Category Of Comodules Research Articles

Tor-tilting and weak silting theories in comodule categories

In this paper, we introduce and investigate Tor-tilting comodules and weak silting comodules, providing a generalization of the established tilting and silting theories in the comodule category. Our exploration yields a comprehensive understanding of their fundamental properties, including their associated torsion theories. We delve into not only the relations between Tor-tilting and weak silting comodules but also the connections among them and (co) tilting as well as (co) silting comodules. Additionally, we define Tor-tilting coflat comodules and Tor-tilting coflat dimensions. Finally, we explore Tor-tilting and weak silting objects in the morphism category, demonstrating their equivalence.

Homotopy Quotients and Comodules of Supercommutative Hopf Algebras

We study model structures on the category of comodules of a supercommutative Hopf algebra A over fields of characteristic 0. Given a graded Hopf algebra quotient A→B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A \\rightarrow B$$\\end{document} satisfying some finiteness conditions, the Frobenius tensor category D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {D}}$$\\end{document} of graded B-comodules with its stable model structure induces a monoidal model structure on C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document}. We consider the corresponding homotopy quotient γ:C→HoC\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : {\\mathcal {C}} \\rightarrow Ho {\\mathcal {C}}$$\\end{document} and the induced quotient T→HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}} \\rightarrow Ho {\\mathcal {T}}$$\\end{document} for the tensor category T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {T}}$$\\end{document} of finite dimensional A-comodules. Under some mild conditions we prove vanishing and finiteness theorems for morphisms in HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho {\\mathcal {T}}$$\\end{document}. We apply these results in the Rep(GL(m|n))-case and study its homotopy category HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho {\\mathcal {T}}$$\\end{document} associated to the parabolic subgroup of upper triangular block matrices. We construct cofibrant replacements and show that the quotient of HoT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Ho{\\mathcal {T}}$$\\end{document} by the negligible morphisms is again the representation category of a supergroup scheme.

The stable Picard group of finite Adams Hopf algebroids with an application to the [formula omitted]-motivic Steenrod subalgebra [formula omitted]

In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as finite Adams, shedding light on its Picard group. Then, we establish a reduction process through base changes, enabling us to effectively compute the Picard group of the ▪-motivic mod 2 Steenrod subalgebra▪. Our computation shows that ▪ is isomorphic to ▪, where two ranks come from the motivic grading, one from the algebraic loop functor, and the last is generated by the ▪-motivic joker J.

Ranicki–Weiss assembly and the Steenrod construction

It is shown that the Ranicki–Weiss assembly functor, going from chain complex valued presheaves on a simplicial complex to comodules over its Alexander–Whitney coalgebra, factors fully faithfully through the category of comodules over its Steenrod cup- i i coalgebra.

A comonadicity theorem for partial comodules

We show that the category of partial comodules over a Hopf algebra H is comonadic over Vectk and provide an explicit construction of this comonad using topological vector spaces. The case when H is finite-dimensional is treated in detail. A study of partial representations of linear algebraic groups is initiated; we show that a connected linear algebraic group does not admit partiality.

Extensions of the category of comodules of the Taft algebra

We construct a family of non-equivalent pairwise extensions of the category of comodules of the Taft algebra, which are equivalent to representation categories of non-triangular quasi-Hopf algebras.

Comodule theories in Grothendieck categories and relative Hopf objects

We develop the categorical algebra of the noncommutative base change of a comodule category by means of a Grothendieck category S. We describe when the resulting category of comodules is locally finitely generated, locally noetherian or may be recovered as a coreflective subcategory of the noncommutative base change of a module category. We also introduce the category SHA of relative (A,H)-Hopf modules in S, where H is a Hopf algebra and A is a right H-comodule algebra. We study the cohomological theory in SHA by means of spectral sequences. Using coinduction functors and functors of coinvariants, we study torsion theories and how they relate to injective resolutions in SHA. Finally, we use the theory of associated primes and support in noncommutative base change of module categories to give direct sum decompositions of minimal injective resolutions in SHA.

Projectivity and subprojectivity domains in exact categories

We generalize to exact categories some new perspectives on projectivity, which allow us to unify a series of recent concepts from module categories or abelian categories and to obtain automatically dual results. We discuss objects having a minimal subprojectivity domain in exact categories, and we study properties of such domains, including several closure properties. We give applications to (possibly non-abelian) quasi-abelian categories, finitely accessible and exactly definable additive categories, and in particular, to module and comodule categories.

Categories of modules, comodules and contramodules over representations

Abstract We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.

Cohomology of Graded Twisting of Hopf Algebras

Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms.

Pure-direct-objects in categories: transfer via functors

We study transfer of pure-direct-objects via functors between Grothendieck categories. We consider fully faithful functors and adjoint pairs of functors. We give applications to module categories and comodule categories.

Centres, trace functors, and cyclic cohomology

We study the biclosedness of the monoidal categories of modules and comodules over a (left or right) Hopf algebroid, along with their bimodule category centres over the respective opposite categories and a corresponding categorical equivalence to anti Yetter-Drinfel’d contramodules and anti Yetter-Drinfel’d modules, respectively. This is directly connected to the existence of a trace functor on the monoidal categories of modules and comodules in question, which in turn allows to recover (or define) cyclic operators enabling cyclic cohomology.

Koszul duality in higher topoi

We show that there is an equivalence in any $n$-topos $\mathcal{X}$ between the pointed and $k$-connective objects of $\mathcal{X}$ and the $\mathbb{E}_k$-group objects of $\mathcal{X}$. We also show that for any pointed and $k$-connective object $X$ of $\mathcal{X}$ there is an equivalence between the $\infty$-category of modules in $\mathcal{X}$ over the associative algebra $\Omega^k X$, and the $\infty$-category of comodules in $\mathcal{X}$ for the cocommutative coalgebra $\Omega^{k-1}X$. In doing so, we prove several results which may be of broader interest: we show that Lurie's straightening-unstraightening equivalence holds over an $(n-1)$-groupoid in any $n$-topos for $0\leq n\leq\infty$; we show that taking categories of $\mathcal{O}$-algebras, for $\mathcal{O}$ an $\infty$-operad, commutes with truncation; and we show that for a pointed and connected $(n-1)$-groupoid $Y$ there is an equivalence between $\Omega Y$-modules in $\mathcal{X}$ and $\mathcal{X}$-valued presheaves on $Y$.

CS-Rickart and dual CS-Rickart objects in abelian categories

We introduce relative CS-Rickart objects in abelian categories, as common generalizations of relative Rickart objects and extending objects. We study direct summands and (co)products of relative CS-Rickart objects as well as classes all of whose objects are self-CS-Rickart. Corresponding results for dual relative CS-Rickart objects may be automatically obtained by the duality principle. Applications are given to Grothendieck categories and, in particular, to module and comodule categories.

Categorical Koszul duality

In this paper we establish Koszul duality between dg categories and a class of curved coalgebras, generalizing the corresponding result for dg algebras and conilpotent curved coalgebras. We show that the normalized chain complex functor transforms the Quillen equivalence between quasicategories and simplicial categories into this Koszul duality. This allows us to give a conceptual interpretation of the dg nerve of a dg category and its adjoint. As an application, we prove that the category of representations of a quasicategory K is equivalent to the coderived category of comodules over C⁎(K), the chain coalgebra of K. A corollary of this is a characterization of the category of constructible dg sheaves on a stratified space as the coderived category of a certain dg coalgebra.

Graded comodule categories with enough projectives

AbstractIt is well known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper, we prove that the category of connective graded comodules over a connective, graded, flat, finite-type Hopf algebroid has enough projectives. Applications to algebraic topology are given: the Hopf algebroids of stable co-operations in complex bordism, Brown–Peterson homology, and classical mod p homology all have the property that their categories of connective graded comodules have enough projectives. We also prove that categories of connective graded comodules over appropriate Hopf algebras fail to be equivalent to categories of graded connective modules over a ring.

On the monoidal invariance of the cohomological dimension of Hopf algebras

We discuss the question of whether the global dimension is a monoidal invariant for Hopf algebras, in the sense that if two Hopf algebras have equivalent monoidal categories of comodules, then their global dimensions should be equal. We provide several positive new answers to this question, under various assumptions of smoothness, cosemisimplicity or finite dimension. We also discuss the comparison between the global dimension and the Gerstenhaber-Schack cohomological dimension in the cosemisimple case, obtaining equality in the case the latter is finite. One of our main tools is the new concept of twisted separable functor.

Split objects with respect to a fully invariant short exact sequence in abelian categories

We introduce and investigate (dual) relative split objects with respect to a fully invariant short exact sequence in abelian categories. We compare them with (dual) relative Rickart objects, and we study their behaviour with respect to coproducts and classes all of whose objects are (dual) relative split. We also introduce and study (dual) strongly relative split objects. Applications are given to Grothendieck categories, module categories, and comodule categories.

On the graded algebras associated with Hecke symmetries, II. The Hilbert series

Hecke symmetries give rise to a family of graded algebras which represent quantum groups and spaces of noncommutative geometry. The present paper continues the work aiming to understand general properties of these algebras without a restriction on the parameter q of Hecke relation used in earlier results. However, if q is a root of 1, we need a restriction on the indecomposable modules for the Hecke algebras of type A that can occur as direct summands of representations in the tensor powers of the initial vector space V. In this setting, we generalize known results on rationality of Hilbert series. The combinatorial nature of this problem stems from a relationship between the Grothendieck ring of the category of comodules for the Faddeev–Reshetikhin–Takhtajan bialgebra A(R) associated with a Hecke symmetry R and the ring of symmetric functions. We then improve two results on monoidal equivalences of corepresentation categories and on Gorensteinness of graded algebras from a previous article.

Algebraic structures in comodule categories over weak bialgebras

For a bialgebra L coacting on a -algebra A, a classical result states A is a right L-comodule algebra if and only if A is an algebra in the category of right L-comodules. We generalize this to the setting of weak bialgebras H. We prove that there is an isomorphism between the categories of right H-comodule algebras and the category of algebras in . We also recall and introduce the formulaic notion of H coacting on a -coalgebra and on a Frobenius -algebra, respectively, and prove analogous category isomorphisms. Many examples are provided throughout.