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11
A 2Categorical Approach To Change Of Base And Geometric Morphisms II
, 1998
"... We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibi ..."
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Cited by 45 (7 self)
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We introduce a notion of equipment which generalizes the earlier notion of proarrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the Vpro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2category, in such a way that arbitrary functors F: L ✲ K induce equipment arrows relF: relL ✲ relK, spnF: spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locallyfullyfaithful 2functor to the 2category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.
Equivalences of comodules categories for coalgebras over rings
 J. Pure App. Algebra
"... In this article we defined and studied quasifinite comodules, the cohom functors for coalgebras over rings. linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work charac ..."
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Cited by 7 (0 self)
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In this article we defined and studied quasifinite comodules, the cohom functors for coalgebras over rings. linear functors between categories of comodules are also investigated and it is proved that good enough linear functors are nothing but a cotensor functor. Our main result of this work characterizes equivalences between comodule categories generalizing the MoritaTakeuchi theory to coalgebras over rings. MoritaTakeuchi contexts in our setting is defined and investigated, a correspondence between strict MoritaTakeuchi contexts and equivalences of comodule categories over the involved coalgebras is obtained. Finally we proved that for coalgebras over QFrings Takeuchi’s representation of the cohomfunctor is also valid. Keywords:cotensor product, cohom functor, faithfully coflat comodules, equivalent categories, MoritaTakeuchi context. Mathematics subject classification: 16W30
COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE
, 2002
"... Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general ab ..."
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Cited by 3 (3 self)
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Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures, or to get information about the structure of complicated objects by
Noncommutative geometry through monoidal categories I
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
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Cited by 3 (0 self)
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way. Contents
ON BIMEASURINGS
, 2004
"... Abstract. We introduce and study bimeasurings from pairs of bialgebras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler’s universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant ..."
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Abstract. We introduce and study bimeasurings from pairs of bialgebras to algebras. It is shown that the universal bimeasuring bialgebra construction, which arises from Sweedler’s universal measuring coalgebra construction and generalizes the finite dual, gives rise to a contravariant
Hopf Pairings and (Co)induction Functors over Commutative Rings ∗
, 2004
"... (Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine gr ..."
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(Co)induction functors appear in several areas of Algebra in different forms. Interesting examples are the so called induction functors in the Theory of Affine Algebraic Groups. In this paper we investigate Hopf pairings (bialgebra pairings) and use them to study (co)induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings) satisfying the so called αcondition. For those pairings the coinduction functor is studied and nice descriptions of it are obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (Noetherian) ground rings.
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES I
, 2007
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. In all these considerations we lay stress on the role of the monoidal structure, and the difference between this approach and the approach using (in general nonmonoidal) abelian categories as models for categories of quasicoherent sheaves on noncommutative schemes. Contents
Hopf Pairings and Induction Functors over Rings ∗
, 2003
"... The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for ..."
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The so called induction functors appear in several areas of Algebra in different forms. Interesting examples are the induction functors in the Theory of Affine Algebraic groups. In this note we investigate the so called Hopf pairings (bialgebra pairings) and use them to study induction functors for affine group schemes over arbitrary commutative ground rings. We present also a special type of Hopf pairings (bialgebra pairings), satisfying the so called αcondition. For those pairings the induction functor is studied and a nice description of it is obtained. Along the paper several interesting results are generalized from the case of base fields to the case of arbitrary commutative (noetherian) ground rings.
NONCOMMUTATIVE GEOMETRY THROUGH MONOIDAL CATEGORIES
, 2007
"... Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending ..."
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Abstract. After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasicoherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, HopfGalois extensions) can be given geometric meaning extending their geometric interpretations in the commutative case. On the other hand, we show that some constructions in commutative geometry (e.g. faithfully flat descent theory, principal fibrations, equivariant and infinitesimal geometry) can be interpreted as noncommutative geometric constructions applied to commutative objects. For such generalized geometry we define global invariants constructing cyclic objects from which we derive Hochschild, cyclic and periodic cyclic homology (with coefficients) in the standard way. Contents
Picard Group for Corings
, 2007
"... Ph.D. Thesis realized by Mohssin Zarouali Darkaoui under the supervision of ..."
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Ph.D. Thesis realized by Mohssin Zarouali Darkaoui under the supervision of