# Some Trends in Algebra 2013

A conference on module theory and its relations to algebraic geometry, category theory,
commutative algebra, homotopy theory, logic, and representation theory.

# Abstracts

 Speaker Abstract Ana Agore Extending structures for algebras The talk is devoted to the dual of the Hochschild extension problem for associative algebras. Let $A$ be an algebra, $E$ a vector space containing $A$ as a subspace and $V$ a complement of $A$ in $E$. All algebra structures on $E$ containing $A$ as a subalgebra are described and classified by two non-abelian cohomological type objects which are explicitly constructed: ${\mathcal A}{\mathcal H}^{2}_{A} \, (V, \, A)$ will classify all such algebras up to an isomorphism that stabilizes $A$ and ${\mathcal A}{\mathcal H}^{2} \, (V, \, A)$ provides the classification up to an isomorphism of algebras that stabilizes $A$ and $V$. A new product, called the unified product, is introduced as a tool of our approach. Different types of split extensions of algebras are fully described in terms of special cases of unified products: in particular, the classical crossed product and its generalizations are special cases of the unified product. Silvana Bazzoni Cotilting modules and homological epimorphismsWe study the relation between 1-cotilting classes and homological epimorphisms of rings. We show that for a valuation domain $R$ there is a bijective correspondence between cotilting classes and equivalence classes of injective homological ring epimorphisms originating in $R$. Partial results and some open problems will be stated for 1-cotilting modules over arbitrary rings. Clinton Boys The graded representation theory of the alternating groups When new results are proved for symmetric groups, a natural question arises: what do these results imply for the alternating group, the index-2 subgroup of the symmetric group consisting of even permutations? In this talk we discuss the implications of the Brundan-Kleshchev graded isomorphism theorem for the alternating groups. Simion Breaz $\Sigma$-pure injective modules We prove that a right $R$-module $M$ is $\Sigma$-pure injective if and only if $\Add(M)\subseteq \Prod(M)$. Consequently, the homotopy category $\Htp{\Modr R}$ satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply this result to provide new characterizations for right pure-semisimple rings or to give a partial positive answer to a question of G. Bergman. Martin Doubek Generalized BV algebras and master equations in string theory via operads Generalized BV algebras and master equations in string theory via operads abstract: So called string vertices $S$ in closed string theory are required to satisfy the BV equation $d(S)+h\Delta(S)+1/2\{S,S\}=0$ in certain BV algebra with operations $d, \Delta, \{-,-\}$. We show how the solutions are equivalently described in terms of standard construction on the cyclic operad $Com$, thus revealing that the solutions are in fact homotopy algebras. The theory generalizes to operads other than $Com$ and yields master equations for open and open-closed string theories. Mehmet Akif Erdal The Equivariant James Spectral Sequence The James spectral sequence, which has been introduced by Teichner in his Ph.D. thesis, is a generalization of the Atiyah-Hirzebruch spectral seqeunce. It converges to the generalized homology of the Thom spectra and has many geometric applications. Let $G$ be a compact lie group, $U$ be a complete $G$-universe. Let $\xi:E \rightarrow BO_G(U)$ be a $G$-equivariant stable Gauss map. We construct the Thom $G$-spectrum $M\xi$, as a homotopy colimit of a certain functor. This construction allow us to construct the equivariant $RO(G)$-graded version of the James Spectral Sequence, by using the homotopy colimit spectral sequence. Sergio Estrada On the absence of flat objects in finitely accessible categories The category $Qcoh(X)$ of quasi--coherent sheaves on a quasi--compact and quasi--separated scheme $X$ is finitely accessible. Therefore there is the usual categorical notion of flatness on finitely accessible categories. But there is also the standard definition of flatness in $Qcoh(X)$ from the stalks. So it makes sense to wonder the relationship (if any) between these two notions. In the talk we will discuss on the absence of nonzero categorical flat objects in many finitely accessible categories. As particular instance, we show that $Qcoh(\mathbf{P}^n(R)))$ has no other categorical flat objects than zero, where $R$ is any commutative ring. Gabriella D'Este Pedro Antonio Guil Asensio Hereditary pure torsion classes We define the notion of hereditary pure torsion classes over a ring and of pure-injective module relative to them and show that they generalize both the class of flat modules and of definable subcategories. We also prove that the work of Auslander on simple subfunctors of continuous functors can be extended to them to show that they are determined by sets of indecomposable relative pure-injective modules. The obtained results suggest that some type of topology may be associated to these classes of modules. This is a preliminary report of a join work with I. Herzog and P. Rothmaler. Daniel Herden Michal Hrbek Minimal generating sets of modules This is a joint work with Pavel Růžička. Unless it is finitely generated, a module may not possess a generating set minimal with respect to inclusion (e.g. non-zero injective abelian groups). If a module has such generating set, it is called weakly based. We give a lucid characterization of weakly based modules over Dedekind domains and show some applications. We also address modules over simple, perfect, and semiartinian rings. Peter Kalnai Products of small modules A module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal. Joint work with Jan Zemlicka. Derya Keskin Tutuncu Steffen Koenig Quasi-hereditary algebras and boxes Quasi-hereditary algebras arise in Lie theory for instance as Schur algebras and as blocks of the Bernstein-Gelfand-Gelfand category O. Algebras of global dimension at most two are quasi-hereditary, too. Such algebras come with so-called standard (Weyl, Verma) modules, and the category of modules with standard filtration is being studied. We describe this category by first considering its A-infinity structure, then translating this into a differential graded setup and finally identifying it as a category of representations of a box (in the sense of Drozd, Ovsienko and Roiter). As an application we get that up to Morita equivalence, each quasi-hereditary algebra has an exact Borel subalgebra. In this context, Ringel duality becomes a special case of Burt-Butler duality. This is joint work with Julian Kuelshammer and Sergiy Ovsienko. Wieslaw Kubis Homogeneous objects via Katetov functors We shall present the notion of a Katetov functor, leading to special objects with a strong homogeneity property. Recall that an object $x$ is homogeneous with respect to the class of objects $K$ if every two embeddings of a fixed $K$-object into $x$ are conjugate. The name embedding' corresponds to a fixed subcategory whose arrows are supposed to be monic in the original categoryi. Roughly speaking, a covariant functor is Katetov if, together with a fixed natural transformation from the identity, it realizes' all primitive extensions of objects of a fixed category $S$ of `small' objects. Typical examples of $S$ are categories of finitely generated models of a some first-order language. The range of a Katetov functor is then the category of all models that can be presented as colimits of directed systems in $S$. Some applications to categories of cell complexes will be given. Martin Markl Natural operations and Koszul hierarchyThe Koszul hierarchy (aka higher brackets or Koszul brackets) is an explicit construction that, for any commutative associative algebra A with a differential Delta (which is, very crucially, not necessary a derivation), produces a sequence of multilinear maps Phi_n : A x ... x A ---> A (n copies of A) such that (1) the operations Phi_n form a strongly homotopy Lie algebra, and (2) Phi_n = 0 implies Phi_{n+1} = 0 (heredity property) Koszul brackets are used for instance to define higher-order derivations: Delta is a degree n derivation if Phi_{n+1}(Delta) = 0. Higher order derivations play an important role e.g. in BRST approach to closed string field theory. Recently, a similar construction appeared also for associative (non-commutative) algebras. I was able to show that both brackets are given by the twisting by a specific unique automorphism and that they are essentially unique. Consequently, the notion of higher-order derivations is God given, not human invention. The proof is based on careful analysis of a space of natural operations. Here I employed the technique developed in my work with M. Batanin on the Deligne conjecture. As a matter of fact, the core of my theory is a vanilla version of Deligne's conjecture. I plan to focus my talk on this side of the story. Andrew Mathas The graded representation theory of the symmetric groups The representation theory of the symmetric groups is a classical subject which has its roots in the work of Frobenius and Young at the end of the 18th century. In the semisimple case almost everything is known, however, over finite fields the theory is much harder and many basic questions have yet to be answered. In 2008 there was a huge paradigm shift in this field when Brundan and Kleshchev used work of Khovanov-Lauda and Rouquier to show that the group algebras of the symmetric groups carry a natural Z-grading. This grading is intimately connected with canonical bases and the representation theory of the affine special linear groups and it gives new meaning to Ariki's Categorification Theorem, which proved the LLT conjecture. In this talk I will survey these developments, with the aim being to give a taste for the main ideas whilst keeping the technicalities to a minimum. I will explain the KLR grading in terms of the well-understood semisimple representation theory of the symmetric groups and end by stating a conjecture for the dimensions of the irreducible modules of the symmetric groups which comes out of this new approach to this classical subject Gigel Militaru Quasitriangular algebras, braidings on bimodules and central simple algebras We introduce and study a new class of algebras, we call it \emph{quasitriangular algebras}, as a generalization of the classical concepts of Azumaya algebras or central simple algebras. An algebra $A$ over a commutative ring $k$ is called a quasitriangular algebra if there exists an element $R \in A\ot A\ot A$ satisfying certain axioms. We prove that these elements $R \in A\ot A\ot A$, if exists, are in bijective correspondence to the set of all braidings on the monoidal category ${}_A{\mathcal M}_A$ of $A$-bimodules; moreover we prove that any such braiding is a symmetry. The concept of quasitriangular algebra is related to three classical concepts in algebra as follows: (1) a commutative algebra is quasitriangular if and only if $k\to A$ is an \emph{epimorphism} in the category of rings; (2) if the invariants functor $G = (-)^A:\ {}_A{\mathcal M}_A\to {\mathcal M}_k$ is a \emph{separable} functor, then $A$ is quasitriangluar; (3) any Azumaya algebra is a quasitria ngular algebra. George Ciprian Modoi Some applications of deconstructibility in triangulated categories We define deconstructible subcategories in triangulated categories in analogy with the abelian case. We show that a triangulated category with coproducts satisfies Brown representability, provided that itself is deconstructible. We use this result in order to indicate some triangulated categories which satisfy Brown representability, included the opposite of the derived category of a Grothendieck abelian category. Sinem Odabasi Flaviu Pop On direct products and Ext^1 contravariant functorsThis is a joint work with Claudiu Valculescu. We show, using inequalities between infinite cardinals, that, if R is an hereditary ring, the contravariant derived functor Ext^1_R(−,G) commutes with direct products if and only if G is an injective R-module. David Pospisil A classification of compactly generated co-t-structures for commutative noetherian ringsI will present some results from [1]. Namely the classification of compactly generated Hom-orthogonal pairs in a triangulated category and the classification of compactly generated co-t-structures in a derived category of a commutative noetherian ring. [1] J. Stovicek, D. Pospisil, "On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings", submitted to TAMS, 2013, arXiv: 1212.3122 Mike Prest Almost dual pairs and definable classes of modules This is joint work with Akeel Ramadan Mehdi. Lenzing investigated categories of modules with support in (that is, which are in the $\varinjlim$-closure of) categories of finitely presented left modules. Recently H. Holm considered the categories of right modules which arise as (character-)duals of those with support in some category of finitely presented modules. By placing Holm's results in the context of elementary duality on definable subcategories we are able to extend them. In doing so we also prove that dual modules have enough indecomposable direct summands. H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theor., 13 (2010), 543-560. H. Lenzing, Homological transfer from finitely presented to infinite modules, pp. 734-761 in Abelian Group Theory, Lecture Notes in Math., Vol. 1006, Springer-Verlag, 1983. A. R. Mehdi and M. Prest, Almost dual pairs and definable classes of modules, University of Manchester, preprint, 2013, arXiv:1304.4481 Pavel Prihoda Large projective modules over small symmetric groupsI will recall several basic notions from representation theory of finite groups that can be applied to find idempotent ideals of integral group ring of a finite group. Hopefully I will be able to show how it works in some small examples. Oriol Raventos Obstruction theory for the representability of cohomological functors We will describe an obstruction theory that enables us to decide if a cohomological functor defined from a subcategory of a triangulated category to the category of abelian groups is representable. We will also discuss the relation with the transfinite purity of rings and the transfinite Adams representability problem. Adam van Roosmalen Serre functors and derived equivalences This talk is based on joint work with Donald Stanley where we study the role of the Serre functor in the theory of derived equivalences. Let A be a k-linear (k a field) abelian category such that the bounded derived category Db A has a Serre functor, and let H be the heart of a t-structure (D^{\leq 0}, D^{\geq 0}) on Db A. One can verify that if the realization functor Db H --> Db A is a triangle equivalence, then (1) the t-structure is bounded and (2) D^{\leq 0} is closed under the Serre functor. In this talk, we will consider some cases where the converse holds as well. Jiri Rosicky A functorial form of Shelah's singular compactness theorem We present a functorial form of Shelah's singular compactness theorem where "free" objects are those belonging to the image of a functor. As a consequence, we get a cellular form where "free" objects are cellular ones. This form covers all known instances of Shelah's singular compactness theorem, i.e., S-filtered modules, transversals and colourings of graphs. Serap Sahinkaya Alexander Slavik Jan Stovicek Derived equivalences induced by big cotilting modules Big cotilting modules have been originally studied as a well behaved formal generalization of finite dimensional cotilting modules from representation theory. Generalizing results of Happel, Reiten and Smalo, and of Colpi, Gregorio and Mantese, I will show that big cotilting modules are precisely images of injective cogenerators of Grothendieck categories under certain tilting derived equivalences. If time permits, I will discuss more aspects such as that the derived equivalence is induced by a Quillen equivalence and that this induces an equivalence of the corresponding Grothendieck derivators. Carlo Toffalori Some model theory of modules over Bezout domains This is a joint work with Gena Puninski. We develop the model theory of modules over commutative B\'ezout domains, in particular we characterize the domains whose lattice of pp-formulae has no width and give some applictaions to the existence of superdecomposable pure injective modules. Jan Trlifaj Jan Zemlicka Modules with restricted minimum condition A module $M$ satisfies the restricted minimum condition if $M/N$ is artinian for every essential submodule $N$ of $M$. We discuss several properties of this class of modules.