For questions about comodules over corings, coalgebras, bialgebras, Hopf algebras.
Questions tagged [comodules]
15 questions
5
votes
1 answer
Modules over the dual of an infinite dimensional coalgebra
Let $k$ be a field and let $A$ be a finite dimensional (unital, associative, not necessarily commutative) $k$-algebra. The $k$-linear dual of $A$ is a coalgebra, and viceversa, the $k$-linear dual of a finite dimensional coalgebra is an algebra. Let…
57Jimmy
- 6,408
4
votes
0 answers
$H$-comodule structure of $A\otimes_K A$
I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and need some help with something that the paper does not define (and that I'm not able to find anywere). Consider this scenario:
Let $A$ be a $K$-algebra ($K$ is a field),…
Alejandro Bergasa Alonso
- 3,639
4
votes
2 answers
Is the dual of a module naturally a comodule?
This question is basically an extension of the following fact: given a finite dimensional, associative, unital $k$-algebra $A$, then the vector dual $A^*$ is a coassociative, counital coalgebra with coproduct $\Delta : A^* \rightarrow A^*\otimes…
Chris Young
- 83
3
votes
0 answers
Regular functions on torsors
Let $X \rightarrow Y$ be a torsor for a linear algebraic group $G$ (i.e. $X$ is a principal $G$-bundle over $Y$). Assume also that both $X$, and $Y$ are affine. What can be said about $\mathbb{C}[X]$ considered as $G$-module?
Assume now that we are…
Asav
- 293
3
votes
1 answer
How exactly does the coaction on the comodule X*⊗X work?
I'm struggling a bit with Sweedler notation. Let $(H,∆,ε,S,m,u)$ be a Hopf algebra over a commutative ring $k$ and let $X,Y$ be right $H$-comodules which are finitely generated projective as $k$-modules (if you like, let $k$ be a field and let $X,Y$…
3
votes
1 answer
Counterexample to the fundamental theorem of comodules
The Fundamental Theorem of Comodules (aka the Finiteness Theorem for Comodules) states that if $\Bbbk$ is a field, then any element of a comodule over a $\Bbbk$-coalgebra lies in a finite-dimensional subcomodule. I know that this is a distinctive…
Ender Wiggins
- 2,982
2
votes
0 answers
Example of a quasi-finite Comodule that is not finitely cogenerated
Let $k$ be a field. Let $C$ be a coassociative and counital coalgebra over $k$. Takeuchi defines the notion of quasi-finite comodule as follows: a left $C$-comodule $M$ is quasi-finite if the induced functor $M\otimes -:Vect_k \rightarrow LComod_C$…
2
votes
0 answers
Isomorphism between two Hopf algebras
Let $k$ be a field, over which we consider algebras and coalgebras. A $k$-coalgebra is a comonoid object in $k$-modules, and a $k$-algebra is a monoid object in $k$-modules. A $k$-bialgebra is equivalently a monoid object in $k$-coalgebras or a…
user900250
2
votes
1 answer
The kernel of a morphism of co-rings is a co-ideal
I would like to show that
The Kernel of a coring morphism $\phi:C\rightarrow C'$ between two $R$-corings $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ is a coideal.
The only point that I can't show is that $$\Delta(\text{Ker}(\phi))\subset…
user405156
- 1,619
2
votes
0 answers
Construction of "braided" Hopf algebras
Assume $(H,m,\eta, \Delta, \epsilon, S,r)$ to be a coquasitriangular Hopf-Algebra over $\mathbb C$.
The category $C(H)$ of $H$-comodules is braided monoidal.
Now consider a coaction $\delta: H \rightarrow H \otimes_{\mathbb C} H$ of $H$ on itself…
SeHa
- 231
1
vote
1 answer
Subcomodule structure on pure submodule
A number of sources state that the following:$^\ast$ that if $C$ is an $R$-coalgebra ($R$ a commutative ring), with $M$ a right $C$-comodule, and $\iota:K\to M$ a $C$-pure submodule, such that $\rho_M(K)\subseteq K\otimes C$, then $K$ inherits a…
Blunka
- 965
1
vote
0 answers
Prove $A$ is $K[G]$-comodule algebra
I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and there's something I'm trying to prove for what I may need some help.
More precisely, is this statement from Example 1.2:
"Dually, $A$ is a $K[G]$-comodule algebra if and…
Alejandro Bergasa Alonso
- 3,639
1
vote
1 answer
Is a coalgebra comodule cosemisimple if and only if every subcomodule is a direct summand?
It is well known that if $ R $ is a ring, then every $ R $-module $ M $ is semisimple (that is, $ M $ is the direct sum of simple $ R $-modules) if and only if every submodule of $ M $ is a direct summand. Is it true that if $ C $ is a coalgebra…
calm
- 477
0
votes
0 answers
Comodule structure of $H_{\ast}(Q_nS^0)$ over mod 2 dual Steenrod algebra $\mathcal{A}_{\ast}$
$Q_nS^0$ is the $n$-th component of infinite loop space, in other words $Q_nS^0=\varinjlim_k(\Omega^kS^k)_n$, where $(\Omega^kS^k)_n$ refers to homotopy classes of maps $(S^k,\ast)\to (S^k,\ast)$ of degree $n$.
$H_{\ast}(Q_nS^0,\mathbb{Z}/2)$ is a…
Siyuan Yin
- 157
0
votes
0 answers
Change of scalars for comodules as adjunctions?
Let $k$ be a commutative ring and $f: C \to C'$ be a homomorphism of $k$-coalgebras (for simplicity, we can suppose that it is surjective so that $f(C) = C'$). There is a functor:
$$f_*: {}^lC-comod \to {}^lC'-comod$$
$$(M, \Delta_M) \mapsto (M,…
Dat Minh Ha
- 524