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.

Total integrals of Doi Hom-Hopf modules

2016 ◽  
Vol 15 (04) ◽  
pp. 1650069
Author(s):  
Shuangjian Guo ◽  
Xiaohui Zhang ◽  
Shengxiang Wang
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Integral Formula ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
Total Integral

Let [Formula: see text] be a monoidal Hom-Hopf algebra, [Formula: see text] a right [Formula: see text]-Hom-comodule algebra and [Formula: see text] a right [Formula: see text]-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of [Formula: see text] in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral [Formula: see text] if and only if any representation of the pair [Formula: see text] is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of [Formula: see text]-Doi Hom-Hopf modules a result of Doi on projectivity of every relative [Formula: see text]-Hopf module as an [Formula: see text]-module.

RELATIVE PROJECTIVITY AND RELATIVE INJECTIVITY IN THE CATEGORY OF DOI–HOPF MODULES

2011 ◽  
Vol 10 (05) ◽  
pp. 931-946 ◽  
Author(s):  
T. GUÉDÉNON
Keyword(s):  
Hopf Algebra ◽  
Fundamental Theorem ◽  
Hopf Module ◽  
Comodule Algebra ◽  
Hopf Modules

Let k be a field, H be a Hopf algebra, A be a right H-comodule algebra and C be a right H-module coalgebra. We extend to the category of (H, A, C)-Doi–Hopf modules a result of Doi on projectivity of every relative (A, H)-Hopf module as an A-module. We also extend the Fundamental Theorem of [C, H]-Hopf modules due to Doi to the category of (H, A, C)-Doi–Hopf modules. Then we discuss relative projectivity and relative injectivity in this category.

scholarly journals THE RELATIVE PICARD GROUP OF A COMODULE ALGEBRA AND HARRISON COHOMOLOGY

2005 ◽  
Vol 48 (3) ◽  
pp. 557-569 ◽  
Author(s):  
S. Caenepeel ◽  
T. Guédénon
Keyword(s):  
Hopf Algebra ◽  
Quotient Group ◽  
Cohomology Group ◽  
Picard Group ◽  
Base Extension ◽  
Comodule Algebra ◽  
Hopf Modules ◽  
K Theory ◽  
Algebra Over A Field

AbstractLet $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible group-like elements of the coring $A\otimes H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a cosemisimple Hopf algebra over a field.

scholarly journals Comodule algebras and 2-cocycles over the (Braided) Drinfeld double

2019 ◽  
Vol 21 (04) ◽  
pp. 1850045 ◽  
Author(s):  
Robert Laugwitz
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Monoidal Category ◽  
Crossed Product ◽  
Braided Monoidal Category ◽  
Drinfeld Double ◽  
Comodule Algebra ◽  
Category Of Modules ◽  
Cohomology Spaces ◽  
Quasitriangular Hopf Algebra

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

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 Co-stability of Radicals and Its Applications to PI-Theory

2016 ◽  
Vol 23 (03) ◽  
pp. 481-492 ◽  
Author(s):  
A. S. Gordienko
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jacobson Radical ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Comodule Algebra ◽  
Graded Polynomial Identities ◽  
Finite Dimensional Lie Algebras ◽  
Algebra Over A Field

We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.

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 On the cohomology of comodules over smash coproducts

2019 ◽  
Vol 18 (10) ◽  
pp. 1950189
Author(s):  
S. Caenepeel ◽  
T. Guédénon
Keyword(s):  
Spectral Sequence ◽  
Spectral Sequences ◽  
Derived Functors

We consider the category of comodules over a smash coproduct coalgebra [Formula: see text]. We show that there is a Grothendieck spectral sequence connecting the derived functors of the Hom functors coming from [Formula: see text]-colinear, [Formula: see text]-colinear and rational [Formula: see text]-colinear morphisms. We give several applications and connect our results to existing spectral sequences in the literature.

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