nLab cotensor product

This page is about cotensor products of comodules over coalgebras. The word “cotensor” is also used for powers in enriched categories, and for the multiplicative disjunction in a polycategory/linearly distributive category.

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Definition

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

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

Definition

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

$$N \Box M \coloneqq \mathrm{ker} (\rho_N \otimes \mathrm{id}_M - \mathrm{id}_N \otimes \rho_M ).$$

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

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

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

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

Properties

Induced comodules and a relation to tensor product

Let now $\mathcal{M}=({}_k\mathrm{Mod},\otimes_k)$ be the symmetric monoidal category of $k$-modules where $k$ is a commutative unital ring.

Let $D$ be another $k$-coalgebra, with coproduct $\Delta_C$. If $D$ is flat as a $k$-module (e.g. $k$ is a field), and $N$ a left $D$- right $C$-bicomodule, then the cotensor product $N \Box M$ is a $D$-subcomodule of the tensor product $N \otimes_k M$. In particular, under the flatness assumption, if $\pi \colon D \rightarrow C$ is a homomorphism of coalgebras (usually surjective in applications), then $D$ is a left $D$-right $C$-bicomodule via $\Delta_D$ and $(\id \otimes \pi) \circ \Delta_D$ respectively, hence

$$\mathrm{Ind}^D_C : C-Comod\to D-comod: \coloneqq D \Box^C -$$

is a functor from left $C$- to left $D$-comodules called the induction (corepresentation) functor for left comodules from $C$ to $D$.

For comodules over commutative Hopf coalgebroids

Proposition

Consider a commutative Hopf algebroid $\Gamma$ over $A$ (def.). Any left comodule $N$ over $\Gamma$ (def.) becomes a right comodule via the coaction

$$N \overset{\Psi}{\longrightarrow} \Gamma \otimes_A N \overset{\simeq}{\longrightarrow} N \otimes_A \Gamma \overset{id \otimes_A c}{\longrightarrow} N \otimes_A \Gamma \,,$$

where the isomorphism in the middle the is braiding in $A Mod$ and where $c$ is the conjugation map of $\Gamma$.

Dually, a right comodule $N$ becoomes a left comodule with the coaction

$$N \overset{\Psi}{\longrightarrow} N \otimes_A \Gamma \overset{\simeq}{\longrightarrow} \Gamma \otimes_A N \overset{c \otimes_A id}{\longrightarrow} \Gamma \otimes_A N \,.$$

Definition

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

$$N_1 \Box_\Gamma N_2 \;\coloneqq\; ker \left( N_1 \otimes_A N_2 \overset{\Psi_{N_1}\otimes_{A} id - id \otimes_A \Psi_{N_2} }{\longrightarrow} \right) \,.$$

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

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

Example

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

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

is called the primitive elements of $N$:

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

Proposition

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

$$N_1 \Box N_2 \;\simeq\; N_2 \Box N_1 \,.$$

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

Lemma

Given a commutative Hopf algebroid $\Gamma$ over $A$, and given $N_1, N_2$ two left $\Gamma$-comodules, such that $N_1$ is projective as an $A$-module, then

1. The morphism

   $$Hom_A(N_1, A) \overset{f \mapsto (id \otimes_A f) \circ \Psi_{N_1}}{\longrightarrow} Hom_A(N_1, \Gamma \otimes_A A) \simeq Hom_A(N_1, \Gamma) \simeq Hom_A(N_1, A) \otimes_A \Gamma$$

   gives $Hom_A(N_1,A)$ the structure of a right $\Gamma$-comodule;

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

   $$Hom_A(N_1, A) \Box_\Gamma N_2 \simeq Hom_\Gamma(N_1, N_2) \,.$$

   Hence in particular

   $$A \Box_\Gamma N_2 \;\simeq\; Hom_\Gamma(A,N_2)$$

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

Remark

In computing the second page of $E$-Adams spectral sequences, the second statement in lemma 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.

Related concepts

• a special case of the cotensor product as a functor of one variable is the induced comodule construction
• The derived functor of cotensoring is called Cotor.

References

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.

Basic homological algebra of cotensor products for coalgebras over a field is advanced in

• Yukio Doi, Homological coalgebra, J. Math. Soc. Japan 33:1 (1981) 31–50 MR597479 euclid

(Co)flatness conditions play major role for study of cotensor products when coalgebras are over commutative rings, or in the categories of bimodules (corings),

• Leonid Positselski, Homological algebra of semimodules and semicontramodules: Semi-infinite homological algebra of associative algebraic structures, arxiv/0708.3398
• T. Brzeziński, R. Wisbauer, Corings and comodules, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

For a nonaffine extension of the sections of associated quantum vector bundle, using localization theory see

• Zoran Škoda, Coherent states for Hopf algebras,

  Lett. Math. Phys. 81 (2007), no. 1, 1–17. (arXiv:math.QA/0303357)

In Hopf algebra theory motivated by algebraic topology, cotensor products appear as early as in

• John Milnor, John Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 1965 211–264

and along with its derived functor Cotor used in computation of $E^2$-term of a spectral sequence of a fibration in

• Samuel Eilenberg, John. C Moore, Homology and fibrations. I. Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966) 199–236 MR203730 doi

The authors mention that they learned the notion from Mac Lane who knew it earlier in more general contexts.

A textbook account is in

• Doug Ravenel, section A1.1 of Complex cobordism and stable homotopy groups of spheres, 1986/2003

An important problem is that the cotensor product of bicomodules is in general (even for $\mathcal{M}={}_k\mathrm{Mod}$) 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

• Tomasz Maszczyk, Noncommutative geometry through monoidal categories I, arXiv:0707.1542