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.

Derived Categories: A Quick Tour

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Abelian Category ◽  
Derived Categories ◽  
Derived Category ◽  
Homotopy Category ◽  
Intermediate Step ◽  
Spectral Sequences ◽  
Separate Section ◽  
Derived Functors ◽  
Left And Right

This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.

scholarly journals Cohomology of Modules Over -categories and Co--categories

2019 ◽  
Vol 72 (5) ◽  
pp. 1352-1385
Author(s):  
Mamta Balodi ◽  
Abhishek Banerjee ◽  
Samarpita Ray
Keyword(s):  
Hopf Algebra ◽  
Module Category ◽  
Spectral Sequences ◽  
Locally Finite ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
Derived Functors

AbstractLet $H$ be a Hopf algebra. We consider $H$-equivariant modules over a Hopf module category ${\mathcal{C}}$ as modules over the smash extension ${\mathcal{C}}\#H$. We construct Grothendieck spectral sequences for the cohomologies as well as the $H$-locally finite cohomologies of these objects. We also introduce relative $({\mathcal{D}},H)$-Hopf modules over a Hopf comodule category ${\mathcal{D}}$. These generalize relative $(A,H)$-Hopf modules over an $H$-comodule algebra $A$. We construct Grothendieck spectral sequences for their cohomologies by using their rational $\text{Hom}$ objects and higher derived functors of coinvariants.

The natural transformations between T-th order prolongation of tangent and cotangent bundles over Riemannian manifolds

2015 ◽  
Vol 69 (1) ◽  
Author(s):  
Mariusz Plaszczyk
Keyword(s):  
Riemannian Manifold ◽  
Vector Bundle ◽  
Riemannian Manifolds ◽  
Present Note ◽  
Cotangent Bundles ◽  
Natural Transformations ◽  
The One

AbstractIf (M,g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T* M given by v → g(v,−) between the tangent TM and the cotangent T* M bundles of M. In the present note first we generalize this isomorphism to the one J

scholarly journals CALCULATING BAR-NATAN'S CHARACTERISTIC TWO KHOVANOV HOMOLOGY

2006 ◽  
Vol 15 (10) ◽  
pp. 1335-1356 ◽  
Author(s):  
PAUL R. TURNER
Keyword(s):  
Homology Theory ◽  
Khovanov Homology ◽  
Spectral Sequences ◽  
Link Homology ◽  
Characteristic Two

We investigate Bar-Natan's characteristic two Khovanov link homology theory studying both the filtered and bi-graded theories. The filtered theory is computed explicitly and the bi-graded theory analysed by setting up a family of spectral sequences. The E2-pages can be described in terms of groups arising from the action of a certain endomorphism on 𝔽2-Khovanov homology. Some simple consequences are discussed.

The theory of semi-functors

1993 ◽  
Vol 3 (1) ◽  
pp. 93-128 ◽  
Author(s):  
Raymond Hoofman
Keyword(s):  
General Theory ◽  
Natural Transformation ◽  
Lambda Calculus ◽  
Adjoint Functor ◽  
Typed Lambda Calculus ◽  
The World ◽  
Natural Transformations ◽  
Theoretical Characterization

The notion ofsemi-functorwas introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion ofsemi natural transformationplays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion ofnormal semi-adjunctionas defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.

scholarly journals On Some Local Cohomology Spectral Sequences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6197-6293 ◽  
Author(s):  
Josep Àlvarez Montaner ◽  
Alberto F Boix ◽  
Santiago Zarzuela
Keyword(s):  
Local Cohomology ◽  
Sufficient Conditions ◽  
Inverse System ◽  
Spectral Sequences ◽  
Local Cohomology Modules ◽  
Decomposition Formula ◽  
Derived Functors ◽  
Finite Partially Ordered Set ◽  
The Right ◽  
Decomposition Theorems

Abstract We introduce a formalism to produce several families of spectral sequences involving the derived functors of the limit and colimit functors over a finite partially ordered set. The 1st type of spectral sequences involves the left derived functors of the colimit of the direct system that we obtain by applying a family of functors to a single module. For the 2nd type we follow a completely different strategy as we start with the inverse system that we obtain by applying a covariant functor to an inverse system. The spectral sequences involve the right derived functors of the corresponding limit. We also have a version for contravariant functors. In all the introduced spectral sequences we provide sufficient conditions to ensure their degeneration at their 2nd page. As a consequence we obtain some decomposition theorems that greatly generalize the well-known decomposition formula for local cohomology modules of Stanley–Reisner rings given by Hochster.

scholarly journals Modified Traces and the Nakayama Functor

2021 ◽  
Author(s):  
Taiki Shibata ◽  
Kenichi Shimizu
Keyword(s):  
Existence And Uniqueness ◽  
Natural Transformation ◽  
Tensor Category ◽  
Abelian Category ◽  
Module Category ◽  
Module Structure ◽  
Exact Functor ◽  
Abelian Categories ◽  
Natural Transformations ◽  
Uniqueness Criteria

AbstractWe organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor Σ on a finite abelian category ${\mathscr{M}}$ M , we introduce the notion of a Σ-twisted trace on the class $\text {Proj}({\mathscr{M}})$ Proj ( M ) of projective objects of ${\mathscr{M}}$ M . In our framework, there is a one-to-one correspondence between the set of Σ-twisted traces on $\text {Proj}({\mathscr{M}})$ Proj ( M ) and the set of natural transformations from Σ to the Nakayama functor of ${\mathscr{M}}$ M . Non-degeneracy and compatibility with the module structure (when ${\mathscr{M}}$ M is a module category over a finite tensor category) of a Σ-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.

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,-).

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