the partial group algebra and pointedout its connections with groupoids.

The generalization of the study of partial actions for the context of Hopf algebras was firstdone in [3]. This became a starting point for further investigation of partial actions and co-actions of Hopf algebras. One of the main results related to partial actions of Hopf algebras isthe globalization theorem, which states that every partial action of a Hopf algebra on a unitalalgebra can be viewed as a restriction of a global action to a unital ideal [2].

As in the case of groups, partial actions of Hopf algebras are closely related to their partialrepresentations. The concept of a partial representation of a Hopf algebra first appeared in[2], but there, only partial actions on right ideals were considered, therefore, partial represen-tations were presented in an assymetric way. As classical representations are closely relatedto modules, the natural questions to be answered with partial representations are, given aHopf algebra H , what should be a partial H module, and what properties has the category ofpartial H modules. For example, it is a classic result that the category of modules over a bial-gebra H has the structure of a monoidal category and the algebra objects in such a categoryare what we call H module algebras [4]. Therefore, it is natural to expect that, in the caseswhen the category of partial H modules has a monoidal structure, the algebra objects shouldcoincide with the partial H module algebras introduced in [3].

The aim of this talk is to present the basic properties of partial representations of Hopf al-gebras as well to introduce the category of partial modules over a Hopf algebra. Given a Hopfalgebra H , one can construct a universal algebra Hpar such that each partial representationof H can be factorized by an algebra morphism from Hpar. For the case of co-commutativeHopf algebras, we show that the algebra Hpar has the structure of a Hopf algebroid, thisenables one to endow the category of partial H-modules with a monoidal structure. Consid-ering the partial smash products of commutative Frobenuis separable objects in this monoidalcategory, we have a new class of examples of weak Hopf algebras. We present some exam-ples of categories of partial modules. In particular, partial modules over the dual of the groupalgebra of a finite abelian group G give rise to partial G gradings, which appeared first in [1].We show some interesting examples of partial G gradings.

29

References[1] E.R. ÁLVARES; M.M.S. ALVES; E. BATISTA, Partial Hopf module categories, Journal

of Pure and Applied Algebra 217 (2013) 1517-1534.

[2] M.M.S. ALVES; E. BATISTA, Enveloping Actions for Partial Hopf Actions, Comm. Al-gebra 38 (2010), 2872-2902.

[3] S. CAENEPEEL; K. JANSSEN, Partial (co)actions of Hopf algebras and partial Hopf-Galois theory, Comm. Algebra 36 (2008), 2923-2946.

[4] S. DASCALESCU; C. NASTASESCU; S. RAIANU, Hopf Algebras: An Introduction, Mar-cel Dekker Inc.(2001).

[5] M. DOKUCHAEV; R. EXEL, Associativity of Crossed Products by Partial Actions, En-veloping Actions and Partial Representations, Trans. Amer. Math. Soc. 357(5) (2005)1931-1952.

[6] M. DOKUCHAEV; R. EXEL; P. PICCIONE, Partial Representations and Partial GroupAlgebras, J. Algebra 226 (2000) 505-532.

[7] R. EXEL, Partial Actions of Groups and Actions of Semigroups, Proc. Am. Math. Soc.126 (12) (1998) 3481-3494.

1Departamento de Matemática,Universidade Federal de Santa CatarinaFlorianópolis, Sc, [email protected]

2Departamento de MatemáticaUniversidade Federal do ParanáCuritiba, PR, [email protected]

3Faculteit IngenieurswetenschappenVrije Universiteit BrusselBrussels, [email protected]

4Departément de MathematiquesUniversité Libre de BruxellesBrussels, [email protected]

30


The Congruence Subgroup Problem in Group RingsMauricio Caicedo, Ángel del Río

We present some results on the following problem: Classify the finite groups G such thatthe Congruence Subgroup Problem has a negative solution in the integer group ring of G buthas a positive solution in the integer group ring ofG/N for every non trivial normal subgroupN of G. For that we use the known results on the Congruence Subgroup Problem on speciallinear groups, Amitsur and Banieqbal classification of finite subgroup of the linear group ofdegree 2 over D for D a division algebra.

Departamento de Matemáticas,Universidad de Murcia,30100 Murcia, [email protected]@um.es

31


A note on split Malcev algebrasAntonio J. Calderón1, Manuel Forero1, José M. Sánchez-Delgado2

We study the structure of split Malcev algebras of arbitrary dimension over an alge-braically closed field of characteristic zero. We proved that any of such algebras is of theform M = U +

∑j Ij , with U a subspace of an abelian Malcev subalgebra, and any Ij a well

described ideal of M satisfying [Ij , Ik] = 0 if j 6= k.

Under certain conditions, the simplicity ofM is characterized, it is indicated thatM is thedirect sum of the family of its minimal ideals, each one being a simple split Malcev algebra.Finally, it is shown the expression M = L ⊕ (

⊕iMi), where L is a semisimple split Lie

algebra and each one Mi is a simple non-Lie Malcev algebra (seven dimensional over theircentroid).

References[1] CALDERÓN, A.J. On split Lie algebras with symmetric root systems, Proc. Indian. Acad.

Sci, Math. Sci. 118, 351–356. (2008)

[2] ELDUQUE, A. On semisimple Malcev algebras, Proc. Amer. Math. Soc. 107, no. 1, 73–82. (1989)

[3] KUZ’MIN, E.N. Malcev algebras and their representations, Algebra and Logic 7, 233–244. (1968)

[4] MALCEV, A.I. Analytic loops, Mat. Sb (N.S.) (36) 78, no. 3, 569–575. (1955)

[5] SAGLE, A.A. Malcev algebras, Trans. Amer. Math. Soc. 101, 426–458. (1961)

[6] STUMME, N. The structure of locally finite split Lie algebras, J. Algebra. 220, 664–693.(1999)

1Departamento de MatemáticasUniversity of Cádiz11510 Puerto Real, Cádiz, [email protected]@hotmail.com

2Departmento de Álgebra, Geometría y TopologíaUniversity of Málaga29080 Campus de Teatinos, Málaga, [email protected]

32


On the feedback classification problem over commutativerings

Miguel V. Carriegos,

The study of feedback classification of linear control systems is motivated by its applica-tions [1], [5] and goes back to the seminal works of Kalman, Casti, Brunovsky [2], and others.The study of constant control systems may be performed just by using linear algebra of realor complex vector spaces. The feedback classification of linear control systems over finitedimensional vector spaces is solved by Kalman’s Decomposition together with BrunovskyTheorem: The set of feedback classes of control systems over a n-dimensional vector spaceis in bijective correspondence with the set of partitions of integer n.

The talk deals with the feedback classification problem for linear systems over commu-tative rings. Main classification result is recently stated [3]: Feedback classes over a finitelygenerated projective R-module X are given by all possible decompositions of X on the form

X ∼= Z1 ⊕ (Z2)2 ⊕ · · · ⊕ (Zn)n

Thus the feedback classification problem over R is determined by combinatorics on themonoid P(R) of isomorphism classes of finitely generated projective R-modules.

References[1] J.W. BREWER; J.W. BUNCE; F.S. VANVLECK, Linear dynamical systems over com-

mutative rings, Marcel Dekker, 1986.

[2] P.A. BRUNOVSKY, A classification of linear control systems, Kybernetika 3, 1970.

[3] M.V. CARRIEGOS, Enumeration of classes of linear systems via equations and via par-titions in an ordered abelian monoid, Linear Algebra and Its Applications 438, 2013.

[4] J.A. HERMIDA, Linear algebra over commutative rings,Handbook of Algebra III, Else-vier, 2003.

[5] A.A. MAILYBAEV, Uncontrollability for linear autonomous multi-input dynamical sys-tems depending on parameters, SIAM Journal of Control and Optimization 42(4), 2003.

Departamento de MatemáticasUniversidad de LeónCampus de Vegazana, 24071 Leó[email protected]

33


Drazin invertibility for rings and its generalizationsJianlong Chen1

Joint work with J. Cui, Y. Liao, Z. Wang, H. Zhu, and G. Zhuang

The notion of Drazin inverse in rings and semigroups was introduced by Drazin in 1958,which attracted widely concern in ring theory because a Drazin invertible element coincideswith a strongly π-regular element. As a generalization of Drazin inverses, some authorsintroduced concepts of generalized Drazin inverses in Banach algebras and rings.

In this talk, we obtain some results on (generalized) Drazin invertibility of differenceand product of elements in a ring. In addition, we define a class of new generalized Drazininverses, so called pseudo Drazin inverses, and properties of Drazin inverses are extended topseudo Drazin inverses.

1. Over a ring, we study Drazin invertibility of difference and product of elements undersome conditions. In particular, some equivalents on Drazin invertibility of differences andproduct of two idempotents are established. Moreover, under the (weakly) commutative con-dition, we investigate additive properties of Drazin inverse of elements, and present necessaryand sufficient conditions for Drazin invertibility of the sum or difference of two elements.

2. Additive and multiplicative property of generalized Drazin invertibility of elements ina ring are studied. In particular, Cline’s formula and Jacobson’s lemma for the generalizedDrazin inverse are obtained.

3. Motivated by strongly π-regular elements and quasipolar elements, we introduce theconcept of pseudopolar elements. This concept can be used exactly to define a pseudo Drazininverse in associative rings and Banach algebras. Some properties of pseudo Drazin inversesare obtained in associative rings and Banach algebras. We also connect pseudopolar ringswith strongly π-regular rings, semiregular rings, uniquely strongly clean rings and uniquelybleached local rings.

1Department of Mathematics,Southeast University,Nanjing 210096,ChinaEmail:[email protected]

34


GORENSTEIN CONDITIONS OVER TRIANGULARMATRIX RINGS

Manuel Cortés-Izurdiaga, Blas Torrecillas

A ring is left Gorenstein if the classes of left modules with finite projective dimensionand with finite injective dimension coincide and the injective and projective left finitisticdimensions are finite. Let A and B be rings and U a (B,A)-bimodule such that BU hasfinite projective dimension and UA has finite flat dimension. In the talk we shall characterize

when the ring T =(A 0U B

)is left Gorenstein and, over such rings, when a left T -

module is Gorenstein projective or Gorenstein injective. We shall give some applications ofthese results: we shall characterize when T is left CM-free and we shall study when everyGorenstein projective left T -modules is a direct sum of λ-generated modules for some infinitecardinal number λ.

Departament of Mathematics.University of Almería.E-04071, Almería, [email protected]

35


Mitchell Lemma and Gabriel-Popescu TheoremSeptimiu Crivei1, Constantin Nastasescu2, Laura Nastasescu3

We give a generalization of the Mitchell Lemma, and we show that it is a key lemma thatcan be used in order to deduce in a unified easier way several important results, including theUlmer Theorem and the generalized Gabriel-Popescu Theorem. Using the Mitchell Lemmaand the theory of one-sided exact categories, we generalize the Gabriel-Popescu Theoremfrom Grothendieck categories to certain (non-abelian) subcategories of an AB5 category. Asan application we give a new perspective on some results related to Tilting Theory.

References[1] S. BAZZONI; S. CRIVEI, One-sided exact categories, J. Pure Appl. Algebra 217 (2013),

377–391.

[2] S. CRIVEI; C. NASTASESCU, The Gabriel-Popescu Theorem revisited, preprint 2013.

[3] S. CRIVEI; C. NASTASESCU; L. NASTASESCU, A generalization of the MitchellLemma: The Ulmer Theorem and the Gabriel-Popescu Theorem revisited, J. Pure Appl.Algebra 216 (2012), 2126–2129.

1Faculty of Mathematics and Computer Science“Babes-Bolyai" UniversityStr. Mihail Kogalniceanu 1, 400084 Cluj-Napoca, [email protected]

2“Simion Stoilow” Institute of MathematicsRomanian AcademyP.O. Box 1-764, 014700 Bucharest, [email protected]

3Faculty of Mathematics and Computer ScienceUniversity of BucharestStr. Academiei 14, 010014 Bucharest, [email protected]

36


Generic objects for G-simple algebrasOfir David

Let G be an abelian group, F a field of characteristic zero, and let A be a the matrixalgebra Mn(F ) with some G-grading. Denote by C the class of all G-graded algebras Bsuch that Z(B) = K is a field extension of F and B ⊗K Kalg ∼= A⊗F Kalg = Mn(Kalg)where Kalg is the algebraic closure of K. In particular we get that any algebra in C is acentral simple algebra.

A representing object for C is an algebra B ∈ C such that every B′ ∈ C is a specializationof B. If C has a representing object B, one can show that there are properties such that ifsatisfied by B then they are also satisfied by all the elements in C. For example, if C is theclass of central simple algebra and the representing object is aG-crossed product, then all thealgebras in C are G-crossed product with the same group G. In particular there is a strongconnection between the rationality condition of the fraction field of B and the class C.

In this talk we will show how to construct such representing objects using the polynomialidentities of A. More precisely, letting F 〈XG〉 be the free algebra over noncommuting in-determinates XG = xg,i | g ∈ G, i ∈ N, and letting IdG(A) be the ideal in F 〈XG〉 bethe ideal of graded polynomial identities of A, our representing object will be the relativelyfree algebra F 〈XG〉/IdG(A) after some central localization. An important ingredient in theproof is the following theorem: If A,B are G-simple algebras where G is abelian, then thereis a graded embedding A →G B if and only if there is an inclusion IdG(B) ⊆ IdG(A).

Department of MathematicsTechnion - Israel Institute of TechnologyHaifa, 32000 - [email protected]

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Modules, bimodules and complexes with a rigid structureGabriella D’Este

I will present some results on tilting-type objects, either modules, bimodules or com-plexes, often with the property that the morphisms between indecomposable modules arevector spaces of dimension at most one. As we shall, the combinatorial nature of these ob-jects often plays an important role in quite different situations. Indeed, this may happen bydealing with global and classical tilting modules and/or bimodules, in the sense of Brennerand Butler [1] , for instance with the tilting bimodule constructed by Happel and Ringel atthe end of [4]. However, the strategy of looking at the underlying vector spaces and linearmaps seems to be useful also by dealing with partial tilting complexes T •, in the sense ofRickard [5] ), and with indecomposable right bounded complexes X• (of projective moduleswith morphisms up to homotopy), orthogonal to T • with all their shifts.

References[1] BRENNER S., BUTLER C.M., Generalizations of the Bernstein-Gelfand-Ponomarev re-

flection functors, Springer LMN 832 (1980), 103-169.

[2] D’ESTE, G., Looking for right bounded complexes, Groups and Model Theory: A Con-ference in Honor of Rüdiger Göbel’s 70th Birthday, CONM 576 AMS (2012), 41-56.

[3] D’ESTE G., A personal survey on recent and less recent results on Tilting Theory, inpreparation.

[4] HAPPEL D., RINGEL C. M., Tilted algebras, Trans. Amer. Math. Soc. 274 (1982) ,399-443.

[5] RICKARD J., Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989),436-456.

Department of MathematicsUniversity of MilanoVia Saldini 50, 20133 Milano, [email protected]

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Rings over which every simple module has an injective coverYılmaz Durgun, Engin Büyükasık

Throughout this note, R is a ring with an identity element and all modules are unitalR-modules. MR denotes a right R-module. For an R-module M , the character moduleHomZ(M,Q/Z) is denoted by M+, the dual module HomR(M,R) is denoted by M∗, andδM : M →M∗∗ stands for the evaluation map. M is said to be torsionless (resp. reflexive) ifδM is a monomorphism (resp. an isomorphism). ByN ≤M , we mean thatN is a submoduleof M .

Given a right R−module N and M ≤ N , M is said to be s-pure in N if for any simpleleft R-module S, the induced homomorphism M ⊗ S → N ⊗ S is monomorphism. Similarto well known notion of absolutely pure (or FP-injective) module, a right R−module M issaid to be absolutely s-pure if it is s-pure in every extension of it. The notion of absolutelys-pure module was introduced by I. Crivei in [5]. Recently, s-pure submodules have beenstudied further in [1].

A right R−module M is said to be simple-projective if for any simple right R-moduleS, every homomorphism S → M factors through a finitely generated free right R-module.The notion of simple-projective module introduced in [6], in order to investigate when everysimple module have a projective envelope. Simple-projective modules and a generalizationof these modules have been studied by Parra and Rada in [7].

A submodule N of a right R-module M is said to be neat in M if for any simple rightR-module S, every homomorphism f : S → M/N can be lifted to a homomorphism g :S → M . Neat submodules recently studied in [4], where Fuchs characterize the integraldomains over which s-pure and neat submodules concide. Similar to flat modules, a rightR−module N is said to be neat-flat if for any epimorphism M → N , the induced mapHom(S,M) → Hom(S,N) is surjective for any simple right R-module S, that is, everyshort exact sequence ending with M is a neat-exact sequence.

Neat-flat modules are studied in [2], where it is shown that the notions of simple-projectiveand neat-flat modules coincide. The importance of absolutely s-pure and neat-flat modulesis the fact that they are homological objects of some certain proper classes (in the sense of[3]) induced by simple R-modules. This fact allows one to investigate the injectivity andprojectivity of the covers and envelopes of simple modules by using absolutely s-pure andneat-flat modules, respectively.

In this note, we study the properties of absolutely s-pure and of neat-flat modules. Wealso establish connections between absolutely s-pure and neat-flat modules. For a right N -ring i.e. the rings whose simple right modules are finitely presented, we prove that a leftR-module M is absolutely s-pure if and only if Ext1R(Tr(S),M) = 0 for each simple rightR-module S; a right R-module M is neat-flat if and only if M+ is absolutely s-pure. Fora commutative N -ring, we prove that, every absolutely s-pure left R-module is injective ifand only if R noetherian and every neat-flat R-module is flat. In particular, a domain R isDedekind if and only if every absolutely s-pure R-module is injective.

We obtain a characterization of right Kasch rings as follows. R is a right Kasch ring ifand only if the injective hull of every simple right R-module is neat-flat if and only if for

39

every free left R-module F , F+ is neat-flat. As a consequence, R is a right Kasch and aright

∑-CS ring if and only if R is a right QF ring. For a right N -ring, we show that, every

simple right R-module is torsionless if and only if RR is left absolutely s-pure if and only ifR is a right Kasch ring; R is a right

∑-CS ring if and only if every pure injective neat-flat

right R-module is projective if and only if every absolutely s-pure left R-module is injectiveand R is right perfect.

Finally, we study on enveloping and covering properties of absolutely s-pure and neat-flatmodules. For a right N -ring R, we show that every quotient of any injective left R-moduleis absolutely s-pure if and only if every left R-module has a monic absolutely s-pure cover ifand only ifR is a right PS ring; R is a right Kasch ring if and only if every leftR-module hasan epic absolutely s-pure cover. For commutative rings, we show that every simple modulehas a monic injective cover if and only if every module has a monic absolutely s−pure coverif and only if a simple R-module S is either injective or Hom(E,S) = 0 for every injectiveR-module E.

References[1] E. Büyükasık and Y. Durgun, Cofinitely coclosed submodules, submitted (2013).

[2] E. Büyükasık and Y. Durgun, Neat-coprojective modules, submitted (2013).

[3] D.A.Buchsbaum, A note on homology in categories, Ann. of Math. (2) 69 (1959), 66–74.

[4] L. Fuchs, Neat submodules over integral domains, Period. Math. Hungar. 64 (2012), no.2, 131–143.

[5] I.Crivei, s-pure submodules, Int. J. Math. Math. Sci. 4 (2005), 491–497.

[6] L. Mao, Modules characterized by their simple submodules, Taiwanese J. Math. 15(2011), no. 5, 2337–2349.

[7] R. Parra and J. Rada, Projective envelopes of finitely generated modules, Algebra Colloq.18 (2011), no. Special Issue No.1, 801–806.

Department of MathematicsIzmir Institute of TechnologyGülbahçeköyü, 35430, Urla/Izmir, [email protected]@iyte.edu.tr

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On the Involutive Yang-Baxter Property in Finite GroupsFlorian Eisele

A finite group G is called involutive Yang-Baxter (IYB for short) if there is some ZG-module M which admits a bijective 1-cocycle χ : G −→ M . This property can also becharacterized in terms of the existence of a particular one-sided ideal contained in the aug-mentation ideal of the group ring ZG. It is an open problem to characterize those finite groupswhich are IYB. It has long been known that such a group has to be solvable, and there is todate no known example of a solvable group which isn’t IYB. So it might well be that all ofthem are. In this talk I will report on some results of ongoing research, both theoretical andcomputational, which provide evidence for the conjecture that all solvable groups are IYB.

Department of MathematicsVrije Universiteit BrusselPleinlaan 21050 [email protected]

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Gorenstein injective envelopesEdgar Enochs1, Alina Iacob2

We consider commutative noetherian rings. We prove that if the character modules ofGorenstein injective modules are Gorenstein flat then the class of Gorenstein injective mod-ules is enveloping. In particular this shows the existence of the Gorenstein injective envelopesover rings with dualizing complexes.

1Department of MathematicsUniversity of KentuckyPostal address: 719 Patterson Office Tower, Lexington KY [email protected]

2Department of Mathematical SciencesGeorgia Southern UniversityPostal address: 65 Georgia Ave. Room 3008, Statesboro, GA [email protected]

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Purity in categories of sheavesSergio Estrada1 Sinem Odabası2,

The notion of purity has been extensively used in relative homological algebra in modulecategory. And there are several equivalent notions of purity depending on the monoidal orthe finitely accessible structure of R-Mod (R any ring with identity). But this is no longerthe situation if we extend the definitions to an arbitrary category. This is the case of themonoidal category Qco(X) of quasi-coherent sheaves over a scheme X . This is a finitelyaccessible category in most practical cases and it behaves similarly to a module category inmany aspects. In this talk, we shall consider several definitions of purity on Qco(X) comingfrom module category, and investigate the interlacing among them. Some applications on theexistence of pure injective envelopes will be also discussed.

1Departamento de Matemática AplicadaUniversidad de Murcia30100 Campus de Espinardo, Murcia, [email protected]

2Departamento de MatemáticasUniversidad de Murcia30100 Campus de Espinardo, Murcia, [email protected]

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On semidualizing and tilting adjoint pairs. ApplicationsJ. R. García Rozas

The purpose of this talk is to introduce the concept of right and left semidualizing adjointpair of functors and study their main properties. These concepts generalize semidualizingmodules and it allows to consider semidualizing comodules, graded modules etc. We alsointroduce tilting adjoint pair of functors as a particular kind of semidualizing ones. We showgeneralized tilting theorem in this general setting and give some applications to tilting theoryin the category of comodules over a coalgebra.

Departamento de MatemáticasUniversidad de Almería04120 Almerí[email protected]

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Cleft extensions, integrals and crossed products in a weaksetting∗

Ramón González Rodríguez,

Keywords: Sweedler Cohomology, cleft extension, weak crossed product, weak Hopfalgebra

Mathematics Subject Classification 2010: 18D10, 16T05.

The theory of crossed products for Hopf algebras arise as a generalization of the classicalsmash products and by the results obtained by Doi and Takeuchi in [6] we know that everycleft extension B → A induces a crossed product B]σH where σ : H ⊗ H → B is asuitable convolution invertible morphism (a normal 2-cocycle). Also, the reverse result holdsbecause Blattner and Montgomery proved in [4] that, if B]σH is a crossed product, theextension B → B]σH is cleft. Moreover, in [5] Doi showed that there exists a bijectionbetween the isomorphism classes of H-cleft extensions B of A and equivalence classes ofcrossed systems for H over A with a fixed action. If H is cocommutative the equivalenceis described by H2

ϕZ(A)(H,Z(A)), the second group of Sweedler cohomology [7] , where

Z(A) is the center of A.The aim of this talk is to extend the preceding results to the cocommutative weak Hopf

algebra setting. To do it, we introduce the notion of H-cleft extension for a weak Hopfalgebra H and we prove that this kind of extensions are examples of weak cleft extensionsas the ones defined in [1]. Also, under cocommutative conditions, we can assume that theassociated cleaving morphism is a total integral and it is possible to identify the set of crossedsystems in a weak setting as the set of weak crossed products induced by a weak left actionand a convolution invertible twisted normal 2-cocycle. Then, as a consequence, we obtaina result that assures the following: If (A, ρA) be a right H-comodule algebra, there existsa bijective correspondence between the equivalence classes of H-cleft extensions AH →A and the equivalence classes of crossed systems for H over AH where AH denotes thesubalgebra of coinvariants in the weak setting. Finally, using the second cohomology groupof the cohomology of algebras over weak Hopf algebras that we have developed in [2], weprove the weak version of the result obtained by Doi about the characterization of equivalenceclasses of crossed systems using Sweedler cohomology.

The results that will be presented are part of a joint work with J.N. Alonso and J.M.Fernández (see [2] and [3]).∗Research supported by Ministerio de Ciencia e Innovación: MTM2010-15634

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References[1] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, A.B. Rodríguez

Raposo, Weak C-cleft extensions, weak entwining structures and weak Hopf algebras, J.of Algebra, 284, 2005, 679-704.

[2] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, Cohomology ofalgebras over weak Hopf algebras, math.QA, arXiv:1206.3850, 2012.

[3] J.N. Alonso Álvarez, J.M. Fernández Vilaboa, R. González Rodríguez, Integrals andcrossed products over weak Hopf algebras, math.QA, arXiv:1207.5363, 2012

[4] R.J. Blattner, S. Montgomery, Crossed products and Galois extensions of Hopf algebras,Pacific J. of Math., 137, 1989, 37-54.

[5] Y. Doi, Equivalent crossed products for a Hopf algebra, Comm. in Algebra, 17, 1989,3053-3085.

[6] Y. Doi, T. Takeuchi, Cleft comodule algebras for a bialgebra, Comm. in Algebra, 14,1986, 801-817.

[7] M. Sweedler, Cohomology of algebras over Hopf algebras, Trans. of the Amer. Math.Soc., 133, 1968, 205-239.

Departamento de Matemática Aplicada IIUniversidade de VigoE.T. telecomunicación, Campus Universitario Lagoas-Marcosende, 36310, Vigo,Pontevedra, [email protected]

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RINGS WHOSE PURE-INJECTIVE RIGHT MODULESARE DIRECT SUMS OF LIFTING MODULES

Pedro A. Guil Asensio1, Derya Keskin Tütüncü2

Let R be an associative ring with identity. A module M is called lifting if for everysubmodule N of M there exists a direct sum decomposition M = M1 ⊕M2 with M1 ≤ Nand N ∩ M2 small in M2. A module M is called extending if every submodule of M isessential in a direct summand of M . Recall that a ring R is said to be of finite type whenthere exists a finite set of indecomposable right R-modules such that any other right moduleis isomorphic to a direct sum of copies of them. In this case, R is left and right artinianand there also exists a finite set of indecomposable left R-modules such that any other leftmodule is isomorphic to a direct sum of copies of them. And a ring R is of right local typewhen every indecomposable right R-module is local. In this talk we announce the following:

Theorem: The following are equivalent for a ring R:

1. Every right R-module is a direct sum of lifting modules.

2. Every pure-injective right R-module is a direct sum of lifting modules.

3. R is of finite type and right local type.

Corollary: The following are equivalent for a ring R:

1. R is both sided serial and artinian.

2. Every left and every right R-module is a direct sum of lifting modules.

3. Every left and every right pure-injective R-module is a direct sum of lifting modules.

4. Every left and every right R-module is a direct sum of extending modules.

1Departamento de MathematicasUniversidad de Murcia30100 Espinardo, Murcia, [email protected]

2Department of MathematicsHacettepe University06800 Beytepe, Ankara, [email protected]

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Gorenstein right derived functors of −⊗− with respect tosemidualizing modules

Jiangsheng Hu1, Dongdong Zhang2, Nanqing Ding1

We study Gorenstein right derived functors of−⊗− with respect to semidual-izing modules. As applications, some new criteria for a semidualizing moduleto be dualizing are given provided that R is a ring with a dualizing complex.

Keywords: Gorenstein homological dimensions; derived functors; semidu-alizing modules; dualizing modules.

Mathematics Subject Classification 2010: 18G10; 18G15; 18G20.

1Department of MathematicsNanjing UniversityNanjing 210093, [email protected]; [email protected]

2Department of MathematicsZhejiang Normal UniversityJinhua 321004, [email protected]

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Group rings of finite strongly monomial groups: centralunits and primitive idempotents

Eric Jespers1, Gabriela Olteanu2, Ángel del Río3, Inneke Van Gelder1

For a finite group G we denote by U(ZG) the unit group of the integral group ring ZG.Its group of central units is denoted by Z(U(ZG)).

In [1] we have proved that the group generated by the so-called generalized Bass unitscontains a subgroup of finite index in Z(U(ZG)) for any arbitrary finite strongly monomialgroup G. No multiplicatively independent set of generators for such a subgroup was ob-tained. We call such a set a virtual basis. However, we obtained an explicit description ofa virtual basis of Z(U(ZG)) when G is a finite abelian-by-supersolvable group (and thus astrongly monomial group) such that every cyclic subgroup of order not a divisor of 4 or 6is subnormal in G. We present an extension of these results on the construction of a virtualbasis of Z(U(ZG)) to a class of finite strongly monomial groups containing the metacyclicgroups G = Cqm oCpn with p and q different primes and Cpn acting faithfully on Cqm . Ourproof makes use of strong Shoda pairs and the description of the Wedderburn decompositionof QG obtained by Olivieri, del Río and Simón in [2].

In [3] a complete set of matrix units (and in particular, of orthogonal primitive idem-potents) of each simple component in the rational group algebra QG is described for finitenilpotent groups G. As an application one obtains a factorization of a subgroup of finite in-dex of U(ZG) into a product of three nilpotent groups, and one explicitly constructs finitelymany generators for each of these factors. We present a description of a complete set ofmatrix units for a class of finite strongly monomial groups containing the finite metacyclicgroups Cqm o Cpn with Cpn acting faithfully on Cqm . For the latter groups we obtain asan application of these results (and the earlier results on central units) again an explicit con-struction of finitely many generators of three nilpotent subgroups that together generate asubgroup of finite index in U(ZG).

References[1] E. JESPERS; Á DEL RÍO; G. OLTEANU; I. VAN GELDER, Central units of integral

group rings, Proc. Amer. Math. Soc. (in press).

[2] A. OLIVIERI; Á. DEL RÍO; J. J. SIMÓN, On monomial characters and central idempo-tents of rational group algebras, Communications in Algebra 32 (2004), no. 4, 1531 -1550.

[3] E. JESPERS; G. OLTEANU; Á. DEL RÍO, Rational group algebras of finite groups: fromidempotents to units of integral group rings, Algebr. Represent. Theory 15 (2012), no. 2,359 - 377.

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[4] E. JESPERS; G. OLTEANU; Á. DEL RÍO; I. VAN GELDER, Group ringsof finite strongly monomial groups: central units and primitive idempotents,http://arxiv.org/abs/1209.1269.

1Department of MathematicsVrije Universiteit BrusselPleinlaan 2, 1050 Brussels, [email protected], [email protected]

2Department of Statistics-Forecasts-MathematicsBabes-Bolyai UniversityStr. T. Mihali 58-60, 400591 Cluj-Napoca, [email protected]

3Departamento de MatemáticasUniversidad de Murcia30100 Murcia, [email protected]

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THE CO-DIMENSION SEQUENCE OF MATRICESYakov Karasik,

One of the most fruitful invariants in PI theory (introduced by Regev in the 70’s) is the co-dimension sequence attached to a T−ideal of identities and its asymptotic behavior. An im-portant example is the co-dimension sequence of the mm matrices over algebraically closedfield F . Very little is known regarding generators of its T−ideal of identities (exceptm = 2),however its co-dimension sequence asymptotics was calculated by Regev using results ofFormanek, Procesi and Razmyslov about Invariants theory and Hilbert series.

In this talk I will survey the main ideas which led to the above result. Furthermore, Iwill show how to implement them in order to calculate asymptotically the (G−graded) co-dimension sequence of matrix algebras with elementary G−grading. It is worth mentioningthat crossed products are a key example of algebras with elementary G−grading.

This work was done in collaboration with Yuval Shpigelman.

Department of MathematicsTechnion Israel Institute of TechnologyHaifa [email protected]

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On Coretractable ModulesDerya Keskin Tütüncü, Berke Kalebogaz

Let R be a ring and let M be an R-module. The module M is called coretractable ifHom(M/N,M) is nonzero for all proper submodule N of M . Recall that a module M is aKasch module if every simple module in σ[M ] can be embedded in M . Amini, Ershad andSharif proved that RR is a Kasch module if and only if RR is a coretractable module (see[2]). In this work we generalize this result as follows:

Theorem: Let MR be a finitely generated self-generator module. Then M is coretractable ifand only if it is Kasch.

Then we study rings whose all right modules are coretractable.Theorem: For a ring R the following are equivalent:

1. Every right R-module is coretractable.

2. R is right perfect and every right R-module is small coretractable.

3. R is right perfect and for every rightR-moduleM , there exists a nonzero f ∈ Hom(P,M)such that P/Kerf is a small coretractable module, where P is the projective cover ofM .

4. R is right perfect and for all right R-modules M and X , Hom(X,M) = 0 if and onlyif Hom(P,M) = 0, where P is the projective cover of X .

5. All torsion theories on R are cohereditary.

We also prove that being coretractable is a Morita invariant property.We will call M mono-coretractable if for every submodule N of M there is a monomor-

phism from M/N to M . We show that coretractable modules are a proper generalizationof mono-coretractable modules. And we investigate some properties of mono-coretractablemodules.

References[1] F. W. ANDERSON; K. R. FULLER, Rings and Categories of Modules, Springer-Verlag,

New York, 1974.

[2] B. AMINI; M. ERSHAD; H. SHARIF, Coretractable Modules, J. Aust. Math. Soc. 86(2009), 289-304.

[3] T. ALBU; R. WISBAUER, Kasch Modules,in Advances in Ring Theory (eds. S.K. Jainand S.T. Rizvi), Birkhäuser, Basel, 1-16, 1997.

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[4] Y. BABA; K. OSHIRO, Classical Artinian Rings and Related Topics, World ScientificPublishing Co. Pte. Ltd., 2009.

[5] J. CLARK; C. LOMP; N. VANAJA; R. WISBAUER, Lifting Modules, Frontiers in Math-ematics, Birkhäuser, 2006.

[6] A. GHORBANI, Co-Epi-Retractable Modules and Co-Pri Rings, Comm Algebra 38(2010), 3589-3596.

[7] T. Y. LAM, Lectures on Modules and Rings, Graduate Texts in Mathematics, Vol:139,Springer - Verlag, 1998.

[8] W. K. NICHOLSON; M.F. YOUSIF, Quasi-Frobenius Rings, Cambridge UniversityPress, Cambridge, 2003.

[9] P. F. SMITH, Modules with many homomorphisms, J. Pure and Appl. Algebra 197 (2005),305-321.

[10] Y. TALEBI; N. VANAJA, The Torsion Theory Cogenerated by M -Small Modules,Comm. Algebra 30 (3) (2002), 1449-1460.

Department of MathematicsHacettepe University06800 Beytepe, Ankara, [email protected]@hacettepe.edu.tr

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ABSOLUTE CO-SUPPLEMENT AND ABSOLUTECO-COCLOSED MODULES

Derya Keskin Tütüncü1, Sultan Eylem Toksoy2

A moduleM is called an absolute co-coclosed (absolute co-supplement) module if when-ever M ∼= T/X the submodule X of T is a coclosed (supplement) submodule of T . Ringsfor which all modules are absolute co-coclosed (absolute co-supplement) are precisely de-termined. We also investigate the rings whose (finitely generated) absolute co-supplementmodules are projective. We show that a commutative domain R is a Dedekind domain if andonly if every submodule of an absolute co-supplement R-module is absolute co-supplement.We also prove that every extension of an absolute co-coclosed module by an absolute co-coclosed module is absolute co-coclosed.

Mathematics Subject Classification 2010: Primary 16D10, Secondary 06C05.

Keywords: absolute co-supplement (co-coclosed) module, supplement (coclosed) submod-ule.

References[1] F. W. ANDERSON; K. R. FULLER, Rings and Categories of Modules, New York:

Springer, 1974.

[2] J. CLARK; D. KESKIN TÜTÜNCÜ; R. TRIBAK, Supplement submodules of injectivemodules, Comm. Algebra. 39 (2011) 4390–4402.

[3] J. CLARK; C. LOMP; N. VANAJA; R. WISBAUER, Lifting Modules, Supplements andProjectivity in Module Theory, Frontiers in Math. Boston: Birkhäuser, 2006.

[4] N. V. DUNG; D. V. HYUNH; P. F. SMITH; R. WISBAUER, Extending Modules,, PitmanResearch Notes in Mathematics Series UK: Longman Scientific and Technical, 1994.

[5] S. E. ERDOGAN (TOKSOY), Absolutely Supplement and Absolutely ComplementModules M. Sc. dissertation, Izmir Institute of Technology, 2004. Electronic copy:http://library.iyte.edu.tr/tezler/master/matematik/T000339.pdf.

[6] L. GANESAN; N. VANAJA, Modules for which every submodule has a unique coclo-sure, Comm. Algebra. 30 (5) (2002) 2355–2377.

[7] A. I. GENERALOV, The ω-cohigh purity in a categories of modules, Math. Notes 33(5-6) 402–408. Translated from Russian from Mat. Zametki 33 (5) (1983) 758–796.

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[8] V. N. GERASIMOV; I. I. SAKHAEV, A counter example to two hypotheses on projectiveand flat modules, Sib.Mat. Zh. 25 (6) (1984) 31–35. Translated from Russian from Sib.Math. J. 24 (1984) 855–859.

[9] K. R. GOODEARL, Ring Theory: Nonsingular Rings and Modules, New York andBasel: Marcel Dekker Inc., 1976.

[10] T. Y. LAM, Lectures on Modules and Rings, New York, Berlin, Heidelberg: Springer,1999.

[11] E. MERMUT, Homological Approach to Complements and Supplements, Ph.D. disser-tation, Dokuz Eylül University, 2004.

[12] A. P. MISHINA; L. A. SKORNYAKOV, Abelian groups and modules, American Mathe-matica Society Translations Series 2 107 (1976). Translated from Russian from Abelevygruppy i moduli, Izdat. Nauka (1969).

[13] A. MOHAMMED; F. L. SANDOMIERSKI, Complements in projective modules, J. Alge-bra 127 (1989) 206–217.

[14] S. M. MOHAMED; B. J. MÜLLER, Continuous and Discrete Modules, London Math.Soc. Lecture Notes Series 147, Cambridge, 1990.

[15] E. G. SKLYARENKO, Relative homological algebra in categories of modules, RussianMath. Surveys 33 (3) (1978), 97–137.

[16] Y. TALEBI; N. VANAJA, The torsion theory cogenerated by M -small modules, Comm.Algebra 30 (3) (2002) 1449–1460.

[17] R. B. JR. WARFIELD, Serial rings and finitely presented modules, J. Algebra 37 (1975)187–222.

[18] H. ZÖSCHINGER, Projektive Moduln mit endlich erzeugtem Radikalfaktormodul, Math.Ann. 255 (1981) 199–206.

[19] H. ZÖSCHINGER, Schwach-injective moduln, Periodica Mathematica Hungarica 52 (2)(2006) 105–128.

1Department of MathematicsHacettepe University06800 Beytepe, Ankara, [email protected]

2Department of MathematicsIzmir Institute of Technology35430 Urla, Izmir, [email protected]

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Discontinuous actions on H2 ×H2

Ann Kiefer1, Ángel del Río2, Eric Jespers1

The main goal is the investigation on the unit group of an order O in a rational groupring QG of a finite group G. In particular we are interested in the unit group of ZG. Formany finite groups G a specific finite set B of generators of a subgroup of finite index inU(ZG) has been given. The only groups G excluded in this result are those for which theWedderburn decomposition of the rational group algebra QG has a simple component that iseither a non-commutative division algebra different from a totally definite quaternion algebraor a 2× 2 matrix ring M2(D), where D is either Q, a quadratic imaginary extension of Q ora totally definite rational division algebraH(a, b,Q).

In some of these cases, up to commensurability, the unit group acts discontinuously on adirect poduct of hyperbolic 2- or 3-spaces. The aim is to generalize the theorem of Poincaréon fundamental domains and group presentations to these cases. For the moment we havedone this for the Hilbert Modular Group, which acts on H2 × H2, in joint work with A. delRío, E. Jespers.

1Departments of MathematicsVrije Universiteit BrusselPleinlaan, 2 1050 [email protected]@vub.ac.be

2Department of MathematicsUniversidad de MurciaMurcia 30100, Españ[email protected]

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Varieties of algebras of polynomial growthDaniela La Mattina,

Let A be an associative algebra over a field F of characteristic zero and let cn(A), n =1, 2, . . . , be its sequence of codimensions. It is well known that the sequence of codimensionsof a PI-algebra either grows exponentially or is polynomially bounded.

In this note we are interested in the case of polynomial growth. For this case a celebratedtheorem of Kemer characterizes the algebras whose sequence of codimensions is polynomi-ally bounded as follows. Let G be the infinite dimensional Grassmann algebra over F andlet UT2 be the algebra of 2 × 2 upper triangular matrices. Then cn(A), n = 1, 2, . . ., ispolynomially bounded if and only if G,UT2 /∈ var(A), where var(A) denotes the variety ofalgebras generated by A.

In the setting of G-graded algebras, where G is a finite group, the sequence of gradedcodimensions is polynomially bounded if and only if vargr(A) does not contain a finite listof G-graded algebras. The list consists of group algebras of groups of order a prime number,the infinite dimensional Grassmann algebra and the algebra of 2×2 upper triangular matriceswith suitable gradings. Such algebras generate the only varieties of G-graded algebras of al-most polynomial growth, i.e., varieties of exponential growth such that any proper subvarietygrows polynomially.

In this note, we completely classify all subvarieties of the G-graded varieties of almostpolynomial growth by giving a complete list of finite dimensional G-graded algebras gener-ating them.

Dipartimento di Matematica e InformaticaUniversità di [email protected]

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Dynamical method in constructive algebraHenri Lombardi

Hilbert’s program was an attempt to save Cantorian mathematics through the use of for-malism. From this point of view, too abstract objects (not having a clear semantics) arereplaced by their formal descriptions. Their hypothetical existence is replaced by the non-contradiction of their formal theory. However, Hilbert’s program in its original finitist formwas ruined by the incompleteness theorems of Godel.

Poincaré’s program “Never lose sight of the fact that every proposition concerning in-finity must be the translation, the precise statement of propositions concerning the finite.” iseven more ambitious than Hilbert’s program.

Bishop’s book (1967) “Foundations of Constructive Analysis” is a kind of realization ofthe Poincaré’s program.

But also a realization of Hilbert’s program, when one replaces finitist requirements byless stringent requirements, constructive ones.

The Computer Algebra software D5 was invented in order to deal with the algebraicclosure of an explicit field, even when the algebraic closure is impossible to construct.

This leads to the general idea to replace too abstract objects (without actual existence) ofCantorian mathematics by finite approximations, and thereby obtain constructive proofs forthe existence of concrete objects when they appear i, the conclusion of a theorem.

We shall illustrate these ideas with some examples.

References[1] E. BISHOP. Foundations of Constructive Analysis. McGraw Hill, (1967).

[2] T. COQUAND; H. LOMBARDI, A logical approach to abstract algebra.. (survey) Math.Struct. in Comput. Science 16 (2006), 885–900.

[3] J. DELLA DORA; C. DICRESCENZO; D. DUVAL. About a new method for computing inalgebraic number fields In Caviness B.F. (Ed.) EUROCAL ’85. Lecture Notes in Com-puter Science 204, 289–290. Springer (1985).

[4] G. DÍAZ-TOCA; H. LOMBARDI. Dynamic Galois Theory. Journal of Symbolic Compu-tation. 45, (2010), 1316–1329.

[5] H. LOMBARDI; C. QUITTÉ. Algèbre Commutative, Méthodes Constructives. Calvage &Mounet, (2011).

Départment de MathématiquesUniversité de Franche-ComtéF-25030 Besançon [email protected]

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On the heart associated with a torsion pairFrancesca Mantese

Given an associative ring R and a torsion pair (T ,F) in the category of right R-modules,the heart H(T ,F) of the t-structure associated with (T ,F) is an abelian subcategory of thebounded derived category Db(R). In this talk we analyze categorical properties ofH(T ,F),showing how they are related with relevant properties of (T ,F). In particular we deal withthe problem of whenH(T ,F) is equivalent to a category of modules.

References[1] R.COLPI; F. MANTESE; A. TONOLO, Cotorsion pairs, torsion pairs, and Σ-pure-

injective cotilting modules, Journal of Pure and Applied Algebra 214 (2010) 519-525.

[2] R. COLPI; F. MANTESE; A. TONOLO, When the heart of a faithful torsion pair is amodule category, Journal of Pure and Applied Algebra 215 (2011) 2923-2936.

[3] F. MANTESE; A. TONOLO, On the heart associated with a torsion pair, Topology andits Applications, 159 (2012) 2483-2489.

Department of Computer ScienceUniversity of VeronaStrada le Grazie 15, I-37134 Verona, [email protected]

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Torsion pairs through Giraud and co-Giraud subcategoriesFrancesco Mattiello, Riccardo Colpi, Luisa Fiorot

An abelian category with a distinguished Giraud subcategory is the data (D, C, l, i) oftwo abelian categories D and C and two adjoint functors

Ci

// Dloo

(with l left adjoint of i) such that l is exact and i fully faithful. The kernel of the functor l,i.e., the full subcategory S of D whose objects S in S satisfy l(S) ∼= 0, defines a localizingSerre class on D and, moreover, the category C is equivalent to the quotient category D/S.The dual notion is that of a distinguished co-Giraud subcategory (D, C, j, r). Giraud andco-Giraud subcategories arise in many fields of algebra. The relevance of this notions comesfrom the Yoneda embedding of the category of left R-modules as a Giraud (resp. co-Giraud)subcategory of the Functor category - i.e., the category of contravariant (resp. covariant)functors from the finitely presented R-modules to the Abelian groups (see [5]) - as well asfrom the celebrated Gabriel-Popescu theorem, which asserts that (up to equivalence) everyGrothendieck category is a Giraud subcategory of a category of leftR-modules, for a suitablychosen ring R (see [3]).

The concept of torsion pair was formally introduced by Dickson [2] in 1966, and has thenbecome a powerful tool for the study of localization in various contexts.

In this work we discuss an interesting interplay between torsion pairs and Giraud (resp.co-Giraud) subcategories. More precisely, given any abelian category D and a Giraud (resp.co-Giraud) subcategory C ofD, we prove that any torsion pair onD gives rise to a torsion pairon the subcategory C and, conversely, any torsion pair on C can be extended to a torsion pairon D. This technique eventually provides a one-to-one correspondence between the class ofall torsion pairs (T ,F) on C and the torsion pairs (X ,Y) on D satisfying natural properties.

There is an interesting connection between a torsion pair in an abelian category and t-structures (in the sense of [1]) in its derived category, also related to tilting theory. An abeliancategory with a given torsion pair gives rise to a new abelian category, called the heart of thetorsion pair, in fact endowed with a distinguished torsion pair, see the work [4] for furtherdetails. The main application of our work concerns hearts of torsion pairs. First of all, ifD is an abelian category with a distinguished Giraud subcategory C such that i admits aright derived functor Ri, and (X ,Y) is a torsion pair on D satisfying a certain "compatibilitycondition" then, denoted by HC and HD the associated hearts, there exists a distinguishedGiraud subcategory (HD,HC , lH, iH). Conversely, under suitable assumptions, any Giraud(resp. co-Giraud) subcategory of the heart of a torsion pair is itself the heart of a (non-trivial)torsion pair.

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References[1] A. A. BEILINSON, J. BERNSTEIN, AND P. DELIGNE, Faisceaux pervers, Analysis and

topology on singular spaces, I (Luminy, 1981), Astérisque, vol. 100, Soc. Math. France,Paris, 1982, pp. 5-171.

[2] S. E. DICKSON, A torsion theory for Abelian Categories, Trans. Amer. Math. Soc. 121(1966), 223-235.

[3] P. GABRIEL, AND N. POPESCU, Caractérisation des catégories abéliennes avec généra-teurs et limites inductives exactes, C. R. Acad. Sci. Paris, 258 (1964), 4188-4190.

[4] D. HAPPEL, I. REITEN, AND S. SMALØ, Tilting in abelian categories and quasitiltedalgebras, Mem. Amer. Math. Soc. 120 (1996), no. 575, viii+88.

[5] I. HERZOG, Contravariant functors on the category of finitely presented modules, Isr. J.Math. 167, 347-410 (2008).

Dipartimento di MatematicaUniversità degli Studi di PadovaVia Trieste 63, 35121, [email protected]@[email protected]

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Cleanness of group algebrasPaula Murgel Veloso1 Álvaro Pérez Raposo2,

An element of a ring is clean if it is the sum of a unit and an idempotent. A ring is cleanif every element in it is clean. If the representation of an element in this way is unique, theelement is said to be uniquely clean, and the ring is uniquely clean if all its elements are so.

The property of cleanness was formulated by Nicholson [1] in the course of his study ofexchange rings, for both are closely related: clean rings are always exchange rings and theconverse is true when idempotents are central in the ring. Uniquely clean rings were studiedlater [2] showing that they are a sort of generalization of Boolean rings.

In the realm of group rings these properties have been studied with the aim to characterizethe rings R and groups G such the group ring RG is clean or uniquely clean.

At first the focus was set on uniquely clean group rings [3], which is a quite restrictiveproperty and leaves no much room for groups and rings. For instance, a necessary conditionfor RG to be uniquely clean is R to be itself uniquely clean and G to be a 2-group. Thiscondition is also sufficient if the group is taken among locally finite groups or solvable groups.

Recently Wang and You [4] studied the property of cleanness of group rings getting niceresults when the ring of coefficients R is commutative and the group G is a p-group. If p isin the Jacobson radical of R, then RG is clean if and only if R is clean.

In this communication we show results about the cleanness property in group algebrasKG, where K is a field. But there is a key general result about clean rings of Camillo andYu [5] in which they establish that semiperfect rings without an infinite set of orthogonalidempotents are clean rings. Therefore, since every finite dimensional K algebra is in thissituation, it is clean, and we must go to infinite dimensional group algebras to find exampleswhich are not clean.

References[1] W.K. NICHOLSON, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc.

229 (1977), 269–278.

[2] W.K. NICHOLSON; Y. ZHOU, Rings in which elements are uniquely the sum of an idem-potent and a unit, Glasg. Math. J. 46 (2004), 227–236.

[3] J. CHEN; W.K. NICHOLSON; Y. ZHOU, Group rings in which every element is uniquelythe sum of a unit and an idempotent, Journal of Algebra 306 (2006), 453–460.

[4] X. WANG; H. YOU, Cleanness of the group ring of an Abelian p-group over a commu-tative ring, Alg. Colloq. 19 (2012), 539–544.

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[5] V.P. CAMILLO; H.P. YU, Exchange rings, units and idempotents, Comm. Alg. 22(1994), 4737–4749.

1Departamento de AnáliseUniversidade Federal FluminenseRio de Janeiro, [email protected]

2Departamento de Matemática AplicadaUniversidad Politécnica de MadridMadrid, [email protected]

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Construction of minimal non-abelian left group codesGabriela Olteanu1, Inneke Van Gelder2

In previous work of Jespers, Olteanu, del Río and Van Gelder was described a completeset of orthogonal primitive idempotents in each Wedderburn component of a semisimplegroup algebra FG for various classes of finite groups G using strong Shoda pairs and thedescription of the Wedderburn decomposition of QG obtained by Olivieri, del Río and Simónin [1].

Making use of the computation of primitive idempotents in finite group algebras, in [2]we provide algorithms to construct minimal left group codes for a large class of groups andfields. As an illustration of our methods, we give alternative constructions to some best linearcodes over F2 and F3.

We implemented out methods using a programming language provided by the computeralgebra system GAP and we included them in the GAP package Wedderga [3].

References[1] A. OLIVIERI, Á. DEL RÍO, J.J. SIMÓN, On monomial characters and central idempo-

tents of rational group algebras, Comm. Algebra 32 (4), 1531–1550 (2004).

[2] G. OLTEANU, I. VAN GELDER, Construction of inimal non-abelian left group codes,submitted. http://arxiv.org/abs/1302.3747

[3] O. BROCHE, A. KONOVALOV, A. OLIVIERI, G. OLTEANU, Á. DEL RÍO, I. VANGELDER, Wedderga - Wedderburn Decomposition of Group Algebras, Version 4.5.1(2013). http://www.gap-system.org/Packages/wedderga.html

1Department of Statistics-Forecasts-MathematicsBabes-Bolyai UniversityStr. T. Mihali 58-60, 400591 Cluj-Napoca, [email protected]

2Department of MathematicsVrije Universiteit BrusselPleinlaan 2, 1050 Brussels, [email protected]

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Precovering and preenveloping idealsFuruzan Ozbek

Given two classes of R-modules, we say that (F , C) is a cotorsion pair if F and C areorthogonal complement of each other with respect to the Ext functor. A significant result ofcotorsion theory proven by [1] Eklof & Trlifaj is that if (F , C) is cogenerated by a set, thenit is complete. Recently the cotorsion pairs of ideals (I,J ), where I,J are subfunctors ofHomR, have been of interest [2]. In this talk we will look at a few results motivated by Eklof& Trlifaj argument for an ideal I when it is generated by a set.

References[1] Eklof P.C., Trlifaj J. (2001). How to make Ext vanish. Bull. London Math. Soc. 33:31-41.

[2] Asensio P.A.G., Fu X.H., Herzog I. and Torecillas B. (2012). Ideal Approximation The-ory. Preprint.

Department of MathematicsUniversity of Kentucky715 Patterson Office Tower Lexington KY 40506-0027 [email protected]

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Hearts of t-structures which are Grothendieck categories∗

Carlos Parra1, Manuel Saorín2

T-structures on triangulated categories were introduced in the early eighties by Beillison,Berstein and Deligne in their study of the perverse sheaves on an algebraic or an analytic va-riety (see [2]). The main discovery of this concept was the existence of an abelian category,called the heart of the t-structure, which allowed the development of a homology theory whatis intrinsic to the triangulated category.

In [1], Alonso, Jeremías and Saorín classify all the compactly generated t-structures onthe derived category of a commutative noetherian ring R . They described such t-structuresin terms of decreasing filtrations by supports of Spec(R). We study when the heart of sucha t-structure is equivalent to a Grothendieck category or to a module category, for somedecreasing filtrations. In fact, we show that if the filtration is eventually constant, then theheart is a Grothendieck category. In case R is a connected ring and the filtration is bounded,then the heart is equivalent to a category of module if and only if the t-structure is a translationof the canonical t-structure.

References[1] L. ALONSO; A. JEREMÍAS; M. SAORÍN, Compactly generated t-structures on the de-

rived category of a Noetherian ring, Journal of Algebra, 324 (2010), 313-346.

[2] A. BEILINSON; J. BERNSTEIN; P. DELIGNE, “Faisceaux Pervers”. Analysis andtopology on singulas spaces, I, Luminy 1981, Astèrisque. 100. Soc. Math. France, Paris.(1982), 5-171.

[3] R. COLPI; E. GREGORIO; F. MANTESE, On the Heart of a faithful torsion theory,Journal of Algebra, 307 (2007), 841-863.

[4] A. LIDIA; P. DAVID; S. JAN, J. TRLIFAJ, Tilting, cotilting, and spectra of commuta-tive noetherian rings, to appear on Transactions of the Amer. Math. Soc. (Available onhttp://arxiv.org/abs/1203.0907).

∗Carlos Parra is supported by a grant from the Universidad de los Andes (Venezuela) and Manuel Saorín issupported by research projects from the fundación SENECA of Murcia and the Spanish Ministry of Education, witha part of FEDER funds.

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[5] F. MANTESE; A. TONOLO, On the heart associated with a torsion pair, Topology andits Applications, 159 (2012), 2483-2489.

1Departamento de MatemáticasUniversidad de los AndesMérida, [email protected]

2Departamento de MatemáticasUniversidad de MurciaAptdo. 4021, 30100 Espinardo, Murcia, [email protected]

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Recollements of module categoriesChrysostomos Psaroudakis1, Jorge Vitória2

Recollements of abelian categories are exact sequences of abelian categories with niceproperties. They are related to torsion-theoretical structures and localisations of abelian cat-egories. In this talk we will focus on recollements in which the middle term is a modulecategory. In this case, it is easy to see that, up to equivalence, the recollement is determinedby a ring epimorphism. Using this fact, we will classify, up to equivalence, recollements ofabelian categories in which all the three terms are equivalent to module categories. This isjoint work with Chrysostomos Psaroudakis.

1Department of mathematicsUniversity of Ioannina45110 Ioannina, [email protected]

2Department of MathematicsUniversity of BielefeldPostfach 10 01 31, D-33501, Bielefeld, [email protected]

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The CoDimension Sequence of Finite Dimension G-SimpleAlgebras

Yuval Shpigelman

Precise knowledge of the identities of finite dimensional algebras in general seems to bea very hard task. Of course, knowing a set of generators of a given T -ideal would be a majorstep ahead but even then it’s not clear how to determine explicitly whether a polynomial is oris not generated by the given set. With this point of view it is natural (and many times moreeffective) to study general invariants attached to T -ideals of the free algebra. One of them(introduced by Regev in the 70’s) is cn(A)- the codimension sequence attached to the T -idealof identities corresponding to an algebra A. In the 80’s Regev, using results of Formanek,Procesi and Razmyslov in Invariants theory and Hilbert series, calculated asymptotically thecodimension sequence of m × m matrices over algebraically closed field of characteristiczero. Inspired by Regev’s ideas, we pushed further his result and calculated asymptoticallycGn (A) - the codimension sequence of matrix algebras A with elementary G-grading. More-over, we used our calculations to prove that if A is a finite dimensional G- simple algebra,then the polynomial part of cGn (A)’s asymptotics has degree −dimFAe−1

2 (this was conjec-tured by E.Aljadeff, A.Giambruno and D.Haile).

In this talk I will present some of the ideas from Regev’s work, and show how one cangeneralize them to tackle the elementary grading case. The lecture is based on a joint workwith Yakov Karasik

Department of MathematicsTechnion Israel Institute of TechnologyHaifa [email protected]

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Deconstructibility, approximations, and locally free modulesAlexander Slávik

Deconstructible classes provide a vital framework for approximation theory, since allsuch classes are precovering provided they are closed under transfinite extensions, cf. [4].However, as shown by Eklof and Shelah in [2], deconstructibility of the class of all Whiteheadgroups and even its precovering property are both independent of ZFC. Furthermore, the classof all flat Mittag-Leffler modules over a non-right perfect ring is not deconstructible by [3]and not precovering under the further assumption of the ring being countable, cf. [1].

We employ set-theoretic arguments to extend these results to classes of modules arisingfrom infinite dimensional tilting theory: the class of all locally F-free modules induced bya non-

∑-pure split tilting module is shown not to be precovering. In particular, we show

that the class of all locally Baer modules is not precovering for any countable hereditary artinalgebra of infinite representation type, cf. [6].

References[1] S. BAZZONI; J. ŠTOVÍCEK, Flat Mittag-Leffler modules over countable rings, Proc.

Amer. Math. Soc. 140 (2012), 1527–1533.

[2] P. C. EKLOF; S. SHELAH, On the existence of precovers, Illinois J. Math. 47 (2003),173–188.

[3] D. HERBERA; J. TRLIFAJ, Almost free modules and Mittag-Leffler conditions, Advancesin Math. 229 (2012), 3436–3467.

[4] M. SAORIN; J. ŠTOVÍCEK, On exact categories and applications to triangulated adjointsand model structures, Advances in Math. 228 (2011), 968–1007

[5] J. ŠAROCH; J. TRLIFAJ, Kaplansky classes, finite character, and ℵ1-projectivity, ForumMath. 24 (2012), 1091–1109.

[6] A. SLÁVIK; J. TRLIFAJ, Approximations and locally free modules, preprint,arXiv:1210.7097.

Matematicko-fyzikální fakultaUniverzita [email protected]

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T-structures on hereditary categoriesMaria José Souto Salorio,

Joint work with Ibrahim Assem and Sonia Trepode.Let H be a connected abelian hereditary k−category. We study t-structures (U,U⊥[1])

on Db(H) and characterize the split t-structures. In particular, we look for necessary andsufficient conditions for t-structures (U,U⊥[1]) on Db(H) to be induced by torsion pairs(T ,F) inH.

References[AST] ASSEM I., SOUTO SALORIO, M. J., TREPODE, S., Ext-projectives in suspended

subcategories, J. Pure and Appl. Algebra. 212 (2008) 423-434.

[HRS] HAPPEL, D., REITEN, I., SMALØ, S. O., Tilting in abelian categories and qua-sitilted algebras, Mem. Amer. Math. Soc. 120 (1996), 575, 88 pp.

[Ri] RINGEL, C. M., Triangulated hereditary categories, Compositio Mathematica (Toappear).

[ST] SOUTO SALORIO, M. J., TREPODE, S., T-structures on the bounded derived categoryof the Kronecker algebra. Applied Categorical Structures 20 (2011) 513-529

Facultad Informática University Coruña.Campus de Elviña.15071 A Coruña, Spain [email protected]

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The role of big cotilting modules in derived equivalencesJan Stovicek,

Big cotilting modules have been originally studied as a well behaved formal generaliza-tion of finite dimensional cotilting modules in representation theory. Here I show that cotilt-ing modules have a very clear role in the construction of derived equivalences. Every bigcotilting module C over a ring R induces a derived equivalence of Mod-R to a Grothendieckcategory H with a tilting object, and C is sent to an injective cogenerator of H under thisequivalence. This extends results from [2] and allows to interpret classical derived equiva-lences of categories of sheaves to categories of representations, such as [1], in the context ofcotilting theory.

Keywords: cotilting module, derived equivalence, derivator

Mathematics Subject Classification 2010: 16D90, 18E30

References[1] A. A. BEILINSON, Coherent sheaves on Pn and problems in linear algebra. Functional

Anal. Appl. 12(3), 212–214 (1979).

[2] R. COLPI, E. GREGORIO, F. MANTESE, On the heart of a faithful torsion theory. J.Algebra 307(2), 841–863 (2007).

Department of AlgebraFaculty of Mathematics and PhysicsCharles University in PragueSokolovska 83, 186 75 Praha 8, Czech [email protected]

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Derived dualities induced by a 1-cotilting bimoduleAlberto Tonolo,

Let R and S be two arbitrary associative rings. We denote by R-Mod and Mod-S thecategories of left R-modules and of right S-modules.

Morita and Azumaya studied the category dualities given by the contravariant Hom func-tors induced by Morita bimodules, i.e., bimodules RWS such that RW and WS are injectivecogenerators and R = EndWS , S = EndRW . Müller proved that the modules in the domainand the range of these dualities coincide with the linearly compact modules.

The 1-cotilting modules generalize injective cogenerators: they are modules which areinjective exactly on the subcategory they cogenerate. The cotilting theory studies the du-alities induced by the contravariant functors HomR(−, U), HomS(−, U) and Ext1R(−, U),Ext1S(−, U) associated to a (faithfully balanced) 1-cotilting bimodule RUS .

Recently, in a joint paper with Francesca Mantese [1], we have studied the dualities in-duced by a cotilting bimodule in the framework of the derived categories of modules overan arbitrary associative ring. This is the correct setting for understanding, both at the levelof the derived categories and of the module categories, the nature of these dualities. Consid-ering the total derived functors RHom(−,RU) and RHom(−, US) and their cohomologiesis indeed possible to evaluate the interplay of the functors HomR(−, U), HomS(−, U) andExt1R(−, U), Ext1S(−, U).

The aim of this talk is to give some recent results [2], obtained in collaboration withFrancesca Mantese, on the duality induced by the total derived functors RHom(−,RU)and RHom(−, US). In particular, generalizing the Müller result, we give a characteriza-tion of the complexes in the domain and in the range of the dualities induced by the functorsRHom(−,RU) andRHom(−, US) employing a suitable notion of linear compactness.

References[1] F. Mantese and A. Tonolo. Reflexivity in derived categories. Forum Math., 22((6)):1161–

1191, 2010.

[2] F. Mantese and A. Tonolo. Derived dualities induced by a 1-cotilting bimodule. 2013.

Dipartimento di MatematicaUniversità di Padovavia Trieste 63, 35131 Padova, [email protected]

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A∞-algebras, E∞-algebras, and Bousfield LocalizationDavid White,

In a monoidal model categoryM it is often advantageous to study the associative, com-mutative,A∞, andE∞ objects. In this talk we’ll discuss how well these objects are preservedby Bousfield localization, which is the correct notion of localization for model categories.The idea of Bousfield localization is to take a set of maps S and pass fromM to a new modelstructure on the same category where S is contained in the weak equivalences.

An example due to Mike Hill in the context of equivariant spectra demonstrates thatthis process can fail to preserve E∞ algebras, and this was an obstacle in his work on theKervaire Invariant One problem. In this talk I’ll give an example showing that localizationneed not even preserve monoidal structure. I’ll then give conditions on M and the mapsbeing localized under which localization preserves monoidal structure. Further conditionswill guarantee that it preserves the various types of algebraic objects of interest. When wespecialize to the model category of equivariant spectra we recover the theorem of [2] whichwas used to fix the Kervaire Invariant proof.

References[1] CARLES CASACUBERTA, JAVIER J. GUTIÉRREZ, IEKE MOERDIJK AND RAINER

VOGT, Localization of algebras over coloured operads, Proceedings of the LondonMathematical Society 101 (2010), 105-136

[2] MICHAEL A. HILL, MICHAEL J. HOPKINS, Equivariant Multiplicative Closure,arXiv:1303.4479, 2013

[3] PHILIP HIRSCHHORN, Model categories and their localizations, volume 99 of Math-ematical Surveys and Monographs. American Mathematical Society, Providence, RI,2003.

[4] MARK HOVEY, Model Categories, Math. Surveys and Monographs, vol. 63, Amer.Math. Soc., Providence, 1999.

[5] DAVID WHITE, Bousfield Localization and Commutative Monoids, Wesleyan UniversityPhD thesis, 2013

Mathematics DepartmentWesleyan UniversityExley Science Center 655265 Church StreetMiddletown, CT, USA [email protected]

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Forms of path algebrasAdi Wolf,

Let K/F be a field extension, and let A be a finite dimensional F -algebra. We considerthe question of classifying the K/F -forms of A, i.e., find up to F -isomorphism all the F -algebrasB such thatA⊗FK ∼= B⊗FK. WhenK/F is a Galois extension andA = Mn (F )or A = Fn these are known to be the central simple F -algebras of degree n and the etaleF -algebras of dimension n, respectively (see [1], chapter VII).

Let Γ be an acyclic quiver and consider the pointed setA = A (K/F, Γ) ofF -isomorphismclasses of K/F -forms of the path algebra A = FΓ. Let K/F be a (finite or profinite) Galoisextension with a Galois group G. Then by Galois descent A is isomorphic to the pointedset of 1-cocycles of G with values in AutK−alg (KΓ). Denote by SΓ the group of directedgraph automorphisms of Γ. Then G acts on it trivially and we have an injection of G-groupsSΓ → AutK−alg (KΓ). Using Guil-Asensio and Saorin results on decomposition of theautomorphim group of a path algebra ([2]), we show that this injection induces a natural iso-morphism in cohomology H1 (G ,SΓ) ∼= H1 (G ,AutK−alg (KΓ)). This leads us to definethe notion of combinatorial forms of a quiver and to show that the K/F -forms of FΓ aregeneralised path algebras of combinatorial forms of Γ.

If time permits, we will then consider a rooted tree Γ and discuss some properties of theinfinite family of algebras A

(F/F, Γ

), such as versal torsors and cohomological invariants.

This talk is based on a phd work which is done under the supervision of Prof. Eli Aljadeff,Technion.

References[1] M.A. KNUS, A. MERKURJEV, M. ROST, J.-P. TIGNOL, Book of involutions, AMS

Colloquium Publications, Vol. 44 (1998).

[2] F. GUIL-ASENSIO; M. SAORIN, The automorphism group and the Picard group of amonomial algebra, Comm. Algebra 27, No. 2 (1999), 857-887.

Department of MathematicsTechnion - Israel Institute of TechnologyHaifa, Israel [email protected]

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On classical rings of quotients of duo ringsMichal Ziembowski,

Throughout this talk all rings are associative with identity. Recall that a ring R is rightduo (respectively left duo) if any right (resp. left) ideal of R is a two-sided ideal. If R is leftand right duo, then we say that R is a duo ring. In [1] A.J. Diesl et al. posed the following:If a ring R is duo, is the right classical ring of quotients Qrcl(R) duo? In this talk we want toshow that there exists a duo ring R such that its right classical ring of quotients Qrcl(R) is leftduo and not right duo. Using mentioned construction we will built up a duo ring with rightclassical ring of quotients which is neither right nor left duo.

Keywords: duo rings, classical right ring of quotients

Mathematics Subject Classification 2010: 16S85, 16U20

References[1] A.J. DIESL, C.Y. HONG, N.K. KIM, P.P. NIELSEN, Properties which do not pass to

classical rings of quotients. J. Algebra 379, 208–222 (2013).

Faculty of Mathematics and Information SciencesWarsaw University of Technology00-662 Warsaw, [email protected]

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