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Tarantino, Marco (2018) Recollements from exact model structures and heart constructions in triangulated categories. [Ph.D. thesis]

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Abstract (italian or english)

We consider a complete hereditary cotorsion pair (A,B) in a Grothendieck category G such that A contains a generator of finite projective dimension. The derived category D(B) of the exact category B is defined as the quotient category of the category Ch(B), of unbounded cochain complexes with terms in B, modulo the subcategory tilde(B) consisting of acyclic complexes with terms and cycles in B.

We prove that there are two recollements anologous to the classical one, with middle term being respectively D(B) and K(B). We study also some cases where there is a recollement involving both K(B) and D(B). Simmetrically, we prove analogous results for the exact category A.

We also introduce the notion of Nakaoka contexts in additive categories as couples of torsion pairs t_1=(T_1,F_1) and t_2=(T_2,F_2) such that T_2 is included in T_1. We give a set of axioms for a Nakaoka context in order to ensure that the heart, i.e. the intersection between T_1 and F_2, is Abelian. Then, we inspect the properties of Nakaoka contexts in Abelian and triangulated categories.

In particular, given a t-structure t_1 in a triangulated category, we are able to find a bijection between the Nakaoka contexts (t_1,t_2) with Abelian heart and the cohereditary torsion pairs in the heart of t_1.

Abstract (a different language)

Consideriamo una cotorsion pair completa ed ereditaria (A,B) in una categoria di Grothendieck G tale che A contenga un generatore di dimensione proiettiva finita. La categoria derivata D(B) della categoria esatta B è definita come il quoziente fra la categoria Ch(B), dei complessi illimitati a termini in B, e la categoria tilde(B) dei complessi aciclici con termini e cicli in B.

Dimostreremo che vi sono due recollement analoghi al caso classico, ove il termine mediano è sostituito rispettivamente da D(B) e K(B). Studieremo anche alcuni casi dove vi sia un recollement che coinvolga entrambi K(B) e D(B). In maniera simmetrica, dimostreremo gli analoghi risultati per la categoria esatta A.

Introdurremo inoltre la nozione di contesto di Nakaoka in categorie additive come una coppia di torsion pair t_1=(T_1,F_1) e t_2=(T_2,F_2) tali che T_2 sia incluso in T_1. Dato un contesto di Nakaoka, saremo in grado di produrre un insieme di assiomi che, se soddisfatti, garantiscano l'abelianità del suo cuore, cioé dell'intersezione fra T_1 e F_2. Infine studeremo le proprietà dei contesti di Nakaoka in categorie Abeliane e triangolate.

In particolare, data una t-struttura t_1 in una categoria triangolata, saremo in grado di trovare una biiezione fra i contesti di Nakaoka (t_1,t_2) il cui cuore sia abeliano e le torsion pair coereditarie nel cuore di t_1.

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EPrint type:Ph.D. thesis
Tutor:Bazzoni, Silvana
Supervisor:Bazzoni, Silvana
Ph.D. course:Ciclo 31 > Corsi 31 > SCIENZE MATEMATICHE
Data di deposito della tesi:28 January 2019
Anno di Pubblicazione:30 November 2018
Key Words:Homological Algebra, Model Structures, Algebra Omologica, Strutture di Modello
Settori scientifico-disciplinari MIUR:Area 01 - Scienze matematiche e informatiche > MAT/02 Algebra
Struttura di riferimento:Dipartimenti > Dipartimento di Matematica
Codice ID:11729
Depositato il:06 Nov 2019 12:23
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