Some Researches On The Torsion Theories And Recollements Of Extriangulated Categories

In 2016,Nakaoka and Palu introduced the notion of extriangulated categories.Extriangulated category is a simultaneous generalization of exact category and triangulated category.The study of extriangulated category has become a hot topic in category theory.This dissertation concentrates on torsion theories and recollements of extriangulated categories.This dissertation mainly includes five parts.In the introduction,we introduce the background and research status related to this dissertation,and systematically explain the main results and framework.In Chapter One,we list the main notions and some known conclusions involved in this dissertation.In Chapter Two,we mainly study the relations between cotorsion pairs of an extriangulated categories and torsion pairs or(co-)t-structures of its quotient categories.Firstly,we introduce the notions of cohereditary cotorsion pair and projective(injective)cotorsion pair of an extriangulated category,and study their properties.Secondly,we prove that a projective cotorsion pair of a Frobenius extriangulated category can induce a torsion pair of its stable category.This result generalizes the existing work under the framework of triangulated category.Finally,we prove that there is a one-to-one correspondence between cohereditary cotorsion pairs of a Frobenius extriangulated category and t-structures of its stable category by using of the properties of cohereditary cotorsion pairs of extriangulated category and t-structures of triangulated category.Morever,we study the relation between bounded hereditary cotorsion pairs of a Frobenius extriangulated category and bounded co-t-structures of its stable category.In Chapter Three,we study the properties of filtration subcategories and recollements of extriangulated categories.Firstly,we study the relationship between filtration subcategories and(co)torsion pairs in an extriangulated category.Secondly,we study the relationship between filtration subcategories and recollements of extriangulated categories.In particular,let C’,C,C" be three extriangulated categories and C admits a recollement relative to C’and C",we prove that certain wide subcategories of C can induce wide subcategories of C’ and C",and then construct a recollement of the three wide subcategories.Finally,we give the definition of comparison of recollements of extriangulated categories,and study its properties.In particular,let(F’,F,F")be a comparison of recollements of extriangulated categories.We prove that if the functor F in the(F’,F,F")is equivalence,then the functors F’,F" are also equivalences.In Chapter Four,we study how a recollement of extriangulated categories induces a recollement of quotient categories.As an application,we study the relations of torsion pairs or t-structures among three stable categories in a recollement.