Quasi-Frobenius rings were introduced by Nakayama [3] in the study of representations of algebras. Afterwards, Quasi-Frobenius rings played a central role in ring theory, and numerous characterizations were given by various authors. In particular, Ikeda [2] characterized these rings as two sided self-injective and two sided Artinian. There has been a great deal of research devoted to improve Ikeda’s mentioned result by weakening the Artinian condition or the injectivity or both, and this led to the discovery of new concepts of rings and modules such as simple-injectivity and min-injectivity of rings and modules [1, 4, 5]. Nicholson and Yousif [4], dealt with the question of when a simple-injective ring becomes injective. They show that a semiprimary right simple-injective ring is injective. This result generalizes Osofsky’s well-known result [6] that "a left perfect left-right injective ring is a quasi Frobenius ring" to "a left perfect left-right simple-injective ring is a quasi Frobenius ring". This basic approach is concerned with when simple injective rings become injective. Considering this approach, in this talk, we aim to present some new results about simple injective modules using relative min-injectivity. 

This is joint work with Sinem Benli-Göral and Engin Büyükaşık.

[1] Alagöz Y., Benli S., Büyükaşık E., On simple-injective modules, J. Algebra Appl., 22(6) (2023), 2350138.

[2] Ikeda M., A characterization of quasi-Frobenius rings, Osaka Math. J. 4 (1952), 203-209.

[3] Nakayama T., On Frobeniusean algebras I, Ann. of Math. 40 (1939), 611-633.

[4] Nicholson W.K., Yousif M.F. On perfect simple-injective rings, Proc. Amer. Math. Soc. 125 (1997) 979-985. 

[5] Nicholson W. K., Yousif M. F., Mininjective rings, J. Algebra 187 (1997), 548-578.

[6] Osofsky B. L., A generalization of quasi-Frobenius rings, J.Algebra, 4, (1966), 373-389.

In 1992, Amnon Neeman proved that the localizing subcategories of the derived categories of modules over a commutative noetherian ring are classified by subsets of its spectrum. This theorem provides a bridge between homological algebra and stable homotopy theory identifying the spectrum of a commutative ring with the chromatic tower in the homotopical category of spectra. Later, in 2000, in a joint work with Jeremías and Souto, the result was extended to Noetherian schemes where its subsets classify localizing subcategories that are tensor-ideals, a condition automatically satisfied in the ring case. In 2011, Neeman characterized colocalizing subcategories by subsets of the spectrum of the ring by exhibiting a bijection between localizing and colocalizing subcategories. In this talk we will extend this result to Noetherian schemes showing that colocalizing subcategories that are hom-coideals are classified by subsets of the spectrum and are also in bijection with tensor-ideal localizing subcategories. We will discuss these results and the importance of the tensor structure of the derived category of quasi-coherent sheaves on a scheme. This is joint work with Ana Jeremías and Eduardo Loureiro.

Weakly proregular sequences were introduced by Alonso, Jeremías and Lipman in 1997 when they studied local cohomology supported in a closed subscheme in a quasi-compact and separated scheme. When the closed subscheme is weakly proregularly embedded, local cohomology supported in that subscheme behaves as in the Noetherian setting. We now study the case of a weakly proregularly embedded specialization-closed subset in the spectrum of a non-Noetherian ring, aiming to characterize this functor in this setting. This is joint work (in progress) with Leovigildo Alonso and Ana Jeremías. 

We extend a result of B. Bhatt, S. Iyengar and L. Ma about perfectoid rings [4] by introducing a new class of regular rings in the non-Noetherian setting. To achieve this, our main tools will be the use of homological functors and a generalization of the main technical results proved by L. Avramov in [1], [2] and [3].

[1] L. L. Avramov, ‘Flat morphisms of complete intersections’, Dokl. Akad. Nauk SSSR 225(1) (1975) 11–14 (English translation: Sov. Math. Dokl. 16(6) (1975) 1413–1417 (1976)).

[2] L. L. Avramov, ‘Homology of local flat extensions and complete intersection defects’, Math. Ann. 228(1) (1977), 27–37.

[3] L. L. Avramov, ‘Descente des déviations par homomorphismes locaux et génération des idéaux de dimension projective finie’, C. R. Acad. Sci. Paris Sér. I Math. 295(12) (1982), 665–668.

[4] B. Bhatt, S. B. Iyengar and L. Ma, ‘Regular rings and perfect(oid) algebras’, Comm. Algebra 47(6) (2019), 2367–2383."

Various constructions of categories have a universal property expressing the freeness/initiality of the construction within a specific categorical doctrine. Expressed in an algorithmic framework, it turns out that this universal property is in a certain sense a doctrine-specific "ur-algorithm" from which various known categorical constructions/algorithms (including spectral sequences of bicomplexes) can be derived in a purely computational way. This can be viewed as a categorical version of the Curry-Howard correspondence to extract programs from proofs.

A ring is a preadditive category with a single object. In this way the tensor product of small preadditive categories generalises the tensor product of a pair of rings. Likewise a biadditive functor is a generalisation of a bimodule, and similarly their tensor product may be considered. This construction is a coend, and goes back to work of Yoneda. Simson later used this idea to generalise the tensor ring of a bimodule.  

In this talk I will discuss how this language has been arising in ongoing joint work: one project involves standard constructions in cluster-tilting subcategories from higher homological algebra; another involves continuous versions of representations of species.

Taking ultraproducts is an essential tool in model theory, most notably giving a proof of the powerful Compactness Theorem for first-order logic.  However, working in ZFC, the standard ultraproduct construction is not very useful outside of first-order logic.  On the other hand, many interesting algebraic properties are not first-order: torsion modules, locally finite groups, rank 1 valued fields, and more.

We discuss several ways to tweak the ultraproduct construction that make it work in particular nonelementary classes.  We will also describe work to put these ad hoc constructions in a unified categorical framework.

We characterise double byproducts as ordinary byproducts, and show that their deformations by 2-cocycles are double wreath quasi-quantum groups. Also, we present examples of 2-cocycles from almost skew pairings in categories of Yetter-Drinfeld modules and show that various types of quasi-quantum groups known in the literature are of this type (joint work with Daniel Popescu and Blas Torrecillas).

The aim of this talk is to explain the notion of cohomology in the context of small categories. This idea has some important similarities with cellular sheaf cohomology and also it behaves well with some homotopical invariants such as the Svarc genus of a functor between two small categories. We will explain the relevant notions and show how to use them to study some nice properties of categories. 

We introduce two novel complementary notions of the Lefschetz number for a functor from a finite acyclic category to itself and we prove a Lefschetz fixed-object theorem and a Lefschetz fixed-morphism theorem. In order to do so, we use the connection between these type of categories and simplicial structures, such as trisps or delta complexes. Through the use of a pair of functors that, when composed, form the barycentric subdivision, we are not only able to identify fixed objects but also fixed chains of morphisms.

Ok let's continue my talk "Derivation of functors in Banach spaces, what could go wrong?" presented at the conference Functor Categories, Model Theory, and Constructive Category Theory (Málaga, 3-7 July 2023), with the added interest that saying what goes right in the category Ban of Banach spaces and linear continuous operators L is much more demanding than saying what goes wrong: The category of Banach spaces is not an Abelian category, it has no either injective or projective objects or limits. Still, those issues can be amended: The categories of Banach spaces has perfectly defined exact sequences, is as Abelian as it can be, has relatively injective and projective objects and, as long as we restrict ourselves to work with contractive operators, has limits.

Complete abstract.

We study the comparison between the logarithmic and the meromorphic de Rham complexes along a divisor in a complex manifold. We focus on the case of free divisors, starting with the case of locally quasihomogeneous divisors, and we explain how D-module theory can be used for this comparison.

In [1], Stenstrom introduced the notion of a flat object in a locally finitely presented Grothendieck category A. An object F in A is flat if every epimorphism from M to F is pure. In this talk, we will discuss the following question, posed in the earlier work [2] co-authored with Daniel Simson: 

Suppose that A has enough flat objects. Does this imply that A has enough projective objects?

We will expound our attempt to answer this question in the specific case of the category of comodules over a coalgebra. We will also echo several related results by Rump [3], Crivei, Prest and Torrecillas [4], and Estrada and Saorín [5].

References

[1] B. Stenstrom, Purity in functor categories. J. Algebra 8 (1968), 352-361.

[2] J. Cuadra and D. Simson, Flat comodules and perfect coalgebras. Comm. Algebra 35 (2007), 3164-3194.

[3] W. Rump, Flat covers in abelian and in non-abelian categories. Adv. Math. 225 (2010), 1589–1615.

[4] S. Crivei, M. Prest, and B. Torrecillas, Covers in finitely accessible categories. Proc. Amer. Math. Soc. 138 (2010), 1213–1221. 

[5] S. Estrada and M. Saorín, Locally finitely presented categories with no flat objects. Forum Math. 27 (2015), 269-301.

Rump and, independently, Crivei, Prest, and Torrecillas proved that every object in a locally finitely presented Grothendieck category has a flat cover. Here, 'flat' refers to Stenström's notion of flatness. This allows one to build unique up-to-homotopy minimal flat resolutions for any object in such categories. However, as in the category of modules, one would like flat covers to be epimorphisms, or equivalently, for the category to have enough flat objects. In the present talk, we show that locally finitely presented Grothendieck categories with no non-zero flat objects (and hence with no non-zero projective objects) are, unexpectedly, quite abundant. Examples include the category of quasi-coherent sheaves on a projective space, for which, however, there are enough 'geometrical' flat sheaves. This talk is based on joint work with Manuel Saorín.

First of all, I will recall the well known notion of semidirect product both for classical algebraic structures (like groups and rings) and for more recent ones (digroups, left skew braces, heaps, trusses). Then I will deal the concept of semidirect product for an arbitrary algebra $A$ in a variety $\Cal V$ of type~$\Cal F$. Here algebra means in the sense of Universal Algebra. An inner semidirect-product  decomposition $A=B \ltimes\omega$ of $A$ consists of a subalgebra $B$ of $A$ and a congruence $\omega$ on $A$  such that $B$ is a set of representatives of the congruence classes of $A$ modulo $\omega$. An outer semidirect product is the restriction to $B$ of a functor from a suitable category $\Cal C_B$ containing $B$, called the enveloping category of $B$, to the category $\Set_*$ of pointed sets.

In this talk, the theory of infinite powers of ghost ideals is presented. A key tool developed in this theory is an ideal version of Eklof's Lemma. This theory is used to study Generalized Generating Hypothesis. In particular, (1) it is used to show a dual of a result of Xu: if the class of pure projective right R-modules is closed under extensions, then every FP-projective right R-module is pure projective; and (2) it is used to study the ghost ideal in the category of complexes. This is a joint work with S. Estrada, I. Herzog, and S. Odabasi.

For (von Neumann) regular semiartinian rings with primitive factors artinian there is an invariant called dimension sequence (Theorem 2.1 in [1]) formed by slices of socle chain of the ring. The necessary conditions on this invariant was studied for example in [2]. We will focus on how much the dimension sequence determines the ring. In the case of commutative ring if we consider the dimension sequence with finitely many slices all of them being countable, the corresponding ring is (up to isomorphism) given by one construction from the ring of eventually constant sequences. We will also discuss the case for dimension sequence of infinitely many countable slices.

[1] P. Růžička, J. Trlifaj, J. Žemlička: Criteria of Steadiness. Marcel Dekker Abelian Groups, Module Theory, and Topology, 1998.

[2] J. Žemlička: On socle chains of semiartinian rings with primitive factors artinian. Lobachevskii Journal of Mathematics, Volume 37, 2016, Pages 316-322.

The Ziegler spectrum for categories enriched in closed symmetric monoidal Grothendieck categories is introduced. It recovers the classical Ziegler spectrum of a ring. As an application, the Ziegler spectrum as well as the category of generalised quasi-coherent sheaves of a reasonable scheme is defined and studied. It is shown that there is a closed embedding of the injective spectrum of a coherent scheme endowed with the tensor fl-topology (respectively of a noetherian scheme endowed with the dual Zariski topology) into its Ziegler spectrum. It is also shown that quasi-coherent sheaves and generalised quasi-coherent sheaves are related to each other by a recollement.

The computation of lattices of thick subcategories of triangulated categories has emerged as a popular topic and serves as a more achievable analogue of classifying objects. Often one understands such lattices by describing them in terms of some associated topological space. However, in many representation theoretic examples this is not possible. I’ll explain what the obstruction is and introduce possible methods of addressing this issue. This talk is based on joint work with Greg Stevenson.

The motivation of this talk is to study the representations in categories of Lie and Leibniz crossed modules.

On the one hand, we will construct, for crossed modules of Lie algebras or Lie 2-algebras,  the universal enveloping associative 2-algebra such that the category of Lie modules over a crossed module of Lie algebras is isomorphic to the category of modules over that universal enveloping and construct a functor that is left adjoint to the 2-liezation functor. On the other hand, we will construct, for crossed modules of Leibniz algebras or Leibniz 2-algebras,  the universal enveloping associative 2-algebra such that the category of Leibniz modules over a crossed module of Leibniz algebras is isomorphic to the category of modules over that universal enveloping.

We develop a theory of integration with respect to the Lefschetz number in the context of tame topology and o-minimal structures containing the semilinear sets. We also obtain a more general fixed point theorem. Indeed, such results could be interesting from a categorical point of view.

Let K be a field and E be the graph with a vertex v and n. The associated Leavitt path algebra L_K(E) is a perfect left localization of the free algebra in n variables Λ, and the category of finitely presented simple L_K(E)- modules is a quotient category of the finitely presented simple modules over Λ. Applying methods and techniques for the study of simple modules over Leavitt path algebras, we obtain a better understanding of the finitely presented irreducible representation of Λ. As an interesting application, we give  a new  classification of irreducible polynomials in n non-commutative variables.

Flatness can be defined in any locally finitely presented Grothendieck category. In case such a Grothendieck category has enough projectives, any projective object is flat. In 2007, Cuadra and Simson conjectured whether a locally finitely presented Grothendieck category with enough flat objects has enough projective objects. Even though significant results have been achieved in the category of left comodules over a coalgebra, the conjecture still resists in its generality. In this talk, we present an attempt to face the conjecture by looking at those Grothendieck categories which fit as the right-most term of a recollement of a commutative ring.

Assume that K is an algebraically closed field and R, A are locally bounded K-categories. Moreover, assume that G is a group of K-linear automorphisms of R, acting freely on the objects of R, and let F:R->A=R/G be the associated Galois G-covering. Recall that in case G is torsion-free, the push-down functor F_l:mod(R)->mod(A) is a Galois G-precovering of module categories. We show that this situation lifts to the level of categories of finitely presented 

functors. More precisely, the group G acts freely on the category F(R) of finitely presented functors (which is Krull-Schmidt) and there exists a functor Phi:F(R)->F(A) which is a Galois G-precovering (of functor categories). Hence it makes sense to adapt the well-known terminology of Dowbor-Skowroński and define the functors of the first kind, lying in the image of Phi, and that of the second kind - the remaining ones. We show that even in the nicest situations (when R and A are simply connected and representation-finite) the functor Phi may not be dense and thus the functors of the second kind exist. Last but not least, we show some applications of the above facts to the theory of Krull-Gabriel dimension. 

The center of a monoid is always a commutative monoid.  This statement has a categorified version: the center of a monoidal category is always a braided monoidal category, called its Drinfeld center. This talk is motivated by the following question: can we find useful categorical tools for the construction of objects in the Drinfeld center?

For this, we show how to construct induced (op)lax monoidal functors between Drinfeld centers from a given monoidal adjunction for which the so-called projection formula holds. These induced functors can then be used for the construction of internal (co)algebra objects in the Drinfeld center. We also discuss when these induced functors are Frobenius monoidal functors.

This is joint work (in progress) with Johannes Flake (Universität Bonn) and Robert Laugwitz (University of Nottingham).

The tensor-embedding of a module category into the functor category allows us to see the pp-type of an element in a module - the set of all pp formulas that it satisfies in that module - as the analogue, and generalisation, of the annihilator of the element.  Esssentially, and via elementary duality, we can identify pp-types with arbitrary subfunctors of the forgetful functor.

I will explain this and also give various applications which show how pp-types can be used.

Hopf braces are recent mathematical objects introduced by I. Angiono, C. Galindo and L. Vendramin in [AGV] and obtained through a linearisation process from skew braces, which give rise to non-degenerate, bijective and not necessarily involutive solutions of the Quantum Yang-Baxter Equation (see [GV]). As was proven in \cite[Corollary 2.4]{AGV}, cocommutative Hopf braces are also relevant from a physical standpoint because they also induce solutions of the above-mentioned equation.

On the one hand, a well-known result for Hopf braces is their strong relationship with invertible 1-cocycles due to the fact that both categories are equivalent (see [AGV, Theorem 1.12] and [GRR, Theorem 3.2]). These objects are no more than coalgebra isomorphism between Hopf algebras, related to each other through a module-algebra structure, and satisfying a weaker condition than being algebra morphism.

On the other hand, R. González Rodríguez in [RGON] introduced the notions of module over a Hopf brace and Hopf module over a Hopf brace, obtaining a categorical equivalence between the base braided monoidal category C and the category of Hopf modules over a Hopf brace, also known by the Fundamental Theorem of Hopf modules for Hopf braces.

Therefore, considering the aforementioned precedents, the aim of this talk is going to be giving a suitable notion of module over a invertible 1-cocycle in such a way that the categorical equivalence between Hopf braces and invertible 1-cocycles remains valid between their module categories.

Bibliography.

[AGV] I. Angiono, C. Galindo and L. Vendramin, Hopf braces and Yang-Baxter operators, Proc. Am. Math. Soc. 145 (5) (2017) 1981-1995.

[VRBAMod] J. M. Fernández Vilaboa, R. González Rodríguez, B. Ramos Pérez and A. B. Rodríguez Raposo, Modules over invertible 1-cocycles, Turk. J. Math. 48 (2) (2024) 248-266.

[RGON] R. González Rodríguez, The fundamental theorem of Hopf modules for Hopf braces, Linear Multilinear Algebra 70 (20) (2022) 5146-5156.

[GRR] R. González Rodríguez and A. B. Rodríguez Raposo, Categorical equivalences for Hopf trusses and their modules, Preprint (2023).

[GV] L. Guarnieri and L. Vendramin, Skew braces and the Yang–Baxter equation, Math. Comput. 86 (307) (2017) 2519-2534.

We’ll discuss a way of generalising species that allows for semilinear representations. Here we mean species in the sense of Dlab and Ringle, generalising in the sense of Simson, and semilinearity in the sense of Bennett-Tennenhuis and Crawley-Boevey. First, we’ll quickly cover species and some motivation to generalise them. Then we’ll examine the generalisation and compare two possible constructions when working with quivers. Along the way we will discuss the role of semilinearity. We’ll conclude with at least one example and more if time permits. Work in progress with Raphael Bennett-Tennenhuis.   

Universal structures were introduced in the work of R. Fraïssé and B. Jonsson as structures that embed all other members (of appropriate cardinality) of certain classes of structures. The concept has been modified to elementary embeddings in the work of Vaught and Morley, which applies only to model classes of complete theories. In this work, joint with Anand Pillay, we consider an intermediate sort of embedding, pure embeddings. Namely, define a structure M to be pure-universal in a class K if M is in K and purely contains an isomorphic copy of every structure in K that has cardinality not exceeding that of M.

Most of the literature is concerned with the existence of universal structures in certain cardinalities. We ask instead for when certain specific structures are universal. Specifically we ask when free R-modules of infinite rank are pure-universal in the class of flat R-modules. The main result is an answer to this question using the theory of Bass modules as presented at this conference two years ago.

Motivation will be presented that stems from the model theory of universal-algebraic varieties in order to put this work into this more general perspective of equational classes of algebraic structures.

Spectral curves are algebraic curves associated to commutative subalgebras of rings of ordinary differential operators (ODOs). Their origin is linked to the Korteweg-de Vries equation and to seminal works on commuting ODOs by I. Schur and, Burchnall and Chaundy. They allow the solvability of the spectral problem Ly=E y, for an algebraic parameter E and an algebro-geometric ODO L, whose centralizer is known to be the affine ring of a abstract spectral curve. In this talk, differential resultants we will used to effectively compute the defining ideal of the spectral curve, defined by the centralizer of a third order differential operator L, with coefficients in an arbitrary differential field of zero characteristic. Our results establish a new framework appropriate to develop a Picard-Vessiot theory for spectral problems. I am presenting joint work with Maria-Ángeles Zurro, as part of the project ``Algorithmic Differential Algebra and Integrability" (ADAI), from the Spanish MICINN, PID2021-124473NB-I00.

In this talk we will speak about the notion of (support) τ-tilting subcategories in abelian categories with enough projective objects. After recalling their definition, we show a bijection between support τ-tilting subcategories and τ-torsion-cotorsion triples and we will explain some of the consequences this bijection.

Time permitting we will explain how the notion of support τ-tilting subcategories relates with classical τ-tilting theory and we will show an application of our results to quiver representations.

This talk is based in a joint work with J. Asadollahi and S. Sadeghi https://arxiv.org/abs/2208.12703

The approximation classes of modules that arise as components of cotorsion pairs are tied up by Salce's duality. For general approximation classes, possibilities of dualization depend on closure properties of these classes. While some results are easily dualized, other dualizations require large cardinal principles, and some fail in ZFC, with counterexamples provided by classes of aleph1-projective modules. We show that Vopěnka's Principle implies that each covering class of modules closed under homomorphic images is of the form Gen (M) for a module M, and the latter property restricted to classes generated by aleph1-free abelian groups implies Weak Vopěnka's Principle.           

In this talk we  give two methods to turn a non- semisimple tensor category  into a semisimple category  with the Green ring unchanged. As an application, we study the deformations of the representation categories of some semigroups. 

Motivated by Birkhoff’s problem, Ringel and Schmidmeier initiated the modern study of monomorphism categories. Their original work on submodule categories has been developed rapidly and generalized considerably, as many applications were found in various mathematics subjects, such as Gorenstein homological algebras, the study of weighted projective lines and cotorsion theory. Recently, Eshraghi and Hafezi studied the connections between morphism categories and functor categories, showing a correspondence of almost split sequences between these categories. In this talk, we are going to revisit the Auslander-Reiten theory of monomorphism categories using this correspondence.