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We study relative homological algebra and relative Hochschild cohomology. We dualise the construction in [Cib+21b] for a ring extension B ⊆ A to construct a long nearly exact sequence for the relative Hochschild cohomology HH∗(A|B), the Hochschild cohomology HH∗(A) and the Hochschild cohomology HH∗(B,A). Parallel to this we also study corings and the associated Cartier cohomology and Hochschild cohomology. Given an A-coring C and its right algebra R we have induced maps ExtiA(M, N) → ExtiR(R⊗A M, R⊗A N) by the induction functor. We characterise the vanishing of the Hochschild cohomology of the coring in terms of these induced maps being isomorphisms for degrees greater than or equal to one.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Some of the diagram lemmas of Homological Algebra, classically known for abelian categories, are not characteristic of the abelian context; this naturally leads to investigations of those non-abelian categories in which these diagram lemmas may hold. In this Thesis we attempt to bring together two different directions of such investigations; in particular, we unify the five lemma from the context of homological categories due to F. Borceux and D. Bourn, and the five lemma from the context of modular semi-exact categories in the sense of M. Grandis. / AFRIKAANSE OPSOMMING: Verskeie diagram lemmata van Homologiese Algebra is aanvanklik ontwikkel in die konteks van abelse kategorieë, maar geld meer algemeen as dit behoorlik geformuleer word. Dit lei op ’n natuurlike wyse na ’n ondersoek van ander kategorieë waar hierdie lemmas ook geld. In hierdie tesis bring ons twee moontlike rigtings van ondersoek saam. Dit maak dit vir ons moontlik om die vyf-lemma in die konteks van homologiese kategoieë, deur F. Borceux en D. Bourn, en vyflemma in die konteks van semi-eksakte kategorieë, in die sin van M. Grandis, te verenig.

Neste trabalho, estudamos a cohomologia de de Rham e métodos para os seus cálculos. Finalizamos com aplicações da cohomologia de de Rham em teoremas da topologia / In this work we study the de Rham cohomology and methods for its calculations. We close it with applications of the Rham cohomology in theorems from topology

This thesis consists of three papers about N-complexes and their uses in categorification. N-complexes are generalizations of chain complexes having a differential d satisfying dN = 0 rather than d2 = 0. Categorification is the process of finding a higher category analog of a given mathematical structure. Paper I: We study a set of homology functors indexed by positive integers a and b and their corresponding derived categories. We show that there is an optimal subcategory in the domain of every functor given by N-complexes with N = a + b. Paper II: In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of N-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this in general and present evidence that the axioms of triangulated categories have a simplicial origin. Paper III: Let n be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of n:th cyclotomic integers.

This dissertation examines subfunctors of Ext relative to covering (enveloping) classes and the theory of covering (enveloping) ideals. The notion of covers and envelopes by modules was introduced independently by Auslander-Smalø and Enochs and has proven to be beneficial for module theory as well as for representation theory. The first few chapters examine the subfunctors of Ext and their properties. It is showed how the class of precoverings give us subfunctors of Ext. Furthermore, the characterization of these subfunctors and some examples are given. In the latter chapters ideals, the subfunctors of Hom, are investigated. The definition of cover and envelope carry over to the ideals naturally. Classical conditions for existence theorems for covers led to similar approaches in the ideal case. Even though some theorems such as Salce’s Lemma were proven to extend to ideals, most of the theorems do not directly apply to the new case. It is showed how Eklof & Trlifaj’s result can partially be extended to the ideals generated by a set. In that case, one also obtains a significant result about the orthogonal complement of the ideal. We relate the existence theorems for covering ideals of morphisms by identifying the morphisms with objects in A2 (which is the category of all representations of 2-quiver by R-modules) and obtain a sufficient condition for the existence of covering ideals in a more general setting. We finish with applying this result to the class of phantom morphisms.

Στην παρούσα εργασία παρουσιάζουμε βασικές έννοιες του αντικειμένου της Ομολογιακής Άλγεβρας,όπως αυτές των μακρών ακριβών ακολουθιών, τις επεκτάσεις των modules και τις ομάδες Ext. Στη συνέχεια παρουσιάζουμε τις επεκτάσεις των ομολογιακών και συνομολογιακών συναρτητών, τους παράγωγους συναρτητές, που προκύπτουν μέσω προβολικών και ενριπτικών επιλύσεων αντικειμένων Αβελιανών κατηγοριών. Τέλος χαρακτηρίζουμε τους παράγωγους συναρτητές μέσω της καθολικής τους ιδιότητας. / In this work we present the basic concepts of Homological Algebra, such as the long exact sequences, the extensions of modules and the Ext-groups.Further we present the extensions of the homology and cohomology functors, the derived functors that arise through projective and injective resolutions from objects of an Abelian category. Finally we characterize derived functors through their universal property.

Orientador: Dessislava Hristo Kochloukova / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-11T16:37:34Z (GMT). No. of bitstreams: 1 Cariello_Daniel_M.pdf: 483301 bytes, checksum: 1ec393c194e31aecf24d1e555c11df76 (MD5) Previous issue date: 2008 / Resumo: Esse trabalho é sobre a conjectura FPm para grupos metabelianos, para m = 2, 3. Estudamos a conjectura e o invariante homológico 'SGMA' 'POT .0'(Q,M). Uma das implicações da conjectura FP2 está demonstrada no capítulo 4, para o caso do grupo metabeliano ser extensão cindida. No último capítulo damos um idéia de como estender essa demonstração para demonstrar o mesmo resultado em dimensão 3. / Abstract: This work is about the FPm-conjecture for metabelian groups, when m = 2, 3. We study the conjecture and the homological invariant 'SGMA' 'POT .0'(Q,M). One of the implications of the FP2-conjecture is proved in chapter 4, when the metabelian group is a split extension. In the last chapter, we expose some ideas to extend our proof to dimension 3. / Mestrado / Algebra / Mestre em Matemática

Neste trabalho, estudamos a cohomologia de de Rham e métodos para os seus cálculos. Finalizamos com aplicações da cohomologia de de Rham em teoremas da topologia / In this work we study the de Rham cohomology and methods for its calculations. We close it with applications of the Rham cohomology in theorems from topology

The main theme of the thesis is to present and compare three different viewpoints on semi-abelian homology, resulting in three ways of defining and calculating homology objects. Any two of these three homology theories coincide whenever they are both defined, but having these different approaches available makes it possible to choose the most appropriate one in any given situation, and their respective strengths complement each other to give powerful homological tools. The oldest viewpoint, which is borrowed from the abelian context where it was introduced by Barr and Beck, is comonadic homology, generating projective simplicial resolutions in a functorial way. This concept only works in monadic semi-abelian categories, such as semi-abelian varieties, including the categories of groups and Lie algebras. Comonadic homology can be viewed not only as a functor in the first entry, giving homology of objects for a particular choice of coefficients, but also as a functor in the second variable, varying the coefficients themselves. As such it has certain universality properties which single it out amongst theories of a similar kind. This is well-known in the setting of abelian categories, but here we extend this result to our semi-abelian context. Fixing the choice of coefficients again, the question naturally arises of how the homology theory depends on the chosen comonad. Again it is well-known in the abelian case that the theory only depends on the projective class which the comonad generates. We extend this to the semi-abelian setting by proving a comparison theorem for simplicial resolutions. This leads to the result that any two projective simplicial resolutions, the definition of which requires slightly more care in the semi-abelian setting, give rise to the same homology. Thus again the homology theory only depends on the projective class. The second viewpoint uses Hopf formulae to define homology, and works in a non-monadic setting; it only requires a semi-abelian category with enough projectives. Even this slightly weaker setting leads to strong results such as a long exact homology sequence, the Everaert sequence, which is a generalised and extended version of the Stallings-Stammbach sequence known for groups. Hopf formulae use projective presentations of objects, and this is closer to the abelian philosophy of using any projective resolution, rather than a special functorial one generated by a comonad. To define higher Hopf formulae for the higher homology objects the use of categorical Galois theory is crucial. This theory allows a choice of Birkhoff subcategory to generate a class of central extensions, which play a big role not only in the definition via Hopf formulae but also in our third viewpoint. This final and new viewpoint we consider is homology via satellites or pointwise Kan extensions. This makes the universal properties of the homology objects apparent, giving a useful new tool in dealing with statements about homology. The driving motivation behind this point of view is the Everaert sequence mentioned above. Janelidze's theory of generalised satellites enables us to use the universal properties of the Everaert sequence to interpret homology as a pointwise Kan extension, or limit. In the first instance, this allows us to calculate homology step by step, and it removes the need for projective objects from the definition. Furthermore, we show that homology is the limit of the diagram consisting of the kernels of all central extensions of a given object, which forges a strong connection between homology and cohomology. When enough projectives are available, we can interpret homology as calculating fixed points of endomorphisms of a given projective presentation.