scholarly journals Adjoint functors and derived functors with an application to the cohomology of semigroups

1967 ◽  
Vol 7 (1) ◽  
pp. 25-34 ◽  
Author(s):  
William W Adams ◽  
Marc A Rieffel
Keyword(s):  
Adjoint Functors ◽  
Derived Functors

scholarly journals Localization, Isomorphisms and Adjoint Isomorphism in the Category Comp(A - Mod)

2020 ◽  
Vol 12 (4) ◽  
pp. 65
Author(s):  
Bassirou Dembele ◽  
Mohamed Ben Faraj Ben Maaouia ◽  
Mamadou Sanghare
Keyword(s):  
Commutative Rings ◽  
Adjoint Functors ◽  
Derived Functors

A and B are considered to be non necessarily commutative rings and X a complex of (A - B) bimodules. The aim of this paper is to show that: The functors \overline{EXT}^n_{Comp(A-Mod)}(X,-): Comp(A-Mod) \longrightarrow Comp(B-Mod) and Tor_n^{Comp(B-Mod)}(X,-): Comp(B-Mod) \longrightarrow Comp(A-Mod) are adjoint functors. The  functor S_C^{-1}() commute with  the functors X\bigotimes - , Hom^{\bullet}(X,-) and their corresponding derived functors  \overline{EXT}^n_{Comp(A-Mod)}(X,-) and  Tor_n^{Comp(B-Mod)}(X,-).

Classes of extensions and resolutions

1961 ◽  
Vol 254 (1039) ◽  
pp. 155-222 ◽  
Keyword(s):  
Vector Bundle ◽  
General Theory ◽  
Abelian Category ◽  
Homology Theory ◽  
Spectral Sequences ◽  
Adjoint Functors ◽  
Dedekind Domains ◽  
Derived Functors ◽  
Natural Transformations ◽  
Natural Classes

A class of resolutions of objects of an abelian category determines a theory of derived functors if each morphism between objects extends to a morphism, unique to within homotopies, between their resolutions. This paper is primarily concerned with resolutions canonically associated with certain natural classes of extensions (E-functors), and the known examples are constructed by using pairs of adjoint functors. An inclusion between two E-functors on the same category induces natural transformations between functors derived from their associated resolutions, and other relations exist in the form of invariant exact couples. The relations simplify for the special and frequently occurring class of ‘central’ inclusions of E-functors; in particular the operations of forming satellites of a functor on the two resolutions commute. Amongst various applications the general theory provides generalizations of: results on groups of extensions of modules over Dedekind domains; the Hochschild—Serre spectral sequences in the homology theory of groups; the spectral sequences for coherent algebraic sheaves that determine Ext by means of vector bundle resolutions and affine coverings.

scholarly journals Closed categories generated by commutative monads

1971 ◽  
Vol 12 (4) ◽  
pp. 405-424 ◽  
Author(s):  
Anders Kock
Keyword(s):  
Adjoint Functors ◽  
Closed Category ◽  
Base Category

The notion of commutative monad was defined by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the front- and end-adjunctions are closed transformations. (The terms ‘Closed Category’ etc. are from the paper [2] by Eilenberg and Kelly). In particular, the monad itself is a ‘closed monad’; this fact was also proved in [4].

scholarly journals On the Derived Functors of the Third Symmetric-Power Functor

2009 ◽  
Vol 2 (0) ◽  
pp. 19-39
Author(s):  
Bernhard Köck ◽  
Ramesh Satkurunath
Keyword(s):  
Symmetric Power ◽  
The Third ◽  
Derived Functors ◽  
Power Functor
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