# Faithful flatness of Hopf algebras over coideal subalgebras with a conditional expectation

Julien Bichon

► To cite this version:

Julien Bichon. Faithful flatness of Hopf algebras over coideal subalgebras with a conditional expectation. 2023. hal-03936040

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Preprint submitted on 12 Jan 2023

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# Faithful flatness of Hopf algebras over coideal subalgebras with a bimodule conditional expectation

Julien Bichon

► To cite this version:

Julien Bichon. Faithful flatness of Hopf algebras over coideal subalgebras with a bimodule conditional expectation. 2023. hal-03936040v2

HAL Id: hal-03936040

https://uca.hal.science/hal-03936040v2

Preprint submitted on 26 Aug 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

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FAITHFUL FLATNESS OF HOPF ALGEBRAS OVER COIDEAL SUBALGEBRAS WITH A CONDITIONAL EXPECTATION

JULIEN BICHON

ABSTRACT. Let $H$ be a Hopf algebra and let $A \subset H$ be a right coideal subalgebra. We show that if $A$ is a direct summand in $H$ as a right $A$-module, then $H$ is faithfully flat as a right $A$-module.

# 1. INTRODUCTION

Faithful flatness is an algebraic concept arising from algebraic geometry and closely related to the question of forming “good” quotients. It takes an important role in the theory of group schemes and their noncommutative generalizations, Hopf algebras, and especially in questions related to exact sequences [1].

The question whether a Hopf algebra is faithfully flat over its Hopf subalgebras was raised in Montgomery's book [11]. Schauenburg [15] has shown that the answer is negative in general, but the examples considered there are due to a somewhat pathological behavior of the antipodes, and it is expected that faithful flatness over Hopf subalgebras should hold in the most reasonable situations. Here are some highlights of the known positive results ($A \subset H$ is a Hopf subalgebra in what follows):

(1) First there is the classical result of Cartier that ensures that if $H$ is commutative then it is faithfully flat over its Hopf subalgebras. This is precisely one of the ingredients that ensures that there is a good quotient theory for affine group schemes, see [8, 19, 22].

(2) Cartier's result has been generalized to the case when $A$ is commutative, by Arkhipov-Gaitsgory [2], and to the case when $H$ is Noetherian, residually finite-dimensional and has a polynomial identity, by Skryabin [18].

(3) Pointed Hopf algebras are free over their Hopf subalgebras: this was shown by Radford [14].

(4) The celebrated Nichols-Zoeller theorem [12] ensures that finite-dimensional Hopf algebras are free over their Hopf subalgebras

(5) Chirvasitu [6] has shown that if $H$ is cosemisimple, then it is faithfully flat over its Hopf subalgebras.

The case when $A \subset H$ is only a coideal subalgebra is also of high interest in view of quotient theory, but here the situation is not that we can reasonably expect that faithful flatness holds, since even in the commutative case it is easy to find natural coideal subalgebras over which the Hopf algebra is not faithfully flat (for example $k[x, y] \subset \mathcal{O}(\mathrm{SL}_2(k))$). There are however a number of positive results, possibly by relaxing faithful flatness to flatness, of which we also list a selection.

(1) Masuoka-Wigner [9] have shown that commutative Hopf algebras are flat over their coideal subalgebras, and this was generalized by Skryabin [18] the case when $H$ is Noetherian, residually finite-dimensional and has a polynomial identity.

<footnote n="2010">2010 Mathematics Subject Classification. 16T05, 16E40.</footnote>
FAITHFUL FLATNESS OF HOPF ALGEBRAS OVER COIDEAL SUBALGEBRAS WITH A BIMODULE CONDITIONAL EXPECTATION

JULIEN BICHON

ABSTRACT. We give a direct and self-contained proof that if $H$ is a Hopf algebra and $A \subset H$ is a right coideal subalgebra such $A$ is a direct summand in $H$ as an $A$-bimodule, then $H$ is faithfully flat as a left and right $A$-module.

# 1. INTRODUCTION

The aim of this note is to give a direct and self-contained proof of the following result.

**Theorem 1.1.** Let $H$ be a Hopf algebra and let $A \subset H$ be a right coideal subalgebra. If $A$ is a direct summand in $H$ as an $A$-bimodule, then $H$ is faithfully flat as a left and right $A$-module.

The original author's interest for Theorem 1.1 came from A. Chirvasitu's result [3, Theorem 2.1] on the faithful flatness of a cosemisimple Hopf algebras over its Hopf subalgebras. Indeed, Chirvasitu's proof is divided in two steps: he first proves the crucial fact that a Hopf subalgebra $A \subset H$ of a cosemisimple Hopf algebra $H$ is a direct summand of $H$ as an $A$-bimodule, and then concludes using [3, Proposition 1.4, Proposition 1.6], results that are obtained by a discussion involving an important number of external references [2, 4, 6, 8] and various technologies. It is hoped that the present direct and self-contained proof of Theorem 1.1 will provide an easier access to the proof of Chirvasitu's Theorem.

I wish to thank B. Mesabishvili for useful comments and remarks on previous versions of this note, and S. Skryabin for pointing out his recent paper [7], which shows that the assumption “$A$ is a direct summand in $H$ as an $A$-bimodule” in Theorem 1.1 cannot be weakened to “$A$ is direct summand as a right (or left) $H$-module”.

**Notations and conventions.** We work over a fixed field $k$, and assume that the reader is familiar with the theory of Hopf algebras as e.g. in [5]. If $H$ is a Hopf algebra, as usual, $\Delta$, $\varepsilon$ and $S$ stand respectively for the comultiplication, counit and antipode of $H$. We use Sweedler's notations in the standard way.

The category of left $A$-modules over an algebra is denoted $A\mathcal{M}$, the category of left $C$-comodules over a coalgebra is denoted $C\mathcal{M}$, etc...

As usual, we say that a right $A$-module $M$ is flat if the functor $M \otimes_A - : A\mathcal{M} \to k\mathcal{M}$ is exact, which amounts to say map $M \otimes_A -$ preserves injective maps (monomorphisms), and that $M$ is faithfully flat if it is flat and $M \otimes_A -$ creates exact sequences as well. Left (faithfully) flat $A$-modules are defined similarly. We also say that an algebra extension $A \subset B$ is right or left (faithfully) flat is $B$ is (faithfully) flat as a right or left $A$-module.

If $A \subset B$ is an algebra extension, then $A$ is direct summand in $B$ as a right $A$-module if and only there exists a right $A$-linear map $E : B \to A$ such that $E|_A = \text{id}_A$, and we call such a map $E$ is a right conditional expectation for the extension $A \subset B$. The notion of

<footnote n="2010">2010 Mathematics Subject Classification. 16T05, 16D40.</footnote>
(2) Skryabin [17] has shown that finite-dimensional Hopf algebras are free over their coideal subalgebras.

(3) Compact Hopf algebras are faithfully flat over their coideal *-subalgebras: this was shown by Chirvasitu [7].

The primary motivation for this work was the idea that it would be useful to write a self-contained proof of the above mentioned result [6, Theorem 2.1] of Chirvasitu on the faithful flatness of a cosemisimple Hopf algebra over its Hopf subalgebras. Indeed, Chirvasitu's proof is divided in two steps: he first proves the crucial fact that a Hopf subalgebra $A \subset H$ of a cosemisimple $H$ is a direct summand of $H$ as an $A$-bimodule, and then concludes using [6, Proposition 1.4, Proposition 1.6], results that are obtained by a discussion involving an important number of external references [5, 10, 13, 20]. As a result of the writing of the self-contained proof, we obtain the following new faithful flatness result, in the setting of coideal subalgebras.

**Theorem 1.1.** Let $H$ be a Hopf algebra and let $A \subset H$ be a right coideal subalgebra. If $A$ is a direct summand in $H$ as a right $A$-module, then $H$ is faithfully flat as a right $A$-module.

The condition that $A$ is a direct summand in $H$ as a right $A$-module is equivalent to the existence of a right $A$-linear map $E: H \to A$ such that $E|_A = \text{id}_A$. Such a map $E$ is what we call a right conditional expectation for the extension $A \subset H$.

The work of Takeuchi [20] revealed that the question of faithful flatness is intimately related to the structure of certain categories of relative Hopf modules, and the proof of Theorem 1.1 is based on such a structure theorem, Theorem 3.1 in Section 3. This result is a generalization of [6, Proposition 1.6], involving a notion that we call faithful flatishness, and which is in general a technical consequence of faithful flatness. The key point is that faithful flatishness is an easy consequence of the existence of a conditional expectation, and is sufficient to prove the structure theorem of relative Hopf modules. Several arguments in the proof of Theorem 3.1 are close to classical arguments that one finds here and there in the literature, and it is fair to say that the proof of [16, Theorem 3.7] was particularly inspiring.

The paper is organized as follows: in Section 2 we introduce the notion of a faithfully flatish extension, while in Section 3, after having recalled the adequate framework of relative Hopf modules, we prove Theorem 3.1, from which Theorem 1.1 follows. The final Section 4 discusses some left/right variations and consequences of Theorem 1.1, and an illustration on an example.

**Notations and conventions.** We work over a fixed field $k$, and assume that the reader is familiar with the theory of Hopf algebras as e.g. in [11]. If $H$ is a Hopf algebra, as usual, $\Delta$, $\varepsilon$ and $S$ stand respectively for the comultiplication, counit and antipode of $H$. We use Sweedler's notations in the standard way. The category of left $A$-modules over an algebra is denoted $A\mathcal{M}$, the category of left $C$-comodules over a coalgebra is denoted $C\mathcal{M}$, etc...

## 2. FAITHFULLY FLAT AND FAITHFULLY FLATISH EXTENSIONS

As usual, we say that a right $A$-module $M$ is flat if the functor $M \otimes_A - : A\mathcal{M} \to k\mathcal{M}$ is exact, which amounts to say map $M \otimes_A -$ preserves injective maps (monomorphisms), and that $M$ is faithfully flat if it is flat and $M \otimes_A -$ creates exact sequences as well. We also say that an algebra extension $A \subset B$ is right (faithfully) flat is $B$ is (faithfully) flat as a right $A$-module.
left conditional expectation is defined similarly, and a bimodule conditional expectation is an A-bimodule map $E : B \to A$ such that $E|_A = \text{id}_A$.

## 2. PROOF OF THEOREM 1.1

### 2.1. Preliminary set-up.
We begin by fixing a number of notation and constructions, which will run throughout the section. All this material can be found in [8].

Let $H$ be a Hopf algebra and let $A \subset H$ be a right coideal subalgebra, which means that $A$ is subalgebra of $H$ such that $\Delta(A) \subset A \otimes H$. Let $A^+ = \text{Ker}(\varepsilon) \cap A$ and consider $HA^+$, the left ideal of $H$ generated by $A^+$. It is an immediate verification that $HA^+$ is a coideal in $H$ ($\Delta(HA^+) \subset HA^+ \otimes H + H \otimes HA^+$ and $\varepsilon(HA^+) = 0$), so we can form the quotient coalgebra $C = H/HA^+$ together with the canonical surjection : $\pi : H \to C = H/HA^+$. The coalgebra $C$ has as well a left $H$-module structure induced by $\pi$, so that $C$ is a left $H$-module coalgebra. We therefore consider the category of (relative) Hopf modules $C_H\mathcal{M}$, whose objects are the left $H$-modules and left $C$-comodules $X$ such that the coaction $\alpha_X : X \to C \otimes X$ is left $H$-linear, i.e. in Sweedler notation, we have for any $h \in H$ and $x \in X$,

$$ (h.x)_{(-1)} \otimes (h.x)_{(0)} = h_{(1)} \cdot x_{(-1)} \otimes h_{(2)} \cdot x_{(0)} $$

For a left $A$-module $M$, the induced $H$-module $H \otimes_A M$ has a left $C$-comodule structure given by $(h \otimes_A m)_{(-1)} \otimes (h \otimes_A m)_{(0)} = \pi(h_{(1)}) \otimes h_{(2)} \otimes_A m$ making it into an object of $C_H\mathcal{M}$. This defines the induction functor

$$ L = H \otimes_A - : A\mathcal{M} \to C_H\mathcal{M} \\ M \mapsto H \otimes_A M $$

For an object $X$ in $C_H\mathcal{M}$, let

$$ \text{coC}X = \{x \in X \mid x_{(-1)} \otimes x_{(0)} = \pi(1) \otimes x\} $$

It is immediate to check that $\text{coC}X \subset X$ is a sub-$A$-module and this defines a functor

$$ R = \text{coC}(-) : C_H\mathcal{M} \to A\mathcal{M} \\ X \mapsto \text{coC}X $$

which is right adjoint to $L$. We therefore have a pair of adjoint functors

$$ (2.1) \qquad (L, R) : A\mathcal{M} \stackrel{C_H\mathcal{M}}{\rightleftharpoons} \mathcal{M} $$

whose respective unit and counit are given by

$$ (2.2) \qquad \eta_M : M \to \text{coC}(H \otimes_A M) \qquad \mu_X : H \otimes_A \text{coC}X \to X \\ m \mapsto 1_H \otimes_A m \qquad h \otimes x \mapsto h \cdot x $$

We have now the necessary material to state the following result, which is [3, Proposition 1.6].

**Theorem 2.1.** Let $H$ be a Hopf algebra, let $A \subset H$ be a right coideal subalgebra and let $C = H/HA^+$ be the corresponding quotient coalgebra. The following assertions are equivalent.

(1) The induction functor $A\mathcal{M} \to C_H\mathcal{M}$ is an equivalence of categories.

(2) The extension $A \subset H$ is right faithfully flat.

(3) The above unit and counit morphisms (2.2) are isomorphisms.
We introduce a technical notion, which is in general a consequence of faithful flatness, but which is sufficient in many applications.

**Definition 2.1.** An algebra extension $A \subset B$ is said to be right faithfully flatish if for any left $A$-module $M$, the map

$$M \to \left\{ \sum_i x_i \otimes_A m_i \in B \otimes_A M \mid \sum_i x_i \otimes_A 1_B \otimes_A m_i = \sum_i 1_B \otimes_A x_i \otimes_A m_i \right\}$$
$$m \mapsto 1_B \otimes_A m$$

is an isomorphism.

It is a well-known property that faithfully flat extension are faithfully flatish, see e.g. the second theorem of Section 13.1 in [22]. The following results ensures that faithful flatfishness holds in the context of Theorem 1.1.

**Proposition 2.2.** Let $A \subset B$ be an algebra extension, and assume that $A$ is a direct summand in $H$ as a right $A$-module. Then the extension $A \subset B$ is right faithfully flatish.

*Proof.* Let $E : B \to A$ be a right conditional expectation. The right $A$-linearity of $E$ enables us to define, for any left $A$-module $M$, the map

$$E_M : B \otimes_A M \to M$$
$$x \otimes m \mapsto E(x).m$$

Denote by $X(M)$ the $A$-module on the right in Definition 2.1. It is an immediate verification, using that $E|_A = \text{id}_A$, that $E_{M|X(M)} : X(M) \to M$ is a reciprocal isomorphism to the one in Definition 2.1. $\square$

### 3. RELATIVE HOPF MODULES AND FAITHFUL FLATISHNESS

In this section we show that for a right coideal subalgebra $A \subset H$, faithful flatishness is equivalent to faithful flatness, which will prove Theorem 1.1. This is done in the context of a structure theorem for a certain category of Hopf modules introduced by Takeuchi [20].

**3.1. Preliminary set-up.** We begin by fixing a number of notation and constructions, which will run throughout the section. All this material can be found in [20].

Let $H$ be a Hopf algebra and let $A \subset H$ be a right coideal subalgebra, which means that $A$ is subalgebra of $H$ such that $\Delta(A) \subset A \otimes H$. Let $A^+ = \text{Ker}(\varepsilon) \cap A$ and consider $HA^+$, the left ideal of $H$ generated by $A^+$. It is an immediate verification that $HA^+$ is a coideal in $H$ ($\Delta(HA^+) \subset HA^+ \otimes H + H \otimes HA^+$ and $\varepsilon(HA^+) = 0$), so we can form the quotient coalgebra $C = H/HA^+$ together with the canonical surjection : $\pi : H \to C = H/HA^+$. The coalgebra $C$ has as well a left $H$-module structure induced by $\pi$, so that $C$ is a left $H$-module coalgebra. We therefore consider the category of (relative) Hopf modules $\mathcal{C}_H M$, whose objects are the left $H$-modules and left $C$-comodules $X$ such that the coaction $\alpha_X : X \to C \otimes X$ is left $H$-linear, i.e. in Sweedler notation, we have for any $h \in H$ and $x \in X$,

$$(h.x)_{(-1)} \otimes (h.x)_{(0)} = h_{(1)} \cdot x_{(-1)} \otimes h_{(2)} \cdot x_{(0)}$$

For a left $A$-module $M$, the induced $H$-module $H \otimes_A M$ has a left $C$-comodule structure given by $(h \otimes_A m)_{(-1)} \otimes (h \otimes_A m)_{(0)} = \pi(h_{(1)}) \otimes h_{(2)} \otimes_A m$ making it into an object of $\mathcal{C}_H M$. This defines the induction functor

$$L = H \otimes_A - : A M \to \mathcal{C}_H M$$
$$M \mapsto H \otimes_A M$$
It is immediate that (1) ⇒ (2) since an equivalence of categories is a faithfully exact functor and the exact sequences in $A\mathcal{M}$ and $C_H\mathcal{M}$ are precisely those that are exact in $\kappa\mathcal{M}$. It is clear that (3) ⇒ (1). The arguments we develop to prove Theorem 1.1 will provide as well a proof of (2) ⇒ (3).

**2.2. The canonical isomorphisms.** We will use some “canonical” isomorphisms, that we construct in this subsection.

For a left $H$-module $X$, endow $C \otimes X$ with the tensor product left $H$-module structure and with the left $C$-comodule structure provided by the comultiplication of $C$. In this way $C \otimes X$ becomes an object of $C_H\mathcal{M}$ (in fact $C \otimes X$ is the image of $X$ by the right adjoint to the forgetful functor $C_H\mathcal{M} \to H\mathcal{M}$).

**Proposition 2.2.** Let $A \subset H$ be a right coideal subalgebra and let $X$ be left $H$-module. The canonical map

$$ \kappa_X : H \otimes_A X \to C \otimes X \\ h \otimes_A x \mapsto \pi(h_{(1)}) \otimes h_{(2)} \cdot x $$

is an isomorphism in the category $C_H\mathcal{M}$.

*Proof.* It is a direct verification that $\kappa_X$ is a morphism in $C_H\mathcal{M}$, and that

$$ C \otimes X \to H \otimes_A X \\ \pi(h) \otimes x \mapsto h_{(1)} \otimes_A S(h_{(2)}).x $$

is the inverse isomorphism.

<figure><img src="image_2.png" /></figure>

**2.3. The unit of the adjunction.** We first analyse the unit of our adjunction, starting with a general observation.

**Proposition 2.3.** Let $A \subset B$ be an algebra extension, let $M$ be a left $A$-module $M$, and consider the map

$$ \iota_M : M \to \left\{ \sum_i x_i \otimes_A m_i \in B \otimes_A M \mid \sum_i x_i \otimes_A 1_B \otimes_A m_i = \sum_i 1_B \otimes_A x_i \otimes_A m_i \right\} \\ m \mapsto 1_B \otimes_A m $$

The $\iota_M$ is an isomorphism if one of the following conditions holds:

(1) $A \subset B$ is right faithfully flat;

(2) $A$ is a direct summand in $B$ as a right $A$-module.

*Proof.* It is a well-known result (see e.g. the second theorem of Section 13.1 in [9]) that $\iota_M$ is an isomorphism if $B$ is faithfully flat as a right $A$-module, that we do not reproduce here.

Let $E : B \to A$ be a right conditional expectation. The right $A$-linearity of $E$ enables us to define, for any left $A$-module $M$, the map

$$ E_M : B \otimes_A M \to M \\ x \otimes_A m \mapsto E(x).m $$

For simplicity denote $X(M)$ the space on the right. Let us check that $E_{M|X(M)} : X(M) \to M$ is an inverse isomorphism to $\iota_M$. In one direction it is clear that $E_M \circ \iota_M = \text{id}_M$. To
For an object $X$ in $C_H\mathcal{M}$, let

$$^{\text{co}C}X = \{x \in X \mid x_{(-1)} \otimes x_{(0)} = \pi(1) \otimes x\}$$

It is immediate to check that $^{\text{co}C}X \subset X$ is a sub-$A$-module and this defines a functor

$$R =^{\text{co}C}(-) : C_H\mathcal{M} \to_A \mathcal{M} \\ X \mapsto^{\text{co}C} X$$

which is right adjoint to $L$. We therefore have a pair of adjoint functors

$$(3.1) \qquad (L, R) : A\mathcal{M} \to C_H\mathcal{M}$$

whose respective unit and counit are given by

$$\eta_M : M \to^{\text{co}C}(H \otimes_A M) \qquad \mu_X : H \otimes_A^{\text{co}C} X \to X \\ m \mapsto 1_H \otimes_A m \qquad h \otimes x \mapsto h.x$$

**3.2. Statement of the main result.** We have now the necessary material to state the main result of the section, as follows.

**Theorem 3.1.** Let $H$ be a Hopf algebra, let $A \subset H$ be a right coideal subalgebra and let $C = H/HA^+$ be the corresponding quotient coalgebra. The following assertions are equivalent.

(1) The induction functor $A\mathcal{M} \to C_H\mathcal{M}$ is an equivalence of categories.

(2) The extension $A \subset H$ is right faithfully flat.

(3) The extension $A \subset H$ is right faithfully flatish.

Theorem 1.1 is an immediate consequence of the combination of Theorem 3.1 and Proposition 2.2.

It is immediate that (1) ⇒ (2) since an equivalence of categories is a faithfully exact functor and the exact sequences in $A\mathcal{M}$ and $C_H\mathcal{M}$ are precisely those that are exact in $\kappa\mathcal{M}$, and it has already been mentioned that (2) ⇒ (3). The rest of this section is devoted to the proof of (3) ⇒ (1), which will consist of showing that the unit and counit of the pair of adjoint functors $(L, R)$ in (3.1) are isomorphims, and will be done in several steps.

**3.3. The canonical isomorphisms.** The proof of Theorem 3.1 will require a number of “canonical” isomorphisms, that we construct in this subsection.

For a left $H$-module $X$, endow $C \otimes X$ with the tensor product left $H$-module structure and with the left $C$-comodule structure provided by the comultiplication of $C$. In this way $C \otimes X$ becomes an object of $C_H\mathcal{M}$ (in fact $C \otimes X$ is the image of $X$ by the right adjoint to the forgetful functor $C_H\mathcal{M} \to H\mathcal{M}$).

Our first canonical isomorphism is as follows.

**Proposition 3.2.** Let $X$ be left $H$-module. The canonical map

$$\kappa_X : H \otimes_A X \to C \otimes X \\ h \otimes_A x \mapsto \pi(h_{(1)}) \otimes h_{(2)} \cdot x$$

is an isomorphism in the category $C_H\mathcal{M}$.

*Proof.* It is a direct verification that $\kappa_X$ is a morphism in $C_H\mathcal{M}$, and that

$$C \otimes X \to H \otimes_A X \\ \pi(h) \otimes x \mapsto h_{(1)} \otimes_A S(h_{(2)}).x$$

is the inverse isomorphism. $\square$
prove that $\iota_M \circ E_{M|X(M)} = \mathrm{id}_{X(M)}$, similarly to before, notice that the right A-linearity of $E$ enables us to define the map

$$ E'_M : B \otimes_A B \otimes_A M \to B \otimes_A M \\ x \otimes_A y \otimes_A m \mapsto E(x)y \otimes_A m $$

For $\sum_i x_i \otimes_A m_i \in X(M)$, we have

$$ \sum_i 1_B \otimes_A x_i \otimes_A m_i = \sum_i x_i \otimes_A 1_B \otimes_A m_i $$

and applying $E'_M$ yields

$$ \sum_i x_i \otimes_A m_i = \sum_i E(x_i) \otimes_A m_i = 1_B \otimes_A \left( \sum_i E(x_i) \cdot m_i \right) = \iota_M \circ E_M \left( \sum_i x_i \otimes_A m_i \right) $$

which concludes the proof. $\square$

**Proposition 2.4.** Let $A \subset H$ be a right coideal subalgebra. Assume that $A \subset H$ is right faithfully flat or that $A$ is a direct summand in $H$ as a right $A$-module. Then for any left $A$-module $M$, the unit map

$$ \eta_M : M \to \mathrm{coC}(H \otimes_A M) $$

is an isomorphism.

*Proof.* We consider the canonical isomorphism

$$ \begin{gathered} \kappa'_M = \kappa_{H \otimes_A M} : H \otimes_A (H \otimes_A M) \to C \otimes (H \otimes_A M) \\ h \otimes_A h' \otimes_A m \mapsto \pi(h_{(1)}) \otimes h_{(2)}h' \otimes_A m \end{gathered} $$

from Proposition 2.2. For $\sum_i h_i \otimes_A m_i \in \mathrm{coC}(H \otimes_A M)$, we have

$$ \kappa'_M \left( \sum_i h_i \otimes_A 1_H \otimes_A m_i \right) = \sum_i \pi(1) \otimes h_i \otimes_A m_i = \kappa'_M \left( \sum_i 1_H \otimes_A h_i \otimes_A m_i \right) $$

and the injectivity of $\kappa'_M$ gives

$$ \sum_i h_i \otimes_A 1_H \otimes_A m_i = \sum_i 1_H \otimes_A h_i \otimes_A m_i $$

Our assumption then ensures, by Proposition 2.3, the existence of a unique $m \in M$ such that $\sum_i h_i \otimes_A m_i = 1_H \otimes_A m$. This therefore defines a map $\mathrm{coC}(H \otimes_A M) \to M$, which is clearly an inverse to $\eta_M$. $\square$

*Remark 2.5.* If $A$ is a direct summand in $H$ as a right $A$-module, Proposition 2.4 ensures in particular that $\mathrm{coC}H = A$. Hence in view of Proposition 2.2, we see that $A \subset H$ is coalgebra Galois extension over $C$. Once this is noticed, the shortest way to obtain a proof of Theorem 1.1 is certainly to invoke [1, Proposition 4.4]. I thank B. Mesabishvili for pointing out coalgebra Galois extension in this context.

**2.4. The counit of the adjunction.** We now analyse the counit of our adjunction $(L, R)$. We begin with a lemma, in which we use the following notation: if $X$ is an object of $C_H\mathcal{M}$, we denote $i_X : \mathrm{coC}X \to X$ the natural inclusion map.

**Lemma 2.6.** Let $A \subset H$ be a right coideal subalgebra and let $X$ be an object of $C_H\mathcal{M}$. Assume that $A$ is a direct summand in $H$ as a right $A$-module, and let $E : H \to A$ be a right conditional expectation. Then the map

$$ p_X = (E \otimes_A \mathrm{id}_X) \circ \kappa_X^{-1} \circ \alpha_X : X \to X $$
We will need also a variation on the above canonical isomorphisms, for which we introduce some more notation. For an object $X$ in $\mathcal{C}_H\mathcal{M}$, we consider the following maps:

$$ \delta = \delta_X : X \to C \otimes X \\ x \mapsto x_{(-1)} \otimes x_{(0)} - \pi(1) \otimes x $$

$$ \nabla = \nabla_X : C \otimes X \to C \otimes (C \otimes X) \\ \pi(h) \otimes x \mapsto \pi(h) \otimes x_{(-1)} \otimes x_{(0)} - \pi(h_{(1)}) \otimes \pi(h_{(2)}) \otimes x $$

It is immediate that $\delta$ is left $A$-linear, while $\nabla$ is a morphism in $\mathcal{C}_H\mathcal{M}$.

**Proposition 3.3.** Let $X$ be an object in $\mathcal{C}_H\mathcal{M}$. We have an isomorphism

$$ \tilde{\kappa}_X : H \otimes_A \delta(X) \to \nabla(C \otimes X) \\ h \otimes_A \delta(x) \mapsto \nabla(\pi(h_{(1)})) \otimes h_{(2)}.x $$

making the following diagram commute

$$ \begin{tikzcd} H \otimes_A X \arrow[r, "\mathrm{id}_{H \otimes_A \delta}"] \arrow[d, "\kappa_X"] & H \otimes_A \delta(X) \arrow[d, "\tilde{\kappa}_X"] \\ C \otimes X \arrow[r, "\nabla"] & \nabla(C \otimes X) \end{tikzcd} $$

*Proof.* Consider the map $\kappa_X^0 : H \otimes_A X \to C \otimes (C \otimes X)$ given by the composition

$$ H \otimes_A X \xrightarrow{\mathrm{id}_{H \otimes_A \delta}} H \otimes_A (C \otimes X) \xrightarrow{\kappa_{C \otimes X}} C \otimes (C \otimes X) $$

For $h, h' \in H$ and $x \in X$, we have $\kappa_{C \otimes X}(h \otimes_A \pi(h') \otimes x) = \pi(h_{(1)}) \otimes \pi(h_{(2)}h') \otimes h_{(3)}.x$, hence

$$ \begin{aligned} \kappa_X^0(h \otimes_A x) &= \kappa_{C \otimes X}(h \otimes_A x_{(-1)} \otimes x_{(0)} - h \otimes_A \pi(1) \otimes x) \\ &= \pi(h_{(1)}) \otimes h_{(2)}.x_{(-1)} \otimes h_{(3)}.x_{(0)} - \pi(h_{(1)}) \otimes \pi(h_{(2)}) \otimes h_{(3)}.x \\ &= \nabla(\pi(h_{(1)}) \otimes h_{(2)}.x) \end{aligned} $$

This means that the following diagram commutes

$$ \begin{tikzcd} H \otimes_A X \arrow[r, "\mathrm{id}_{H \otimes_A \delta}"] \arrow[d, "\kappa_X"] & H \otimes_A (C \otimes X) \arrow[d, "\kappa_{C \otimes X}"] \\ C \otimes X \arrow[r, "\nabla"] & C \otimes (C \otimes X) \end{tikzcd} $$

For $x \in \mathrm{Ker}(\delta) = \mathrm{coC}X$, we then have $\kappa_X^0(h \otimes_A x) = 0$, hence, since $\delta(X) \simeq X/\mathrm{Ker}(\delta)$, we get that $\kappa_X^0$ induces the announced map $\tilde{\kappa}_X : H \otimes_A \delta(X) \to \nabla(C \otimes X)$.

To construct the inverse map, consider the composition

$$ (\mathrm{id}_H \otimes_A \delta) \circ \kappa_X^{-1} : C \otimes X \to H \otimes_A \delta(X) $$

In view of the previous commutative diagram, this map coincides with $\kappa_{C \otimes X}^{-1} \circ \nabla$, hence vanishes on $\mathrm{Ker}(\nabla)$, and thus induces

$$ \begin{aligned} \nabla(C \otimes X) &\to H \otimes_A \delta(X) \\ \nabla(\pi(h) \otimes x) &\mapsto h_{(1)} \otimes \delta(S(h_{(2)}).x) \end{aligned} $$

which is clearly an inverse isomorphism to $\tilde{\kappa}_X$.

□
is a projection of $X$ onto $^{\text{co}C}X$. If moreover $E$ is an $A$-bimodule conditional expectation, then $p_X: X \to \text{co}C X$ is $A$-linear, and hence the map $\text{id}_H \otimes_A i_X: H \otimes_A \text{co}C X \to H \otimes_A X$ is injective.

**Proof.** The above map $p_X$ is well-defined since $E$ is right $A$-linear, is an $A$-bimodule map as soon as $E$ is, and for $x \in \text{co}C X$, it is clear that $p_X(x) = x$, since $E(1_H) = 1_A$. Thus one just has to check that $p_X$ has values into $\text{co}C X$, which follows from the commutativity of the following diagram:

<figure><img src="image_3.png" /></figure>

where $\nu_X: H \otimes_A X: H \otimes A(C \otimes X)$ is defined by $\nu_X(h \otimes_A x) = h \otimes_A \pi(1) \otimes x$.

If $E$ is an $A$-bimodule map, then $p_X$ is left $A$-linear and therefore $\text{id}_H \otimes_A p_X$ provides a retraction to $\text{id}_H \otimes_A i_X$, which gives the last statement. $\square$

**Proposition 2.7.** Let $A \subset H$ be a right coideal subalgebra and let $X$ be an object of $\mathcal{C}_H \mathcal{M}$. Consider the following assertions:

(a) $\text{id}_H \otimes_A i_X: H \otimes_A \text{co}C X \to H \otimes_A X$ is injective;

(b) $\mu_X: H \otimes_A \text{co}C X \to X$ is injective;

(c) $\mu_X: H \otimes_A \text{co}C X \to X$ is surjective.

Then we have (a) $\iff$ (b) $\implies$ (c). These assertions hold true if one of the following conditions is satisfied:

(1) $H$ is flat as a right $A$-module;

(2) $H$ is a direct summand in $B$ as an $A$-bimodule.

**Proof.** Consider the map $\delta: X \to C \otimes X$ defined by $\delta(x) = x_{(-1)} \otimes x_{(0)} - \pi(1) \otimes x$. This map is $A$-linear and the sequence of $A$-modules $0 \to \text{co}C X \xrightarrow{i_X} X \xrightarrow{\delta} C \otimes X$ is exact. Applying $H \otimes_A$ yields the sequence

$$H \otimes_A^{\text{co}C} X \xrightarrow{\text{id}_H \otimes_A i_X} H \otimes_A X \xrightarrow{\text{id}_H \otimes_A \delta} H \otimes_A (C \otimes X)$$

that fits in the commutative diagram

<figure><img src="image_4.png" /></figure>

where $\nabla: C \otimes X \to C \otimes (C \otimes X)$ is defined by

$$\nabla(\pi(h) \otimes x) = \pi(h) \otimes x_{(-1)} \otimes x_{(0)} - \pi(h_{(1)}) \otimes \pi(h_{(2)}) \otimes x$$

Since $\kappa_X^{-1} \circ \alpha_X$ is injective, we get (a) $\iff$ (b).

Assume that (a) holds. Then the upper sequence in the above diagram is exact (while the lower row is exact by construction), and it is a simple diagram chasing to check that $\mu_X$ is surjective.
### 3.4. The counit of the adjunction.
We now analyse the counit of our adjunction $(L, R)$.

**Proposition 3.4.** Let $X$ be an object in $C_H\mathcal{M}$. The counit map $\mu_X : H \otimes_A^{\text{co}C} X \to X$ is surjective, and hence the functor $R = \text{co}C(-) : C_H\mathcal{M} \to_A \mathcal{M}$ is faithful. If moreover $H$ is flat as a right $A$-module, then $\mu_X$ is an isomorphism.

*Proof.* Starting with the exact sequence of $A$-modules $0 \to \text{co}C X \xrightarrow{i} X \xrightarrow{\delta} \delta(X) \to 0$ and applying $H \otimes_A$ yields the exact sequence

$$H \otimes_A^{\text{co}C} X \xrightarrow{\text{id}_H \otimes_A^i} H \otimes_A X \xrightarrow{\text{id}_H \otimes_A^\delta} H \otimes_A \delta(X) \to 0$$

that fits in the commutative diagram

$$\begin{tikzcd}
& & H \otimes_A^{\text{co}C} X \arrow[r, "\text{id}_H \otimes_A^i"] & H \otimes_A X \arrow[r, "\text{id}_H \otimes_A^\delta"] & H \otimes_A \delta(X) \arrow[r] & 0 \\
& \big|_{\mu_X} & & \big|_{\kappa_X} & & \\
0 & X & \arrow[r, "\alpha_X"] & C \otimes X & \arrow[r, "\nabla"] & \nabla(C \otimes X) \arrow[r] & 0
\end{tikzcd}$$

where $\kappa_X$ and $\tilde{\kappa}_X$ are the isomorphisms of Proposition 3.2 and Proposition 3.3 respectively and the bottom row is exact. The surjectivity of $\mu_X$ is then obtained from an easy diagram chasing. If $f : X \to Y$ is a morphism in $C_H\mathcal{M}$, the commutativity of the diagram

$$\begin{tikzcd}
& & H \otimes_A^{\text{co}C} X \arrow[r, "\text{id}_H \otimes_A f"] & H \otimes_A^{\text{co}C} Y \\
& \big|_{\mu_X} & & \big|_{\mu_Y} \\
X & f & Y
\end{tikzcd}$$

together with the surjectivity of $\mu_X$ ensures that if $f|_{\text{co}C X} = 0$, then $f = 0$, and hence the functor $R$ is faithful.

Assuming moreover that $H$ is flat as a right $A$-module enables us, as in the proof of [16, Theorem 3.7], to enlarge the previous commutative diagram to the following commutative diagram with exact rows

$$\begin{tikzcd}
& & H \otimes_A^{\text{co}C} X \arrow[r, "\text{id}_H \otimes_A^i"] & H \otimes_A X \arrow[r, "\text{id}_H \otimes_A^\delta"] & H \otimes_A \delta(X) \arrow[r] & 0 \\
& \big|_{\mu_X} & & \big|_{\kappa_X} & & \\
0 & X & \arrow[r, "\alpha_X"] & C \otimes X & \arrow[r, "\nabla"] & \nabla(C \otimes X) \arrow[r] & 0
\end{tikzcd}$$

and again a diagram chasing gives that $\mu_X$ is an isomorphism. $\square$

### 3.5. The unit of the adjunction.
The next step is to study the unit of our adjunction.

**Proposition 3.5.** Assume that $A \subset H$ is right faithfully flatish. Then for any left $A$-module $M$, the unit map

$$\eta_M : M \to \text{co}C(H \otimes_A M)$$

is an isomorphism.

*Proof.* We consider the canonical isomorphism

$$\begin{align*}
\kappa'_M &= \kappa_{H \otimes_A M} : H \otimes_A (H \otimes_A M) \to C \otimes (H \otimes_A M) \\
h \otimes_A h' \otimes_A m &\mapsto \pi(h_{(1)}) \otimes h_{(2)} h' \otimes_A m
\end{align*}$$
If (1) holds, then (a) holds by the definition of flatness, and if (2) holds, Lemma 2.6 ensures that (a) holds. □

The proof of right faithful flatness in Theorem 1.1 is now immediate: if $A$ is a direct summand in $H$ as an $A$-bimodule, Proposition 2.4 ensures that the unit of the adjunction $(L, R)$ is an isomorphism, and Proposition 2.7 ensures that the counit is an isomorphism as well, so $B \subset H$ is right faithfully flat by (3)$\Rightarrow$(2) in Theorem 2.1.

Under the assumption that $A$ is a direct summand in $H$ as an $A$-bimodule, left faithful flatness is shown similarly by considering the right $H$-module quotient coalgebra $D = H/A^+H$, and the category $\mathcal{M}_H^D$, we do not write the details.

Notice as well that Proposition 2.4 and Proposition 2.7 combined together immediately show (2)$\Rightarrow$(3) in Theorem 2.1.

### 3. CONCLUDING REMARK

In the recent paper [7], S. Skryabin shows that for any finite-dimensional nonsemisimple Hopf algebra $A$ there exists a Hopf algebra $H$ containing $A$ as a Hopf subalgebra such that $H$ is not flat over $A$. This shows that the assumption “$A$ is a direct summand in $H$ as an $A$-bimodule” in Theorem 1.1 cannot be weakened to “$A$ is direct summand as a right (or left) $H$-module”, because since a finite-dimensional Hopf algebra is self-injective (projective modules are injective), $A$ is direct summand as a right (or left) $H$-module.

We conclude by summarizing what is known about flatness when $A$ is direct summand as a right $H$-module.

**Proposition 3.1.** Let $A \subset H$ be a right coideal subalgebra. Assume that $A$ is a direct summand in $H$ as a right $A$-module. Then the following assertions are equivalent:

(1) $H$ is faithfully flat as a right $A$-module;

(2) $H$ is flat as a right $A$-module;

(3) For any object $X$ of $C_H\mathcal{M}$, the map $\text{id}_H \otimes_A i_X : H \otimes_A \text{co}C X \to H \otimes_A X$ is injective;

(4) For any object $X$ of $C_H\mathcal{M}$, the map $\mu_X : H \otimes_H \text{co}C X \to X$ is surjective.

*Proof.* By definition (1)$\Rightarrow$(2) and (2)$\Rightarrow$(3), while (3)$\Rightarrow$(4) follows from Proposition 2.7. Assume that (4) holds. Since $A$ is a direct summand in $B$ as a right $A$-module, by Proposition 2.4 we are in the situation of a pair of adjoint functors $(L, R)$ whose unit is an isomorphism and counit is an epimorphism: it is then easy to check that $R$ faithful, and that the counit is a monomorphism as well, so that $L$ and $R$ are inverse equivalences (since we are dealing with categories in which morphisms that are both monomorphisms and epimorphisms are isomorphisms). Hence $H$ is faithfully flat as a right $A$-module. □

### REFERENCES

[1] T. Brzezinski, On modules associated to coalgebra Galois extensions, *J. Algebra* **215** (1999), no. 1, 290-317.

[2] T. Brzezinski, R. Wisbauer, Corings and comodules. London Mathematical Society Lecture Note Series, 309. Cambridge University Press, Cambridge, 2003.

[3] A. Chirvasitu, Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras, *Algebra Number Theory* **8** (2014), no. 5, 1179-1199.

[4] B. Mesablishvili, Monads of effective descent type and comonadicity, *Theory Appl. Categ.* **16** (2006), No. 1, 1-45.

[5] S. Montgomery, Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82. American Mathematical Society, 1993.

[6] P. Nuss, Noncommutative descent and non-abelian cohomology, *K-Theory* **12** (1997), no. 1, 23-74.

[7] S. Skryabin, Failure of flatness over finite-dimensional Hopf subalgebras, arXiv:2506.16292.
from Proposition 3.2. For $\sum_i h_i \otimes_A m_i \in \text{coC}(H \otimes_A M)$, we have

$$ \kappa'_M \left( \sum_i h_i \otimes_A 1_H \otimes_A m_i \right) = \sum_i \pi(1) \otimes_A h_i \otimes_A m_i = \kappa'_M \left( \sum_i 1_H \otimes_A h_i \otimes_A m_i \right) $$

and the injectivity of $\kappa'_M$ gives

$$ \sum_i h_i \otimes_A 1_H \otimes_A m_i = \sum_i 1_H \otimes_A h_i \otimes_A m_i $$

The faithful flatishness assumption then ensures the existence of a unique $m \in M$ such that $\sum_i h_i \otimes_A m_i = 1_H \otimes_A m$. This therefore defines a map $\text{coC}(H \otimes_A M) \to M$, which is clearly an inverse to $\eta_M$. $\square$

3.6. **Proof of Theorem 3.1.** We need a last elementary lemma:

**Lemma 3.6.** Let $(F, G) : \mathcal{C} \to \mathcal{D}$ be a pair of adjoint functors. Assume that the corresponding unit $\eta : 1_\mathcal{C} \to GF$ is an isomorphism and that $G$ is faithful. Then $F$ preserves monomorphisms.

*Proof.* Let $u : X \to Y$ be a monomorphism in $\mathcal{C}$, and let $f, g : V \to F(X)$ be morphisms in $\mathcal{D}$ such that $F(u) \circ f = F(u) \circ g$. Then $GF(u) \circ G(f) = GF(u) \circ G(g)$, and since $GF(u) \circ \eta_X = \eta_Y \circ u$, we obtain, because $\eta_Y$ and $\eta_X$ are isomorphisms and $u$ is a monomorphism, that $G(f) = G(g)$, and we conclude $f = g$ by faithfulness of $G$. $\square$

We can now finish the proof of (3) ⇒ (1) in Theorem 3.1. Assume that $A \subset H$ is right faithfully flatish. Proposition 3.5 then ensures that the unit of the adjunction $(L, R)$ is an isomorphism. Moreover the functor $R$ is faithfully by Proposition 3.4, and hence $L$ preserves monomorphisms by Lemma 3.6, which precisely means that $H$ is flat as a right $A$-module. Then the last statement in Proposition 3.4 ensures that the counit of the adjunction $(L, R)$ is an isomorphism as well, so that $(L, R)$ forms a pair of inverse equivalences of categories.

# 4. CONSEQUENCES AND AN EXAMPLE

## 4.1. Left/right variations.
Theorem 1.1 has an obvious left version:

**Theorem 4.1.** Let $H$ be a Hopf algebra and let $A \subset H$ be a left coideal subalgebra. If $A$ is a direct summand in $H$ as a left $A$-module, then $H$ is faithfully flat as a left $A$-module.

*Proof.* This follows from Theorem 1.1, applied to the right coideal subalgebra $A^{\text{op}} \subset H^{\text{opcop}}$, since a left conditional expectation $E : H \to A$ is precisely a right conditional expectation $H^{\text{op}} \to A^{\text{op}}$, and since $H$ is left $A$-faithfully flat if and only if $H^{\text{op}}$ is right $A^{\text{op}}$-faithfully flat. $\square$

Notice that it is also possible to prove the previous result by adapting Section 3 to this left setting. We leave this to the reader. We have also the following variation of Theorem 1.1:

**Theorem 4.2.** Let $H$ be a Hopf algebra with bijective antipode and let $A \subset H$ be a right coideal subalgebra. If $A$ is a direct summand in $H$ as a left $A$-module, then $H$ is faithfully flat as a left $A$-module.

*Proof.* Apply Theorem 1.1 to the right coideal subalgebra $A^{\text{op}} \subset H^{\text{op}}$ ($H^{\text{op}}$ being a Hopf algebra by bijectivity of the antipode). $\square$
[8] M. Takeuchi, Relative Hopf modules - Equivalences and freeness criteria, *J. Algebra* **60** (1979), no. 2, 452-471.

[9] W.C Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics, 66. Springer-Verlag, 1979.

UNIVERSITÉ CLERMONT AUVERGNE, CNRS, LMBP, F-63000 CLERMONT-FERRAND, FRANCE
*Email address: julien.bichon@uca.fr*
4.2. **Projectivity.** One nice feature of faithful flatness in the setting of coideal subalgebras is that it is often equivalent to projectivity [9]. Here this gives the following result.

**Theorem 4.3.** Let $H$ be a Hopf algebra with bijective antipode and let $A \subset H$ be a left or right coideal subalgebra.

(1) If $A$ is a direct summand in $H$ as a right $A$-module, then $H$ is projective as a right $A$-module.

(2) If $A$ is a direct summand in $H$ as a left $A$-module, then $H$ is projective as a left $A$-module.

*Proof.* Assume first that $A \subset H$ is a right coideal subalgebra. (1) follows from Theorem 1.1 combined with the right version of [9, Theorem 2] (which, as explained after the statement of [9, Theorem 2], follows from the left statement applied to $A^{\text{op}} \subset H^{\text{op}}$). (2) follows from Theorem 4.2 and [9, Theorem 2]. The case when $A \subset H$ is a left coideal subalgebra follows from the right case applied to $A^{\text{op}} \subset H^{\text{opcop}}$. $\square$

4.3. **Example: free wreath products.** We finish the paper by an illustration. For $n \ge 1$, consider the algebra $A_s(n)$ presented by generators $u_{ij}$, $1 \le i, j \le n$, and relations

$$ \sum_{j=1}^{n} u_{ij} = 1 = \sum_{j=1}^{n} u_{ji}, \quad u_{ij}u_{ik} = 0 = u_{ji}u_{ki}, \text{ for } k \neq j, $$

This is the coordinate algebra of Wang's quantum permutation group [21], which can be defined over any field, and has the Hopf algebra structure

$$ \Delta(u_{ij}) = \sum_{k} u_{ik} \otimes u_{kj}, \quad \varepsilon(u_{ij}) = \delta_{ij}, \quad S(u_{ij}) = u_{ji} $$

Let $A$ be a Hopf algebra, and consider $A^{*n}$, the free product algebra of $n$ copies of $A$, which inherits a natural Hopf algebra structure such that the canonical morphisms $\nu_i: A \to A^{*n}$, $1 \le i \le n$, are Hopf algebras morphisms. The free wreath product $A_{*w}A_s(n)$ [3] is the quotient of the algebra $A^{*n} \ast A_s(n)$ by the two-sided ideal generated by the elements:

$$ \nu_k(a)u_{ki} - u_{ki}\nu_k(a), \quad 1 \le i, k \le n, \quad a \in A. $$

The free wreath product $A_{*w}A_s(n)$ admits a Hopf algebra structure given by

$$ \Delta(u_{ij}) = \sum_{k=1}^{n} u_{ik} \otimes u_{kj}, \quad \Delta(\nu_i(a)) = \sum_{k=1}^{n} \nu_i(a_{(1)})u_{ik} \otimes \nu_k(a_{(2)}), $$

$$ \varepsilon(u_{ij}) = \delta_{ij}, \quad \varepsilon(\nu_i(a)) = \varepsilon(a), \quad S(u_{ij}) = u_{ji}, \quad S(\nu_i(a)) = \sum_{k=1}^{n} \nu_k(S(a))u_{ki}. $$

It is immediate that there is an algebra map $E: A_{*w}A_s(n) \to A^{*n}$ given by $E(u_{ij}) = \delta_{ij}$ and $E(\nu_i(a)) = \nu_i(a)$, which thus gives a retraction to the natural map $A^{*n} \to A_{*w}A_s(n)$. Hence $A^{*n}$ stands as left coideal subalgebra of $A_{*w}A_s(n)$, and $E$ furnishes a left and right conditional expectation. Theorem 4.3 therefore ensures that if $A$ has bijective antipode, then $A_{*w}A_s(n)$ is projective as a left and right $A^{*n}$-module. This was shown and used in the proof of [4, Theorem 8.4], using [7], under the additional assumption that $k = \mathbb{C}$ and that $A$ is a compact Hopf algebra.
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[5] T. Brzezinski, R. Wisbauer, Corings and comodules. London Mathematical Society Lecture Note Series, 309. Cambridge University Press, Cambridge, 2003.

[6] A. Chirvasitu, Cosemisimple Hopf algebras are faithfully flat over Hopf subalgebras, Algebra Number Theory 8 (2014), no. 5, 1179-1199.

[7] A. Chirvasitu, Relative Fourier transforms and expectations on coideal subalgebras, J. Algebra 516 (2018), 271-297.

[8] M. Demazure, P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. Masson, Paris, 1970.

[9] A. Masuoka, D. Wigner, Faithful flatness of Hopf algebras, J. Algebra 170 (1994), no. 1, 156-164.

[10] B. Mesabishvili, Monads of effective descent type and comonadicity, Theory Appl. Categ. 16 (2006), No. 1, 1-45.

[11] S. Montgomery, Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82. American Mathematical Society, 1993.

[12] W. D. Nichols, M. B. Zoeller, A Hopf algebra freeness theorem, Amer. J. Math. 111 (1989), no. 2, 381-385.

[13] P. Nuss, Noncommutative descent and non-abelian cohomology, K-Theory 12 (1997), no. 1, 23-74.

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[21] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), no. 1, 195-211.

[22] W.C Waterhouse, Introduction to affine group schemes. Graduate Texts in Mathematics, 66. Springer-Verlag, 1979.

UNIVERSITÉ CLERMONT AUVERGNE, CNRS, LMBP, F-63000 CLERMONT-FERRAND, FRANCE
*Email address: julien.bichon@uca.fr*