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Model theory of comodules
, 2004
"... The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so th ..."
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The purpose of this paper is to establish some basic points in the model theory of comodules over a coalgebra. It is not even immediately apparent that there is a model theory of comodules since these are not structures in the usual sense of model theory. Let us give the definitions right away so that the reader can see what we mean. Fix a field k. A kcoalgebra C is a kvector space equipped with a klinear map ∆ : C − → C ⊗ C, called the comultiplication (by ⊗ we always mean tensor product over k), and a klinear map ɛ: C − → k, called the counit, such that ∆ ⊗ 1C = 1C ⊗ ∆ (coassociativity) and (1C ⊗ ɛ) ∆ = 1C = (ɛ ⊗ 1C)∆, where we identify C with both k ⊗ C and C ⊗ k. These definitions are literally the duals of those for a kalgebra: express the axioms for C ′ to be a kalgebra in terms of the multiplication map µ: C ′ ⊗ C ′ − → C ′ and the “unit ” (embedding of k into C ′), δ: k − → C ′ in the form that certain diagrams commute and then just turn round all the arrows. See [4] or more recent references such as [7] for more. A (right) comodule over the coalgebra C is a kvector space M equipped with a
LOCALIZATIONS FOR CONSTRUCTION OF QUANTUM COSET SPACES
, 2003
"... Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on n ..."
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Viewing comodule algebras as the noncommutative analogues of affine varieties with affine group actions, we propose rudiments of a localization approach to nonaffine Hopf algebraic quotients of noncommutative affine varieties corresponding to comodule algebras. After reviewing basic background on noncommutative localizations, we introduce localizations compatible with coactions. Coinvariants of these localized coactions give local information about quotients. We define Zariski locally trivial quantum group algebraic principal and associated bundles. Compatible localizations induce localizations on the categories of Hopf modules. Their interplay with the functor of taking coinvariants and its left adjoint is stressed out. Using localization approach, we constructed a natural class of examples of quantum coset spaces, related to the quantum flag varieties of type A of other authors. Noncommutative Gauss decomposition via quasideterminants reveals a new structure in noncommutative matrix bialgebras. In the quantum case, calculations with quantum minors yield the structure theorems. Notation. Ground field is k and we assume it is of characteristic zero. If we deal just with one kHopf algebra, say B, the comultiplication is ∆ : B → B ⊗ B, unit map η: k → B, counit ǫ: B → k, multiplication µ: B ⊗ B → B, and antipode (coinverse) is
From Galois Field Extensions to . . .
"... Given a finite automorphism group G of a field extension E ⊃ K, E can be considered as module over the group algebra K[G]. Moreover, E can also be viewed as a comodule over the bialgebra K[G]∗ and here a canonical isomorphism involving the subfield fixed under the action of G arises. This isomorphis ..."
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Given a finite automorphism group G of a field extension E ⊃ K, E can be considered as module over the group algebra K[G]. Moreover, E can also be viewed as a comodule over the bialgebra K[G]∗ and here a canonical isomorphism involving the subfield fixed under the action of G arises. This isomorphism and its consequences were extended and studied for group actions on commutative rings, for actions of Hopf algebras on noncommutative algebras, then for corings with grouplike elements and eventually to comodules over corings. The purpose of this note is to report about this development and to give the reader some idea about the notions and results involved in this theory (without claiming to be comprehensive).
Noncommutative Localization in noncommutative geometry
, 2008
"... The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of “spaces”, locally described by noncommutative rings and their categories of onesided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical t ..."
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The aim of these notes is to collect and motivate the basic localization toolbox for the geometric study of “spaces”, locally described by noncommutative rings and their categories of onesided modules. We present the basics of Ore localization of rings and modules in much detail. Common practical techniques are studied as well. We also describe a counterexample for a folklore test principle. Localization in negatively filtered rings arising in deformation theory is presented. A new notion of the differential Ore condition is introduced in the study of localization of differential calculi. To aid the geometrical viewpoint, localization is studied with emphasis on descent formalism, flatness, abelian categories of quasicoherent sheaves and generalizations, and natural pairs of adjoint functors for sheaf and module categories. The key motivational theorems from the seminal works of Gabriel on localization, abelian categories and schemes are quoted without proof, as well as the related statements of Popescu, Watts, Deligne and Rosenberg. Cohn universal localization does not have good flatness properties, but may be described well at the ring level. We conjecture that the latter feature may be important for gluing in noncommutative geometry whenever the flat descent fails. Cohn localization is here related to the quasideterminants of Gelfand and Retakh; and this may help understanding both subjects.