cotensor product



Given a monoidal category \mathcal{M} and a coalgebra CC in \mathcal{M} denote by C\mathcal{M}^{C} (resp. C resp. {}^{C}\mathcal{M}) the category of right (resp. left) C{C}-comodules; similarly for an algebra EE, denote by E{}_E\mathcal{M} (resp. E\mathcal{M}_E) the category of left E-modules (right EE-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 E B{}_E\mathcal{M}^B for left-right relative (E,B)(E,B)-Hopf modules where EE is a BB-comodule algebra over a bialgebra BB.

Let kk be a commutative unital ring, and let \mathcal{M} be kk-linear (in particular it has zero morphisms).


Given a coalgebra CC in \mathcal{M}, a left CC-comodule (N,ρ N:NNC)(N,\rho_N \colon N\to N\otimes C), a right CC-comodule (M,ρ M:MCM)(M,\rho_M \colon M\to C\otimes M), their cotensor product is an object in \mathcal{M} given by the kernel

NMker(ρ Nid Mid Nρ M). N \Box M \coloneqq \mathrm{ker} (\rho_N \otimes \mathrm{id}_M - \mathrm{id}_N \otimes \rho_M ).

If equalizers exist in \mathcal{M}, this formula extends to a bifunctor

= C: C× C. {}\Box = \Box^{C} \colon \mathcal{M}^{C} \times {}^{C}\mathcal{M} \rightarrow \mathcal{M} \,.

If BB is a bialgebra in \mathcal{M} and EE is a right BB-comodule algebra then the same formula defines a bifunctor

: E B× B E. \Box \colon {}_{E}\mathcal{M}^{B} \times {}^{B}\mathcal{M} \rightarrow {}_{E}\mathcal{M} \,.


Relation to tensor product

Let now =( kMod, k)\mathcal{M}=({}_k\mathrm{Mod},\otimes_k) be the symmetric monoidal category of kk-modules.

Let DD be another kk-coalgebra, with coproduct Δ C\Delta_C. If DD is flat as a kk-module (e.g. kk is a field), and NN a left DD- right CC-bicomodule, then the cotensor product NMN \Box M is a DD-subcomodule of the tensor product N kMN \otimes_k M. In particular, under the flatness assumption, if π:DC\pi \colon D \rightarrow C is a surjection of coalgebras then DD is a left DD- right CC-bicomodule via Δ D\Delta_D and (idπ)Δ D(\id \otimes \pi) \circ \Delta_D respectively, hence Ind C DD C\mathrm{Ind}^D_C \coloneqq D \Box^C - is a functor from left CC- to left DD-comodules called the induction functor for left comodules from CC to DD.

For comodules over commutative Hopf coalgebroids


Consider a commutative Hopf algebroid Γ\Gamma over AA (def.). Any left comodule NN over Γ\Gamma (def.) becomes a right comodule via the coaction

NΨΓ ANN AΓid AcN AΓ, N \overset{\Psi}{\longrightarrow} \Gamma \otimes_A N \overset{\simeq}{\longrightarrow} N \otimes_A \Gamma \overset{id \otimes_A c}{\longrightarrow} N \otimes_A \Gamma \,,

where the isomorphism in the middle the is braiding in AModA Mod and where cc is the conjugation map of Γ\Gamma.

Dually, a right comodule NN becoomes a left comodule with the coaction

NΨN AΓΓ ANc AidΓ AN. N \overset{\Psi}{\longrightarrow} N \otimes_A \Gamma \overset{\simeq}{\longrightarrow} \Gamma \otimes_A N \overset{c \otimes_A id}{\longrightarrow} \Gamma \otimes_A N \,.

Given a commutative Hopf algebroid Γ\Gamma over AA, (def.), and given N 1N_1 a right Γ\Gamma-comodule and N 2N_2 a left comodule (def.), then their cotensor product N 1 ΓN 2N_1 \Box_\Gamma N_2 is the kernel of the difference of the two coaction morphisms:

N 1 ΓN 2ker(N 1 AN 2Ψ N 1 Aidid AΨ N 2). N_1 \Box_\Gamma N_2 \;\coloneqq\; ker \left( N_1 \otimes_A N_2 \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow} \right) \,.

If both N 1N_1 and N 2N_2 are left comodules, then their cotensor product is the cotensor product of N 2N_2 with N 1N_1 regarded as a right comodule via prop. 1.

e.g. (Ravenel 86, def. A1.1.4).


Given a commutative Hopf algebroid Γ\Gamma over AA, (def.), and given NN a left Γ\Gamma-comodule (def.). Regard AA itself canonically as a right Γ\Gamma-comodule Then the cotensor product

Prim(N)A ΓN Prim(N) \coloneqq A \Box_\Gamma N

is called the primitive elements of NN:

Prim(N)={nN|Ψ N(n)=1n}. Prim(N) = \{ n \in N \;\vert\; \Psi_N(n) = 1 \otimes n \} \,.

Given a commutative Hopf algebroid Γ\Gamma over AA, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules , then their cotensor product (def. 2) is commutative, in that there is an isomorphism

N 1N 2N 2N 1. N_1 \Box N_2 \;\simeq\; N_2 \Box N_1 \,.

(e.g. Ravenel 86, prop. A1.1.5)


Given a commutative Hopf algebroid Γ\Gamma over AA, and given N 1,N 2N_1, N_2 two left Γ\Gamma-comodules, such that N 1N_1 is projective as an AA-module, then

  1. The morphism

    Hom A(N 1,A)f(id Af)Ψ N 1Hom A(N 1,Γ AA)Hom A(N 1,Γ)Hom A(N 1,A) AΓ Hom_A(N_1, A) \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow} Hom_A(N_1, \Gamma \otimes_A A) \simeq Hom_A(N_1, \Gamma) \simeq Hom_A(N_1, A) \otimes_A \Gamma

    gives Hom A(N 1,A)Hom_A(N_1,A) the structure of a right Γ\Gamma-comodule;

  2. The cotensor product (def. 2) with respect to this right comodule structure is isomorphic to the hom of Γ\Gamma-comodules:

    Hom A(N 1,A) ΓN 2Hom Γ(N 1,N 2). Hom_A(N_1, A) \Box_\Gamma N_2 \simeq Hom_\Gamma(N_1, N_2) \,.

    Hence in particular

    A ΓN 2Hom Γ(A,N 2) A \Box_\Gamma N_2 \;\simeq\; Hom_\Gamma(A,N_2)

(e.g. Ravenel 86, lemma A1.1.6)


In computing the second page of EE-Adams spectral sequences, the second statement in lemma 1 is the key translation that makes the comodule Ext-groups on the page be equivalent to a Cotor-groups. The latter lend themselves to computation, for instance via Lambda-algebra or via the May spectral sequence.


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 = kMod\mathcal{M}={}_k\mathrm{Mod}) 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

Last revised on May 12, 2016 at 06:44:52. See the history of this page for a list of all contributions to it.