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Gabriella Böhm

Publications and source records attributed to Gabriella Böhm.

46 records · Page 3Linked to original sources
arXiv2008

Morita theory for coring extensions and cleft bicomodules

A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $Σ$ for a pure coring extension $(\cD:L)$ of $(\cC:A)$. It is related to a 2-object subcategory of the category of $k$-linear functors $\Mm^\Cc\to\Mm^\Dd$. Strictness of the Morita context is shown to imply the Galois property of $Σ$ as a $\cC$-comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an $L$-$\cC$ bicomodule $Σ$ -- implying strictness of the associated Morita context -- is introduced. It is shown to be equivalent to being a Galois $\cC$-comodule and isomorphic to $\End^\cC(Σ)\otimes_{L} \cD$, in the category of left modules for the ring $\End^\cC(Σ)$ and right comodules for the coring $\cD$, i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a pure Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules. Cleft extensions by arbitrary Hopf algebroids are described in terms of Morita contexts that do not necessarily correspond to coring extensions.

math.RA↗
arXiv2008

Galois extensions over commutative and non-commutative base

This paper is a written form of a talk. It gives a review of various notions of Galois (and in particular cleft) extensions. Extensions by coalgebras,bialgebras and Hopf algebras (over a commutative base ring) and by corings,bialgebroids and Hopf algebroids (over a non-commutative base algebra) are systematically recalled and compared. In the first version of this paper, the journal version of [15, Theorem 2.6] was heavily used, in two respects. First, it was applied to establish an isomorphism between the comodule categories of two constituent bialgebroids in a Hopf algebroid. Second, it was used to construct a Morita context for any bicomodule for a coring extension. Regrettably, it turned out that the proof of [15, Theorem 2.6] contains an unjustified step. Therefore, our derived results are not expected to hold at the stated level of generality either. In the revised version we make the necessary corrections in both respects. In doing so, we obtain a corrected version of \cite[5, Theorem 4.2] as well, whose original proof contains a very similar error to [15, Theorem 2.6].

math.QA↗
arXiv2008

A Schneider type theorem for Hopf algebroids

Comodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B\subseteq A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the non-commutative base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois property of the extension B\subseteq A and also to the equivalence of the category of relative Hopf modules to the category of B-modules. This extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf algebra. Our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor, a notion introduced and analysed hereby. In the first version of this submission, we heavily used the statement that two constituent bialgebroids in a Hopf algebroid possess isomorphic comodule categories. This statement was based on \cite[Brz3,Theorem 2.6], whose proof turned out to contain an unjustified step. In the revised version we return to an earlier definition of a comodule of a Hopf algebroid, that distinguishes between comodules of the two constituent bialgebroids, and modify the statements and proofs in the paper accordingly.

math.QA↗
arXiv2008

Morita theory of comodules over corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule $Σ$ of an $A$-coring $\cC$. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring $\cC$ or the comodule $Σ$ is finitely generated and projective as an $A$-module. That is, we obtain relations between the category of $\cC$-comodules and the category of firm modules for a firm ring $R$, which is an ideal of the endomorphism algebra $^\cC(Σ)$. For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.

math.RA↗
arXiv2008

Monads and comonads in module categories

Let $A$ be a ring and $\M_A$ the category of $A$-modules. It is well known in module theory that for any $A $-bimodule $B$, $B$ is an $A$-ring if and only if the functor $-\otimes_A B: \M_A\to \M_A$ is a monad (or triple). Similarly, an $A $-bimodule $\C$ is an $A$-coring provided the functor $-\otimes_A\C:\M_A\to \M_A$ is a comonad (or cotriple). The related categories of modules (or algebras) of $-\otimes_A B$ and comodules (or coalgebras) of $-\otimes_A\C$ are well studied in the literature. On the other hand, the right adjoint endofunctors $\Hom_A(B,-)$ and $\Hom_A(\C,-)$ are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of $\Hom_A(B,-)$-comodules is isomorphic to the category of $B$-modules, while the category of $\Hom_A(\C,-)$-modules (called $\C$-contramodules by Eilenberg and Moore) need not be equivalent to the category of $\C$-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. This leads to a deeper understanding and characterisations of algebraic structures such as corings, bialgebras and Hopf algebras. For example, it turns out that the categories of $\C$-comodules and $\Hom_A(\C,-)$-modules are equivalent provided $\C$ is a coseparable coring. Furthermore, a bialgebra $H$ over a commutative ring $R$ is a Hopf algebra if and only if $\Hom_R(H-)$ is a Hopf bimonad on $\M_R$ and in this case the categories of $H$-Hopf modules and mixed $\Hom_R(H,-)$-bimodules are both equivalent to $\M_R$.

math.RA↗
arXiv2008

(Co)cyclic (co)homology of bialgebroids: An approach via (co)monads

For a (co)monad T_l on a category M, an object X in M, and a functor Π: M \to C, there is a (co)simplex Z^*:=ΠT_l^{* +1} X in C. Our aim is to find criteria for para-(co)cyclicity of Z^*. Construction is built on a distributive law of T_l with a second (co)monad T_r on M, a natural transformation i:ΠT_l \to ΠT_r, and a morphism w: T_r X \to T_l X in M. The relations i and w need to satisfy are categorical versions of Kaygun's axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads T_l=T \otimes_R (-) and T_r = (-)\otimes_R T on the category of R-bimodules. The functor Πcan be chosen such that Z^n= T\hat{\otimes}_R... \hat{\otimes}_R T \hat{\otimes}_R X is the cyclic R-module tensor product. A natural transformation i:T \hat{\otimes}_R (-) \to (-) \hat{\otimes}_R T is given by the flip map and a morphism w: X \otimes_R T \to T\otimes_R X is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. Stable anti Yetter-Drinfel'd modules over certain bialgebroids, so called x_R-Hopf algebras, are introduced. In the particular example when T is a module coring of a x_R-Hopf algebra B and X is a stable anti Yetter-Drinfel'd B-module, the para-cyclic object Z_* is shown to project to a cyclic structure on T^{\otimes_R *+1} \otimes_B X. For a B-Galois extension S \to T, a stable anti Yetter-Drinfel'd B-module T_S is constructed, such that the cyclic objects B^{\otimes_R *+1} \otimes_B T_S and T^ {\hat{\otimes}_S *+1} are isomorphic. As an application, we compute Hochschild and cyclic homology of a groupoid with coefficients, by tracing it back to the group case. In particular, we obtain explicit expressions for ordinary Hochschild and cyclic homology of a groupoid.

math.KT↗
arXiv2007

Examples of para-cocyclic objects induced by BD-laws

In a recent paper arXiv:0705.3190, we gave a general construction of a para-cocyclic structure on a cosimplex, associated to a so called admissible septuple -- consisting of two categories, three functors and two natural transformations, subject to compatibility relations. The main examples of such admissible septuples were induced by algebra homomorphisms. In this note we provide more general examples coming from appropriate (`locally braided') morphisms of monads.

math.KT↗
arXiv2007

Pre-torsors and equivalences

Properties of (most general) non-commutative torsors or A-B torsors are analysed. Starting with pre-torsors it is shown that they are equivalent to a certain class of Galois extensions of algebras by corings. It is shown that a class of faithfully flat pre-torsors induces equivalences between categories of comodules of associated corings. It is then proven that A-B torsors correspond to monoidal functors (and, under some additional conditions, equivalences) between categories of comodules of bialgebroids.

math.QA↗
arXiv2003

Internal bialgebroids, entwining structures and corings

The internal bialgebroid -- in a symmetric monoidal category with coequalizers -- is defined. The axioms are formulated in terms of internal entwining structures and alternatively, in terms of internal corings. The Galois property of the coring in question is related to the $\times_R$-Hopf algebra property. The language of entwining structures is used to discuss duality.

math.QA↗