Homological Theory of Representations

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

Topics in the theory of trusses

In 2017 a truss was defined. Thus one can say that the theory of trusses is new and not yet well-established. In recent years trusses start to gain attention due to their connections to ring theory and braces. Braces are closely related to solutions of set-theoretic Yang-Baxter equations, which can lead to applications of trusses in physics. In this thesis, we study connections among groups, heaps, rings, modules, braces and trusses. In the beginning, one can find a description in details of free heaps and coproducts of Abelian heaps. Both constructions are applied to describe a functor from the category of heaps to the category of groups. We establish a connection between unital near-trusses and skew left braces. We show that for a specific choice of congruence on a unital near-truss the quotient is a brace. We also prove that if one localises a regular unital near-truss without an absorber, the result is a skew left brace. In this thesis, one can find many small results on categories of heaps, trusses and modules over a truss. Methods to extend trusses to unital trusses and rings are presented. Then first one allows us to show that a category of modules over a truss is isomorphic with the category of modules over its extension to the unital truss. The second method establishes a deep connection between rings and trusses, i.e. every truss is an equivalence class of some congruence on some specific ring. We present the ring construction. Using this result, we introduce the definition of a minimal extension of a truss into a ring. We construct tensor product and free modules over trusses. The Eilenberg-Watts theorem for modules over trusses is stated and proven. Thus the Morita theory for modules over trusses is developed. The thesis is concluded with results on projectivity and decompositions through a product of the modules.

Galois corings and groupoids acting partially on algebras

Bagio and Paques [Partial groupoid actions: globalization, Morita theory and Galois theory, Comm. Algebra 40 (2012) 3658–3678] developed a Galois theory for unital partial actions by finite groupoids. The aim of this note is to show that this is actually a special case of the Galois theory for corings, as introduced by Brzeziński [The structure of corings, Induction functors, Maschke-type theorem, and Frobenius and Galois properties, Algebr. Represent. Theory 5 (2002) 389–410]. To this end, we associate a coring to a unital partial action of a finite groupoid on an algebra [Formula: see text], and show that this coring is Galois if and only if [Formula: see text] is an [Formula: see text]-partial Galois extension of its coinvariants.

Graded Morita theory over a G-graded G-acted algebra

We develop a group graded Morita theory over a G-graded G-acted algebra, where G is a finite group.

Burnside rings for Real 2-representation theory: The linear theory

This paper is a fundamental study of the Real 2-representation theory of 2-groups. It also contains many new results in the ordinary (non-Real) case. Our framework relies on a 2-equivariant Morita bicategory, where a novel construction of induction is introduced. We identify the Grothendieck ring of Real 2-representations as a Real variant of the Burnside ring of the fundamental group of the 2-group and study the Real categorical character theory. This paper unifies two previous lines of inquiry, the approach to 2-representation theory via Morita theory and Burnside rings, initiated by the first author and Wendland, and the Real 2-representation theory of 2-groups, as studied by the second author.

REPETITIVE EQUIVALENCES AND TILTING THEORY

Let $R$ be a ring and $T$ be a good Wakamatsu-tilting module with $S=\text{End}(T_{R})^{op}$ . We prove that $T$ induces an equivalence between stable repetitive categories of $R$ and $S$ (i.e., stable module categories of repetitive algebras $\hat{R}$ and ${\hat{S}}$ ). This shows that good Wakamatsu-tilting modules seem to behave in Morita theory of stable repetitive categories as that tilting modules of finite projective dimension behave in Morita theory of derived categories.