An analogue of Galois theory will be presented, providing an answer to the longstanding question of constructing such a theory. 

Throughout this talk, we will introduce an action of the skew polynomial function right near-ring K[X; σ, δ] on K based on its evaluation and the concept of the left skew product of functions. This will result in the exploration of the construction of a specific subset, denoted by T (X) ⊂ K[X; σ, δ] with an R−module structure, where R denotes the subring formed by functions that are constant in every orbit of X. This approach facilitates the control of the non-commutativity of K[X; σ, δ]. The final part of this presentation will focus on using the previously mentioned fact to construct a public key exchange protocol that is secure within the Canetti-Krawczyk model.

 We obtain the Serre closure of the category of Banach spaces , namely, the smallest class closed under subspaces, quotients and extension containing all Banach spaces. We will also devote some thought to the Serre closure of Hilbert spaces.

Link to the abstract

In this work we investigate when homological and homotopical properties (approximations, cotorsion pairs, and abelian model structures) of a class of objects  Q  in an abelian categoryare inherited by Qn, the class of objects with Q-(co)resolution dimension at most an integer n. We also give some application in  Qcoh(X), the category of quasicoherent sheaves over a semiseparated Noetherian scheme X.  This is a joint work with Hanane Ouberka.

We study the existence of various kinds of Gorenstein injective envelopes in a general Grothendieck categories through methods from accessibility theory. We characterize when the Gorenstein injective cotorsion pair is complete (in fact perfect) in terms of the existence of Tate trivial generators. As a consequence, we obtain Gorenstein injective envelopes in broad classes of Grothendieck categories, including categories without enough projectives, such as the category of quasi-coherent sheaves on suitable schemes. The talk is based on a joint work with James Gillespie, available at arXiv:2605.02634

A well-known result due to Chase establishes that a ring R is right coherent if and only if every direct product of projective left R-modules is flat. On the other hand, flat modules are known to form the left-hand class of a complete cotorsion pair, and the modules in the right-hand class of such a cotorsion pair are known as cotorsion modules. This flat-cotorsion pair was an essential tool in the proof of the Flat Cover Conjecture see [1].  In this talk, we consider a dual situation to Chase's theorem, i.e. we characterize the class of rings for which every direct sum of injective left R-modules is a cotorsion module. We call these rings left weakly Sigma-cotorsion rings. Among other interesting properties, these rings arise naturally in the study of the (lack of) balance of the right derived functors of the Hom functor associated with flat resolutions.  Noetherian and perfect rings are trivial examples of left weakly Sigma-cotorsion rings, but we present many other nontrivial examples. In fact, we consider the more general situation of rings for which every direct sum of injective modules has cotorsion dimension less than or equal to n, which we call left weakly n-Sigma-cotorsion rings. A major breakthrough in the study of these rings is based on an extension of a result see [2, Theorem 3.3] due to Šaroch and Šťovíček concerning the first-order-theoretic nature of Sigma-cotorsionness.  This talk presents joint work with Manuel Cortés-Izurdiaga and Sergio Estrada (see [3]), together with additional results from [4].

[1] Bican, L., El Bashir, R., Enochs, E.E., All modules have flat covers. In: Bulletin of the London Mathematical Society 33.4 (2001), pp. 385-390. DOI: 10.1017/S0024609301008104

[2] Šaroch, J., Šťovíček, J. Singular compactness and definability for Sigma-cotorsion and Gorenstein modules. In: Selecta Mathematica 26.23 (2020), Article 23. DOI:10.1007/s00029-020-0543-2

[3] Cortés-Izurdiaga, M., Estrada, S., Fresneda, J.M., Weakly Sigma-cotorsion rings. arXiv:2602.11303

[4] Fresneda, J.M., On classical and Gorenstein homological invariants of rings. arXiv:2606.00736 

One of the key problems in constructing schemes for non-commutative rings is that often in non-commutative ring theory there basically are not enough prime ideals. However, the situation changes for non-commutative graded algebras with extra symmetries like, for example, the tensor algebra of a vector space. We introduce and study symmetric projective schemes associated to such algebras as well as their categories of symmetric quasi-coherent sheaves. The latter category is defined in terms of graded representations of the symmetric groups. Symmetric projective schemes have the same topological properties as classical ones and their categories of symmetric quasi-coherent sheaves are closed symmetric monoidal Grothendieck with invertible generators. We also prove that classical projective schemes (resp. the classical category of quasi-coherent sheaves on a projective scheme) are recovered out of symmetric projective schemes (resp. out of symmetric quasi-coherent sheaves). It is worth mentioning that main ideas and methods here originate in stable homotopy theory.

Convolutional codes were originally conceived as vector subspaces of a finite-dimensional vector space over a field of Laurent series having a polynomial basis. Piret and Roos modeled cyclic structures on them by adding a module structure over a finite-dimensional algebra skewed by an algebra automorphism. These cyclic convolutional codes turn out to be equivalent to some right ideals of a skew polynomial ring built from the automorphism. When a skew derivation is considered, serious difficulties arise in defining such a skewed module structure on Laurent series. We discuss some solutions to this problem which involve a purely algebraic treatment of the left skew Laurent series built from a left skew derivation of a general coefficient ring, when possible.

In this talk, we show that the category of modules over a Hopf brace is equivalent to the category of modules over a suitable smash product algebra associated with the brace structure. This equivalence provides a reinterpretation of Hopf brace modules in terms of more classical algebraic objects and establishes connections with Doi–Hopf module theory. We further study the relationship between different notions of Hopf brace modules appearing in the literature by means of the cocommutativity class associated with a Hopf algebra action.

In a broad sense, functors bring clarity to several constructions in several contexts in mathematics. By bundling functors of the same domain and codomain together, one can further consider the categories they form, usually referred to as functor categories. Such categories turn out to be very convenient for understanding phenomena in mathematics, and have been explored from different points of view over the past decades. In this talk, we will discuss one particular perspective, namely, how functor categories can provide a representation theory of (additive) categories. We will then present a few uses of this theory. In particular, we will explain how it can shed light on higher homological algebra, helping to visualize concepts that were previously unseen, such as higher versions of semi-abelian, quasi-abelian and integral categories. Time permitting, we will also speculate on what else there is to be brought to light.

I will explain some results on classification of, not necessarily finitely generated, projective modules that depend on their trace ideals. We will discuss the possible symmetries and asymmetries between traces of projective right R-modules and of projective left R-modules.

This is joint work with Naomi Andrew.

We study the class of weakly Ding injective modules, wDI, over coherent rings.  We show that every weakly Ding injective module is a direct sum of an FP-injective module and a Ding injective module.  We prove that, if wDI is closed under extensions, then it is the right half of a complete hereditary cotorsion pair. We also prove that with the same hypothesis, the following are equivalent: 1. The class wDI is closed under direct limits.\\ 2. The class wDI is covering.\\ 3. The character module of every Ding injective is Gorenstein flat.\\ The equivalent statements 1-3 also imply that the class of Gorenstein flat modules is preenveloping.

Torsion-torsionfree (TTF) triples of modules play a crucial role in the description of Grothendieck categories with finiteness conditions. In order to answer a question posed by Auriel Djament on locally noetherian Grothendieck categories with exact products, we present some interesting results concerning hereditary TTF classes of modules, aiming to identify new sources of such categories from these classes on which to tackle the question.

For an additive functor on module categories (or more generally, on abelian categories with enough projectives or injectives), we construct what we call the left (respectively, right) long fundamental sequence, which ties together the classical homological constructs: the stabilization, the satellites, and the derived functors. For half-exact functors, these long fundamental sequences are exact. Applications include generalizations of the Auslander formulas to arbitrary modules over arbitrary rings. We also establish vast generalizations of the universal coefficient theorems in the form of universal theorems. 

We give a complete classification of factorizations of preordered groups in terms of quasi-projections..

The subprojectivity domains of modules were introduced as a new tool to measure, in some sense, the degree of projectivity of a module, rather than merely determining whether the module is projective or not. In this talk, we present some new results on the subprojectivity domains in relation to approximations (precovers and preenvelopes).

The ultraproduct construction can be used to produce interesting new objects, such as infinitesimals and new kinds of tensor categories.  Taking a cue from the latter, we form ultra- (and reduced) products of categories of modules over arbitrary rings (which can have more than one object).  We look at the effect on, for instance, lattices of pp formulas, Ziegler spectra, definable subcategories.

 Martsinkovsky and Russell proved that their torsion is classical torsion over all commutative domains. Generalizing this to arbitrary (not necessarily commutative) domains, I characterize those over which the same holds true.

This depends on the characterization of MR torsion as a specific pp torsion—as is part of ongoing work with A. Martsinkovsky. The proof uses two dichotomies on (unary) pp formulas I developed in preparatory work: high/bounded and low/cobounded.

We will present some results on self-orthogonal classes in a (Quillen) exact categories that naturally extend to this general setting results on Tilting Theory and Gorenstein Homological Algebra.

Let Λ be a Noetherian algebra  and C a Wakamatsu tilting module with Γ = End_Λ(C). We prove three main results. First, we establish a non-commutative Auslander–Bridger formula relating C-Gorenstein projective dimension and module depth, and apply it to show that finite Gorenstein projective dimension of C implies C is isomorphic to Λ under mild conditions, supporting the Wakamatsu tilting conjecture. Second, when Λ is semilocal, the left and right injective dimensions of C are at most n if and only if the C-Gorenstein projective dimension of every simple left Λ-module (or right Γ-module) is at most n; we also obtain a dual characterization of the projective dimension of C in terms of Auslander projective and Bass injective dimensions of simple modules. Third, when Λ = Γ is local, finiteness of the left and right injective dimensions of C is characterized via cotorsion pairs.

Finiteness of the little finitistic dimension of a finite dimensional algebra B is known to be equivalent to existence of a (possibly infinite dimensional) largest tilting B-module T_f. We use this equivalence to interpret recent surprising results of Cummings concerning the asymmetry of left and right finitistic dimensions of particular triangular matrix algebras B_A built from arbitrary basic finite dimensional algebras A. In particular, we determine the structure of the largest tilting B_A-module T_f in the case when A has finite global dimension.

Stable derivators provide a framework for homotopy theory intermediate between triangulated categories and higher-categorical models: they retain the triangulated structure of homotopy categories, while also including categories of coherent diagrams of arbitrary shapes, allowing for functorial notions of homotopy limits and colimits.  In this talk, we introduce a theory of presentability for stable derivators. Indeed, suppose D is a stable derivator enhancing the triangulated category T, and let α be an infinite regular cardinal. We call D α-presentable when its (homotopically) α-presentable objects form an essentially small class, and every object of T is the homotopy colimit of a coherent α-directed diagram with  α-presentable components; a definition that perfectly mirrors the classical notion of locally α-presentable category.  Our main result shows that this coherent notion recovers exactly the classical "generation" theory of triangulated categories: T is α-well-generated if and only if D is α-presentable. Moreover, the class of α-presentable objects coincides with that of α-compact objects, as defined by Neeman.  The key ingredient is a new Rectification Theorem which replaces suitable incoherent cellular filtrations of an object X in T by coherent directed diagrams in D having homotopy colimit X.  As an application, we extend Lurie's recognition criterion, which identifies the finitely presentable stable ∞-categories C as those whose triangulated homotopy category ho(C) is compactly generated. Indeed, we will show that a stable ∞-category C is α-presentable if and only if ho(C)  is α-well-generated. 

We study the balanced pairs of Cartan-Eilenberg complexes in the category of complexes with the Cartan-Eilenberg exact structure. Some results are given.