On the unitarizability of derived functor modules

1984 ◽  
Vol 78 (1) ◽  
pp. 131-141 ◽  
Author(s):  
Nolan R. Wallach
Keyword(s):  
Derived Functor

On copure projective and cotorsion modules

2019 ◽  
Vol 56 (1) ◽  
pp. 1-12
Author(s):  
Wei Ren ◽  
Duocai Zhang
Keyword(s):  
Noetherian Rings ◽  
Derived Functor ◽  
Flat Modules

Abstract Let R be an IF ring, or be a ring such that each right R-module has a monomorphic flat envelope and the class of flat modules is coresolving. We firstly give a characterization of copure projective and cotorsion modules by lifting and extension diagrams, which implies that the classes of copure projective and cotorsion modules have some balanced properties. Then, a relative right derived functor is introduced to investigate copure projective and cotorsion dimensions of modules. As applications, some new characterizations of QF rings, perfect rings and noetherian rings are given.

scholarly journals Localisation and colocalisation of triangulated categories at thick subcategories

2012 ◽  
Vol 110 (1) ◽  
pp. 59 ◽  
Author(s):  
Hvedri Inassaridze ◽  
Tamaz Kandelaki ◽  
Ralf Meyer
Keyword(s):  
Exact Sequence ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Canonical Map ◽  
Derived Functor ◽  
Homological Functor ◽  
Thick Subcategory

Given a thick subcategory of a triangulated category, we define a colocalisation and a natural long exact sequence that involves the original category and its localisation and colocalisation at the subcategory. Similarly, we construct a natural long exact sequence containing the canonical map between a homological functor and its total derived functor with respect to a thick subcategory.

On Küneth’s correlation and its applications

2019 ◽  
Vol 26 (2) ◽  
pp. 295-301
Author(s):  
Leonard Mdzinarishvili
Keyword(s):  
Exact Sequence ◽  
Hausdorff Space ◽  
Chain Complex ◽  
Abelian Category ◽  
Derived Functor ◽  
Direct System ◽  
Arbitrary Complex ◽  
Formula Group ◽  
Čech Homology ◽  
A Chain

Abstract Let {\mathcal{K}} be an abelian category that has enough injective objects, let {T\colon\mathcal{K}\to A} be any left exact covariant additive functor to an abelian category A and let {T^{(i)}} be a right derived functor, {u\geq 1} , [S. Mardešić, Strong Shape and Homology, Springer Monogr. Math., Springer, Berlin, 2000]. If {T^{(i)}=0} for {i\geq 2} and {T^{(i)}C_{n}=0} for all {n\in\mathbb{Z}} , then there is an exact sequence 0\longrightarrow T^{(1)}H_{n+1}(C_{*})\longrightarrow H_{n}(TC_{*})% \longrightarrow TH_{n}(C_{*})\longrightarrow 0, where {C_{*}=\{C_{n}\}} is a chain complex in the category {\mathcal{K}} , {H_{n}(C_{*})} is the homology of the chain complex {C_{*}} , {TC_{*}} is a chain complex in the category A, and {H_{n}(TC_{*})} is the homology of the chain complex {TC_{*}} . This exact sequence is the well known Künneth’s correlation. In the present paper Künneth’s correlation is generalized. Namely, the conditions are found under which the infinite exact sequence \displaystyle\cdots\longrightarrow T^{(2i+1)}H_{n+i+1}\longrightarrow\cdots% \longrightarrow T^{(3)}H_{n+2}\longrightarrow T^{(1)}H_{n+1}\longrightarrow H_% {n}(TC_{*}) \displaystyle\longrightarrow TH_{n}(C_{*})\longrightarrow T^{(2)}H_{n+1}% \longrightarrow T^{(4)}H_{n+2}\longrightarrow\cdots\longrightarrow T^{(2i)}H_{% n+i}\longrightarrow\cdots holds, where {T^{(2i+1)}H_{n+i+1}=T^{(2i+1)}H_{n+i+1}(C_{*})} , {T^{(2i)}H_{n+i}=T^{(2i)}H_{n+i}(C_{*})} . The formula makes it possible to generalize Milnor’s formula for the cohomologies of an arbitrary complex, relatively to the Kolmogorov homology to the Alexandroff–Čech homology for a compact space, to a generative result of Massey for a local compact Hausdorff space X and a direct system {\{U\}} of open subsets U of X such that {\overline{U}} is a compact subset of X.

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