Algebras and modules
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Geometry on formal duals of algebras
Given a monoidal category and a coalgebra in denote by () the category of right (resp. left) -comodules; similarly for an algebra , denote by (resp. ) the category of left E-modules (right -modules). If the monoidal category is symmetric or there is instead an appropriate distributive law, then there are extensions of this notation to bimodules, bicomodules, relative Hopf modules, entwined modules etc. e.g. Write for left-right relative -Hopf modules where is a -comodule algebra over a bialgebra .
Let be a commutative unital ring, and let be -linear (in particular it has zero morphisms).
Given a coalgebra in , a left -comodule , a right -comodule , their cotensor product is an object in given by the kernel
If equalizers exist in , this formula extends to a bifunctor
If is a bialgebra in and is a right -comodule algebra then the same formula defines a bifunctor
Relation to tensor product
Let now be the symmetric monoidal category of -modules.
Let be another -coalgebra, with coproduct . If is flat as a -module (e.g. is a field), and a left - right -bicomodule, then the cotensor product is a -subcomodule of the tensor product . In particular, under the flatness assumption, if is a surjection of coalgebras then is a left - right -bicomodule via and respectively, hence is a functor from left - to left -comodules called the induction functor for left comodules from to .
For comodules over commutative Hopf coalgebroids
Consider a commutative Hopf algebroid over (def.). Any left comodule over (def.) becomes a right comodule via the coaction
where the isomorphism in the middle the is braiding in and where is the conjugation map of .
Dually, a right comodule becoomes a left comodule with the coaction
Given a commutative Hopf algebroid over , (def.), and given a right -comodule and a left comodule (def.), then their cotensor product is the kernel of the difference of the two coaction morphisms:
If both and are left comodules, then their cotensor product is the cotensor product of with regarded as a right comodule via prop. 1.
e.g. (Ravenel 86, def. A1.1.4).
Given a commutative Hopf algebroid over , (def.), and given a left -comodule (def.). Regard itself canonically as a right -comodule Then the cotensor product
is called the primitive elements of :
(e.g. Ravenel 86, prop. A1.1.5)
Given a commutative Hopf algebroid over , and given two left -comodules, such that is projective as an -module, then
gives the structure of a right -comodule;
The cotensor product (def. 2) with respect to this right comodule structure is isomorphic to the hom of -comodules:
Hence in particular
(e.g. Ravenel 86, lemma A1.1.6)
Cotensor products in noncommutative geometry appear in the role of space of sections of associated vector bundles of quantum principal bundles (which in affine case correspond to Hopf-Galois extensions). See e.g.
- Shahn Majid, Foundations of quantum groups theory, 2nd extended edition, paperback, Cambridge Univ. Press 2000.
For a nonaffine extension of the sections of associated quantum vector bundle, using localization theory see
In Hopf algebra theory, cotensor products appear as early as in
The authors mention that they learned the notion from Mac Lane who knew it earlier in more general contexts.
A textbook account is in
An important problem is that the cotensor product of bicomodules is in general (even for ) not associative, even up to an isomorphism.
Cotensor products play a prominent role in various treatments of Galois theory in noncommutative geometry; a particularly geometric approach is within a version of noncommutative algebraic geometry based on usage of monoidal categories, as sketched in