The Research Of Relative Syzygy Objects And Silting Pair

In this dissertation,we mainly study the homology properties of relative syzygy objects in Ex-triangulated categories and silting pairs in triangulated categories.Firstly,we introduce some research background and significance of Ex-triangulated categories and triangulated categories.Secondly,we introduce the notation of relative syzygy objects in Ex-triangulated categories.Using the method of relative homology,we obtain some properties of relative syzygy objects and its equivalent characterizations,generalize some results in the Abelian categories and triangular categories,and give an equivalent characterization of relative projective dimensions.In Ex-triangulated categories and Recollements R(A’,A,A"),it is proved that the relative syzygy objects in A’ and A" can induce the relative syzygy objects in A。Conversely,the relative syzygy objects in A can induce the relative syzygy objects in A’ and A".Finally,as a generalization of tilting pairs[1],we introduce the concept of silting pairs(semi-tilting pairs).Similar to Bazzoni’s equivalent characterization of tilting modules[2],we give an equivalent characterization of silting pairs.The main result is that there exists a bijection between the equivalence class of silting pairs,special subcategories that satisfy certain conditions,and upper bounded co-t-structures.