3.1.1 General facts about left ideals two-sided coideals and right coideal subrings

As a matter of notation, for $(A,{\mathcal H})$ a bialgebroid and $B\subseteq {\mathcal H}$ , we set

Denote by $\iota \colon B \to {\mathcal H}$ the inclusion. Assume that B is an $A^{\mathrm {o}}$ -subring of ${\mathcal H}$ via t (as we will have later in the paper), that is, B is an $A^{\mathrm {o}}$ -ring via a $\Bbbk $ -algebra extension $t'\colon A^{\mathrm {o}}\to B$ such that $\iota \circ t'= t$ . If we consider the restriction

of $\varepsilon $ to B, then the latter is a right A-linear morphism (left $A^{\mathrm {o}}$ -linear, for the sake of precision) and $B^+ = {\mathcal H}^+ \cap B = \ker (\varepsilon )\cap B = \ker (\varepsilon ')$ .

Remark 3.1 If B is an $A^{\mathrm {o}}$ -subring of ${\mathcal H}$ via t, then we have that

(3.1)

is a short exact sequence of left B-modules, where A has the left B-module structure coming from the restriction of scalars along the inclusion $\iota $ of B into ${\mathcal H}$ . Namely,

(3.2)

for all $a\in A,b\in B$ (see (2.10)). Moreover, (3.1) is a split short exact sequence of right A-modules (in fact, left $A^{\mathrm {o}}$ -modules via $t'$ ), where $t'$ provides an A-linear section for $\varepsilon '$ in view of (B6):

$$ \begin{align*} \varepsilon'\big(t'(a^{\mathrm{o}})\big) = \varepsilon\Big(\iota\big(t'(a^{\mathrm{o}})\big)\Big) = \varepsilon\big(t(a^{\mathrm{o}})\big) = a. \blacktriangle \end{align*} $$

Recall that an $A^{\mathrm {o}}$ -subring B of ${\mathcal H}$ via t is a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring if it admits a right A-linear coaction $\delta \colon {B}{}_{t^{\mathrm {o}}} \to {B}{}_{t^{\mathrm {o}}} \otimes _{A} {}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$ such that the following diagram commutes:

(3.3)

Remark 3.2 It is a well-known fact that if $(A,{\mathcal H})$ is a left bialgebroid, then the category $ {}^{{\mathcal H}}\textsf {Comod}$ of left ${\mathcal H}$ -comodules is a monoidal category with tensor product $\otimes _{A}$ and unit object A, where the ${\mathcal H}$ -coaction on A is provided by the source map (see, e.g., [Reference Böhm3, Theorem 3.18], where the property is stated for the right-hand scenario) and where every left ${\mathcal H}$ -comodule M is a right A-module with action

(3.4)

for all $m \in M$ and $a \in A$ . By applying this result to the co-opposite left bialgebroid $(A^{\mathrm {o}},{\mathcal H}^{\mathrm {co}})$ , we conclude that the category $ \textsf {Comod}{}^{{\mathcal H}}$ of right comodules over the left bialgebroid $(A,{\mathcal H})$ is monoidal too, with tensor product $\otimes _{A^{\mathrm {o}}}$ and unit $A^{\mathrm {o}}$ , where the ${\mathcal H}$ -coaction on $A^{\mathrm {o}}$ is provided by the target map. In this setting, if ${}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}$ is A-flat, then a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring is nothing other than a monoid in $\big ( \textsf {Comod}{}^{{\mathcal H}},\otimes _{A^{\mathrm {o}}},A^{\mathrm {o}}\big )$ .

Recall that a two-sided coideal N in a bialgebroid $(A,{\mathcal H})$ is an A-subbimodule of ${}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$ such that

$$ \begin{align*} \Delta(N) \subseteq \operatorname{Im} \Big( N \otimes_{A} {\mathcal H} + {\mathcal H} \otimes_{A} N\Big) \qquad \text{and} \qquad \varepsilon(N)=0_A, \end{align*} $$

where $\operatorname {Im}(-)$ denotes the canonical image in the tensor product ${}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}} \otimes _{A} {}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$ .

3.1.2 Correspondence between left ideals two-sided coideals and quotient module corings

Proposition 3.3 We have a well-defined bijective correspondence

Proof Let us begin by picking a left ideal two-sided coideal $I\subseteq {\mathcal H}$ and consider the canonical projection $\pi \colon {\mathcal H} \to {\mathcal H}/I$ . Since I is a left ${\mathcal H}$ -submodule of ${\mathcal H}$ , ${\mathcal H}/I$ becomes a left ${\mathcal H}$ -module and, in particular, an $A^{\mathrm {e}}$ -module with ${}_{s} {{\mathcal H}/I}{}_{t^{\mathrm {o}}}$ and $\pi $ is left ${\mathcal H}$ -linear and, in particular, A-bilinear. Since I is a two-sided coideal, ${}_{s} {{\mathcal H}/I}{}_{t^{\mathrm {o}}}$ inherits a structure $\left (\overline {\Delta },\overline {\varepsilon }\right )$ of A-coring in such a way that

are commutative diagrams in ${}_{A}{\textsf {Mod}}_{A} = {}_{A^{\mathrm {e}}}\textsf {Mod}$ . Let us show that both $\overline {\Delta }$ and $\overline {\varepsilon }$ are morphisms of left ${\mathcal H}$ -modules. Set

, for the sake of brevity. The left ${\mathcal H}$ -linearity of $\overline {\Delta }$ is provided by the commutativity of the following diagram:

The left ${\mathcal H}$ -linearity of $\overline {\varepsilon }$ instead is provided by the commutativity of the following diagram:

Summing up, ${\mathcal H}/I$ is a left ${\mathcal H}$ -module coring.

In the opposite direction, assume that we have a surjective morphism of vector spaces $\pi \colon {\mathcal H} \to \overline {{\mathcal H}}$ which is of left ${\mathcal H}$ -module corings and consider $\ker (\pi ) \subseteq {\mathcal H}$ . Since $\pi $ is of left ${\mathcal H}$ -modules, $\ker (\pi )$ is a left ideal in ${\mathcal H}$ . Moreover, being $\pi $ of A-corings, we have that the following diagram with exact rows commutes:

Since $\ker (\pi \otimes _{A} \pi ) = \operatorname {Im}\big (\ker (\pi ) \otimes _{A} {\mathcal H} + {\mathcal H} \otimes _{A} \ker (\pi )\big )$ and since $\varepsilon \big (\ker (\pi )\big ) \subseteq \overline {\varepsilon }\big (\pi \big (\ker (\pi )\big )\big ) = 0_A$ , it follows that $\ker (\pi )$ is also a two-sided coideal.

3.1.3 From left ideal two-sided coideals to right coideal subrings

Proposition 3.4 Under the assumption that ${}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}$ is A-flat, we have a well-defined inclusion-preserving correspondence

Proof Let us begin by picking a left ideal two-sided coideal $I\subseteq {\mathcal H}$ , and consider the canonical projection $\pi \colon {\mathcal H} \to {\mathcal H}/I$ . We already know from Proposition 3.3 that ${\mathcal H}/I$ is an A-coring and that the canonical projection $\pi \colon {\mathcal H} \to {\mathcal H}/I$ is a morphism of left ${\mathcal H}$ -module corings. Therefore, in view of Lemma 2.10, ${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$ is an $A^{\mathrm {o}}$ -subring of ${\mathcal H}$ via t. To finish checking the validity of the first correspondence, we are left to check that it is an ${\mathcal H}$ -comodule with respect to the obvious coaction induced by $\Delta $ . To this aim, recall that ${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$ can be realized as the following equalizer in $\textsf {Mod}_{A}$ (in fact, in ${}_{A^{\mathrm {o}}}\textsf {Mod}$ ):

(3.5)

Since ${}_{s} {{{\mathcal H}}}$ is A-flat, it is enough for us to check that, for every $x\in {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$ , $\sum x_1 \otimes _{A} x_2$ equalizes the pair $\Big ((\pi \circ t) \otimes _{A} {\mathcal H} \otimes _{A} {\mathcal H}, (\pi \otimes _{A}{\mathcal H}\otimes _{A}{\mathcal H})(\Delta \otimes _{A}{\mathcal H})\Big )$ . However,

$$ \begin{align*} (\pi ~\otimes_{A}~ & {\mathcal H} \otimes_{A} {\mathcal H})(\Delta\otimes_{A}{\mathcal H})\left(\sum x_1 \otimes_{A} x_2\right) = (\pi\otimes_{A}{\mathcal H}\otimes_{A}{\mathcal H})(\Delta\otimes_{A}{\mathcal H})\Delta(x) \\ & \stackrel{(2.6)}{=} (\pi\otimes_{A}{\mathcal H}\otimes_{A}{\mathcal H})({\mathcal H}\otimes_{A}\Delta)\Delta(x) = \left(\frac{{\mathcal H}}{I}\otimes_{A}\Delta\right)(\pi\otimes_{A}{\mathcal H})\Delta(x) \\ & \stackrel{(3.5)}{=} \left(\frac{{\mathcal H}}{I}\otimes_{A}\Delta\right)(\pi(1_{\mathcal H})\otimes_{A}x) = \sum \pi(1_{\mathcal H}) \otimes_{A} x_1 \otimes_{A} x_2 \\ & \stackrel{\phantom{(2.6)}}{=} \big((\pi\circ t)\otimes_{A}{\mathcal H}\otimes_{A}{\mathcal H}\big)\left(\sum x_1\otimes_{A}x_2\right), \end{align*} $$

and hence ${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$ is indeed a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring via t.

It is clear that the correspondence is inclusion-preserving because if $I \subseteq J$ , then we have a left ${\mathcal H}$ -linear surjective morphism of A-corings ${\mathcal H}/I \twoheadrightarrow {\mathcal H}/J$ and hence ${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}} \subseteq {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{J}}}$ .

3.1.4 From right coideal subrings to left ideals two-sided coideals

In this section, we consider a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring B of ${\mathcal H}$ and utilize the notation introduced in Section 3.1.1.

Lemma 3.5 Let B be a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring of ${\mathcal H}$ via t, and let $\iota \colon B \to {\mathcal H}$ be the inclusion. Denote by $\pi \colon {\mathcal H} \to {\mathcal H}/{\mathcal H} B^+$ the canonical projection of left ${\mathcal H}$ -modules. Then

(3.6) $$ \begin{align} \pi \circ \iota = \pi \circ \iota \circ t'\circ \varepsilon' = \pi \circ t \circ \varepsilon \circ \iota. \end{align} $$

In particular,

(3.7) $$ \begin{align} \left(\pi \otimes_{A} {\mathcal H}\right) \circ \Delta \circ \iota = (\pi \otimes_{A} {\mathcal H}) \circ (t \otimes_{A} {\mathcal H}) \circ \iota. \end{align} $$

Proof Notice that for all $b \in B$ we have that $b - t'\big (\varepsilon '(b)^{\mathrm {o}}\big ) \in B^+$ and hence ${\iota \Big (b - t'\big (\varepsilon '(b)^{\mathrm {o}}\big )\Big ) \in {\mathcal H} B^+}$ . Therefore,

$$ \begin{align*} \pi\iota(b) = \iota(b) + {\mathcal H} B^+ = \iota\left(t'\left(\varepsilon'(b)^{\mathrm{o}}\right)\right) + \iota\big(b - t'\left(\varepsilon'(b)^{\mathrm{o}}\right)\big) + {\mathcal H} B^+ = \pi\iota\left(t'\left(\varepsilon'(b)^{\mathrm{o}}\right)\right). \end{align*} $$

The right-most equality follows by definition of $t'$ and $\varepsilon '$ . To conclude, observe that

$$ \begin{align*} \left(\pi \otimes_{A} {\mathcal H}\right) \circ \Delta \circ \iota & \stackrel{(3.3)}{=} \left(\pi \otimes_{A} {\mathcal H}\right) \circ \left(\iota \otimes_{A} {\mathcal H}\right) \circ \delta \\ & \stackrel{(3.6)}{=} \left(\pi \otimes_{A} {\mathcal H}\right) \circ (t \otimes_{A} {\mathcal H}) \circ (\varepsilon \otimes_{A} {\mathcal H}) \circ \left(\iota \otimes_{A} {\mathcal H}\right) \circ \delta \\ & \stackrel{(3.3)}{=} \left(\pi \otimes_{A} {\mathcal H}\right) \circ (t \otimes_{A} {\mathcal H}) \circ (\varepsilon \otimes_{A} {\mathcal H}) \circ \Delta \circ \iota \\ & \stackrel{\phantom{(3.3)}}{=} (\pi \otimes_{A} {\mathcal H}) \circ (t \otimes_{A} {\mathcal H}) \circ \iota.\\[-38pt] \end{align*} $$

Proposition 3.6 We have a well-defined inclusion-preserving correspondence

Proof Assume that we have a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring B of ${\mathcal H}$ , where the $A^{\mathrm {o}}$ -ring structure comes from a $\Bbbk $ -algebra extension $t'\colon A^{\mathrm {o}}\to B$ as in Remark 3.1. We may tensor (3.1) by ${\mathcal H}$ over B on the left and eventually apply the Snake Lemma to find the following commutative diagram of left ${\mathcal H}$ -modules, with exact rows:

If we consider , then this is a left ${\mathcal H}$ -ideal by construction. Let us prove that it is also a two-sided coideal.

First of all, let us collect some observations that will be needed afterward. Recall from Corollary 2.8 that we have a natural ${\mathcal H}$ -action on ${\mathcal H} \otimes _{A} {\mathcal H}$ given by

(3.8) $$ \begin{align} {\mathcal H} \otimes ({\mathcal H} \otimes_{A} {\mathcal H}) \xrightarrow{\cdot} {\mathcal H} \otimes_{A} {\mathcal H}, \qquad x \otimes \left(\sum_j h_j \otimes_{A} k_j\right) \mapsto \sum_{j} x_1h_j \otimes_{A} x_2k_j, \end{align} $$

and a natural action of ${\mathcal H}$ on ${\mathcal H}/{\mathcal H} B^+ \otimes _{A} {\mathcal H}$ given by

(3.9) $$ \begin{align} {\mathcal H} \otimes \left(\frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A} {\mathcal H}\right) \xrightarrow{\cdot} \frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A} {\mathcal H}, \quad x \otimes \left(\sum_j \pi(h_j) \otimes_{A} k_j\right) \mapsto \sum_{j} \pi(x_1h_j) \otimes_{A} x_2k_j, \end{align} $$

where $\pi \colon {\mathcal H} \to {\mathcal H}/{\mathcal H} B^+$ is the canonical projection. Moreover, $B\subseteq {\mathcal H}$ is a right ${\mathcal H}$ -comodule with a right A-linear coaction $\delta \colon {B}{}_{t^{\mathrm {o}}} \to {B}{}_{t^{\mathrm {o}}} \otimes _{A} {}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$ such that (3.3) commutes, that is, for every $b \in B$ , we have

(3.10) $$ \begin{align} \sum \iota(b_{0}) \otimes_{A} b_1 = \sum \iota(b)_1 \otimes_{A} \iota(b)_2. \end{align} $$

Therefore, (3.8) induces an action of B on ${\mathcal H} \otimes _{A} {\mathcal H}$ via restriction of scalars along $\iota $ ,

$$ \begin{align*} B \otimes \left({\mathcal H} \otimes_{A} {\mathcal H}\right) & \to {\mathcal H} \otimes_{A} {\mathcal H}, \\ b \otimes \left(\sum_j h_j \otimes_{A} k_j\right) & \mapsto \sum_{j} \iota(b)_1h_j \otimes_{A} \iota(b)_2k_j = \sum_{j} \iota(b_0)h_j \otimes_{A} b_{1}k_j, \end{align*} $$

with respect to which $\Delta \colon {\mathcal H} \to {\mathcal H} \otimes _{A} {\mathcal H}$ becomes left B-linear:

$$ \begin{align*}\Delta\big(\iota(b)h\big) = \sum \iota(b)_1h_1 \otimes_{A} \iota(b)_2h_2 = b \cdot \left(\sum h_1 \otimes_{A} h_2\right). \end{align*} $$

Similarly, (3.9) induces an action of B on ${\mathcal H}/{\mathcal H} B^+ \otimes _{A} {\mathcal H}$ . In particular, (3.8) and (3.9) themselves factor through the tensor product over B (by associativity of the multiplication in ${\mathcal H}$ ):

$$ \begin{align*}{\mathcal H} \otimes_{B} ({\mathcal H} \otimes_{A} {\mathcal H}) \xrightarrow{\mu} {\mathcal H} \otimes_{A} {\mathcal H} \qquad \text{and} \qquad {\mathcal H} \otimes_{B} \left(\frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A} {\mathcal H}\right) \xrightarrow{\mu'} \frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A}{\mathcal H} .\end{align*} $$

Furthermore, we have a well-defined action of B on $B \otimes _{A} {\mathcal H}$ :

$$ \begin{align*} B \otimes \left(B \otimes_{A} {\mathcal H}\right) \to B \otimes_{A} {\mathcal H}, \qquad b \otimes \left(\sum_j b^{\prime}_j \otimes_{A} h_j\right) \mapsto \sum_{j} b_0b^{\prime}_j \otimes_{A} b_{1}h_j \end{align*} $$

and $\delta \colon B \to B \otimes _{A} {\mathcal H}$ becomes left B-linear with respect to this B-module structure and the regular B-action on the domain. By summing up all the informations we collected, we have a commutative diagram of left ${\mathcal H}$ -modules and left ${\mathcal H}$ -module homomorphisms

(3.11)

In view of this, for every $\sum _i h_i \iota (b_i) \in {\mathcal H} B$ , we have

$$ \begin{align*} \begin{aligned} (\pi \otimes_{A} {\mathcal H})\left(\Delta\left(\sum_i h_i \iota(b_i)\right)\right) & \stackrel{(3.11)}{=} \mu' \Big({\mathcal H} \otimes_{A} \big(\left(\pi \otimes_{A} {\mathcal H}\right)\circ \Delta \circ \iota\big)\Big)\left(\sum_i h_i \otimes_{B} b_i\right) \\ & \stackrel{(3.7)}{=} \sum_i h_i \cdot \Big(\pi(1_{\mathcal H})\otimes_{A} \iota(b_i)\Big) \\ & \stackrel{(3.9)}{=} \sum_i \pi\left((h_i)_1\right) \otimes_{A} (h_i)_2\iota(b_i), \end{aligned} \end{align*} $$

and hence for all $\sum _i h_i \iota (b_i) \in {\mathcal H} B^+$ , we have

$$ \begin{align*} (\pi \otimes_{A} \pi)\left(\Delta\left(\sum_i h_i \iota(b_i) \right)\right) = \sum_i \pi\left((h_i)_1\right) \otimes_{A} \pi\Big((h_i)_2 \iota(b_i)\Big) \stackrel{(3.6)}{=} 0. \end{align*} $$

Since, in addition,

$$ \begin{align*} \varepsilon\left(\sum_i h_i \iota(b_i)\right) = \sum_i \varepsilon(h_i \iota(b_i)) \stackrel{(B5)}{=} \sum_i \varepsilon\Big(h_i \iota \big(t'\left(\varepsilon'(b_i)^{\mathrm{o}}\right)\big)\Big) = 0, \end{align*} $$

we have that ${\mathcal H} B^+$ is a two-sided coideal (in view of [Reference Brzeziński and Wisbauer7, Paragraph 17.14], for instance). Again, it is evident that if $B \subseteq \tilde {B}$ , then ${\mathcal H} B^+ \subseteq {\mathcal H} \tilde {B}^+$ . Thus, the correspondence is inclusion-preserving.

3.1.5 The canonical inclusions

Proof Recall that ${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$ can be realized as the following equalizer in $\textsf {Mod}_{A}$ :

(see (3.5)). By (3.7), there exists $\eta _B\colon B \to {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$ such that $\iota \colon B \to {\mathcal H}$ factors as ${j'\circ \eta _B}$ .

Proof Notice that $\overline {{\mathcal H}} = {\mathcal H}/I$ is clearly an object in

. Set $\pi \colon {\mathcal H} \to {\mathcal H}/I$ . If we consider ${\overline {{\mathcal H}}}^{\textsf {co}{\overline {{\mathcal H}}}}$ , then it is not hard to show that the correspondence defined by

is a left B-linear isomorphism, where the B-module structures are both induced by the ${\mathcal H}$ -module structures via restriction of scalars along $\iota \colon B \to {\mathcal H}$ . In fact, for every $\pi (h) \in {\overline {{\mathcal H}}}^{\textsf {co}{\overline {{\mathcal H}}}}$ , we have

$$ \begin{align*}\sum \pi(h)_1 \otimes_{A} \pi(h)_2 = \pi(1) \otimes_{A} \pi(h)\end{align*} $$

and so

$$ \begin{align*} \pi(h) & = \left(\overline{{\mathcal H}} \otimes_{A} \overline{\varepsilon}\right)\left(\sum \pi(h)_1 \otimes_{A} \pi(h)_2\right) = \left(\overline{{\mathcal H}} \otimes_{A} \overline{\varepsilon}\right)\left(\pi(1) \otimes_{A} \pi(h)\right) \\ & = \pi\Big(t\big(\overline{\varepsilon}(\pi(h))^{\mathrm{o}}\big)\Big) = \pi\Big(t\big(\varepsilon(h)^{\mathrm{o}}\big)\Big). \end{align*} $$

This proves that the correspondence is bijective. To prove that it is left B-linear, notice that

$$ \begin{align*}\overline{\varepsilon}\Big(\pi\big(\iota(b)h\big)\Big) = \varepsilon(\iota(b)h) \stackrel{(B5)}{=} \varepsilon(\iota(b)s\varepsilon(h)) \stackrel{(3.2)}{=} b \cdot \varepsilon(h).\end{align*} $$

Since, in addition, we know from a short exact sequence like (3.1) that $A \cong B/B^+$ as left B-module, we may consider the composition

(3.12) $$ \begin{align} \frac{{\mathcal H}}{{\mathcal H} B^+} \cong {\mathcal H} \otimes_{B} \frac{B}{B^+} \cong {\mathcal H} \otimes_{B} A\cong {\mathcal H} \otimes_{B} {\overline{{\mathcal H}}}^{\textsf{ co}{\overline{{\mathcal H}}}} \xrightarrow{\theta_{\overline{{\mathcal H}}}} \overline{{\mathcal H}} = \frac{{\mathcal H}}{I} \end{align} $$

explicitly given by $h + {\mathcal H} B^+ \mapsto \pi (h) = h + I$ for all $h \in {\mathcal H}$ , where $\theta _{\overline {{\mathcal H}}}$ is the $\overline {{\mathcal H}}$ -component of the counit (2.15). Denote it by $\psi $ and consider the commutative diagram

(3.13)

It follows that ${\mathcal H} B^+ \subseteq I$ as claimed.

Corollary 3.10 Let $(A,{\mathcal H})$ be a left bialgebroid such that ${}_{s} {{{\mathcal H}}}$ is A-flat. For I a left ideal two-sided coideal in ${\mathcal H}$ and B a right ${\mathcal H}$ -comodule $A^{\mathrm {o}}$ -subring via t of ${\mathcal H}$ , we have that

$$ \begin{align*}{\mathcal H} B^+ \subseteq I \quad \iff \quad B \subseteq {{\mathcal H}}^{\textsf{co}{\frac{{\mathcal H}}{I}}}.\end{align*} $$

4.1.1 Groupoids and subgroupoids

Recall, for instance, from [Reference Mackenzie26, Chapter I, Definition 1.1], that a groupoid is a small category (i.e., the class of objects is a set) in which every arrow is invertible. We will usually denote a groupoid $\mathscr {G}$ by a pair $\big (\mathscr {G}_0,\mathscr {G}_1\big )$ of sets: the set of objects $\mathscr {G}_0$ and the set of arrows $\mathscr {G}_1$ . A groupoid $\big (\mathscr {G}_0,\mathscr {G}_1\big )$ always comes with an (understood) family of maps

satisfying the expected, reasonable, compatibility conditions. For the sake of simplicity, one often denotes a composition $\bullet (f,g)$ by simple juxtaposition $fg$ . A subgroupoid of a groupoid $\mathscr {G}$ is a subcategory which is also a groupoid.

For a given groupoid $\mathscr {G} = \big (\mathscr {G}_0, \mathscr {G}_1\big )$ , the associated isotropy groupoid (also called inner subgroupoid in [Reference Mackenzie26, Chapter I, Definition 2.4]) is the subgroupoid

, where $\mathscr {G}_x$ is the isotropy group of $x \in \mathscr {G}_0$ , that is to say,

In the isotropy groupoid, the source and the target coincide and we denote them simply by $o\colon \mathscr {G}^{{(i)}} \to \mathscr {G}_0, g \mapsto \sigma (g) = \tau (g)$ . Moreover, there is a canonical morphism of groupoids $\mathscr {G}^{{(i)}} \to \mathscr {G}$ (the inclusion).

4.1.2 Groupoid actions and normal subgroupoids

Recall that a left $\mathscr {G}$ -action of a groupoid $\mathscr {G}$ on a set N consists of two maps $\rho \colon N \to \mathscr {G}_{0}$ and

, satisfying

$$ \begin{align*} \rho(gn) = \tau (g),\qquad e_{\rho(n)} n = n,\qquad g'(gn) = (g'g)n, \end{align*} $$

for all $g , g'\in \mathscr {G}_1$ and $n \in N$ (see [Reference Mackenzie26, Definition 4.1] for the topological analog and [Reference Mackenzie27, Definition 1.6.1] for the differential one). Given a left $\mathscr {G}$ -action on N, one can consider the left translation groupoid

with

as set of arrows and N as set of objects. This is the so-called semi-direct product groupoid. The source and the target of this groupoid are given by

The identity of $n \in N$ is $e_n = (e_{\rho (n)}, n)$ . The composition is given by

$$ \begin{align*} (g,n) (g',n') = (gg', n') \qquad \textrm{whenever} \ g'n'=n. \end{align*} $$

Obviously, the pair

defines a morphism of groupoids.

As for groups, conjugation in $\mathscr {G}$ can be seen as an action of $\mathscr {G}$ on the underlying set of arrow of its isotropy groupoid $\mathscr {G}^{{(i)}}$ . In detail, define the map acting as $\operatorname {Ad}(g,f) = gfg^{-1}$ . One can show that this defines an action, called the adjoint action, so that $\mathscr {G}^{(i)}_1$ becomes a left $\mathscr {G}$ -set.

A normal subgroupoid of $\mathscr {G}$ is a subgroupoid $\mathscr {N} \hookrightarrow \mathscr {G}$ such that:

1. (NG0) $\mathscr {N}_0 =\mathscr {G}_0$ (i.e., $\mathscr {N}$ is wide in the sense of [Reference Mackenzie26, Chapter I, Definition 2.4]);

2. (NG1) $\mathscr {N}_1(x,y)= \emptyset $ , for all $x \neq y $ in $\mathscr {N}_0$ (that is, $\mathscr {N}$ is a subgroupoid of $\mathscr {G}^{(i)}$ );

3. (NG2) for every $g \in \mathscr {G}_1$ , we have

   $$ \begin{align*} \operatorname{Ad}(g)\Big(\mathscr{N}_1\big(\sigma(g), \sigma(g)\big)\Big) \subseteq \mathscr{N}_1(\tau(g),\tau(g)) \end{align*} $$

   as subgroups of $\mathscr {G}_1\big (\tau (g),\tau (g)\big )$ , where $\operatorname {Ad}(g)(f)=\operatorname {Ad}(g,f)$ for all f with $\sigma (f) = \tau (f) = \sigma (g)$ .

Equivalently, a subgroupoid $\mathscr {N} \hookrightarrow \mathscr {G}$ is normal (in the above sense) if and only if $\mathscr {N}$ is a subgroupoid of $\mathscr {G}^{(i)}$ ,

is non empty, and $g\mathscr {N} g^{-1} \subseteq \mathscr {N}$ for all $g \in \mathscr {G}_1$ (see [Reference Paques and Tamusiunas29, Lemma 3.1]). In particular, we denote again by $o \colon \mathscr {N} \to \mathscr {N}_0, n \mapsto \sigma (n) = \tau (n)$ , the common source and target of a normal subgroupoid.

The fact that normal subgroups can be characterized as the invariant subgroups under the adjoint action can be extended to the groupoids context, as the following lemma states.

4.1.3 The orbit space

Let $\mathscr {G}$ be a groupoid, and let N be a left $\mathscr {G}$ -set. For $n \in N$ , one defines the orbit of n under $\mathscr {G}$ to be

(see, e.g., [Reference Mackenzie26, Definition 4.1]). This induces an equivalence relation on N given by

$$ \begin{align*}n \sim n' \quad \Leftrightarrow \quad \exists\, g\in \sigma^{-1}(\rho(n)) \text{ such that } n'= gn \quad \Big( \Leftrightarrow \quad \mathscr{G} n = \mathscr{G} n'\Big).\end{align*} $$

The quotient space with respect to this relation, called the orbit space of N with respect to the action of $\mathscr {G}$ , is denoted by $N/\mathscr {G}$ , and its elements are the orbits $\mathscr {G} n$ for $n \in N$ . See, for instance, [Reference Goehle20, Definition 1.67]. In particular, given a groupoid $\mathscr {G}$ , one can consider the orbit equivalence relation on $\mathscr {G}_0$ with respect to the action of $\mathscr {G}$ given by

In this case, $x \sim y$ if and only if there exists a $g \in \mathscr {G}_1$ such that $\sigma (g) = x$ and $\tau (g) = y$ . The orbit of $x \in \mathscr {G}_0$ is

$$ \begin{align*}\mathscr{G} x = \tau \big( \sigma ^{-1}(x)\big).\end{align*} $$

The set of orbits, called the orbit space, of $\mathscr {G}$ is the quotient set $\mathscr {G}_{0}/\sim $ , which is often denoted by $\mathscr {G}_{0}/\mathscr {G}$ . For further details, we refer the reader to [Reference Goehle20, Section 1.2].

Remark 4.4 For a left $\mathscr {G}$ -set N, the orbit space $N/\mathscr {G}$ of N with respect to the action of $\mathscr {G}$ can also be described as the orbit space of the left translation groupoid

. In fact, for $p \in N$ , we have

Now, we will use this notion to describe the quotient groupoid by a normal subgroupoid as in the classical case of groups. Let us consider a normal subgroupoid $\mathscr {N} \hookrightarrow \mathscr {G}$ . It is clear that $(\mathscr {G}_1, \tau )$ is a left $\mathscr {N}$ -set via the left action

(4.1)

We denote by $\mathscr {G}_1/\mathscr {N}$ the orbit space of this action (or, equivalently, the orbit space of the associated left translation groupoid ) and call it the quotient groupoid (see [Reference Mackenzie26, pp. 9–10]). As a matter of description, the equivalence class of a given arrow $g \in \mathscr {G}_1$ is a “flower with stalk at g”:

Two arrows are equivalent if they “support the same flower.” The action (4.1) is in fact a left one. There is also a right $\mathscr {N}$ -action on $(\mathscr {G}_1, \sigma )$ , which leads to the groupoid and, in principle, to another orbit space $\mathscr {G}_1/\mathscr {N}$ . It is easy to verify that the two orbit spaces coincide, since $\mathscr {N}$ is normal. The following result is expected (see also [Reference Mackenzie26, pp. 9–10]).

Lemma 4.5 ([Reference Barbarán Sánchez and El Kaoutit2, Proposition 2.16])

Let $\mathscr {N} \hookrightarrow \mathscr {G}$ be a normal subgroupoid. Then

have a structure of groupoid such that there is a chain of morphisms

Furthermore, any morphism $\mathscr {G} \to \mathscr {H}$ whose kernel contains $\mathscr {N}$ factors through $\varpi $ .

4.2.1 Commutative Hopf algebroids and Hopf ideals

Definition 4.1 A commutative Hopf algebroid is a cogroupoid object in the category $\mathrm {CAlg}_{\Bbbk }$ of commutative algebras (see, e.g., [Reference Ravenel31]). Thus, a commutative Hopf algebroid consists of a pair of commutative $\Bbbk $ -algebras $\left ( A,\mathcal {H}\right ) $ together with a diagram of algebra maps

(where the A-bimodule structure on ${\mathcal H}$ used to perform the tensor product is ${}_{s} {{\mathcal H}}{}_{t}$ ) such that:

1. (CH1) The datum $({\mathcal H}, \Delta , \varepsilon )$ is an A-coring structure on ${}_{s} {{\mathcal H}}{}_{t}$ . At the level of groupoids, this encodes a unitary and associative composition law between arrows.

2. (CH2) The antipode ${\mathcal S}$ satisfies ${\mathcal S} \circ s=t$ , ${\mathcal S} \circ t = s$ and ${\mathcal S}^2=\operatorname {Id}_{{\mathcal H}}$ , which encode the fact that the inverse of an arrow interchanges source and target and that the inverse of the inverse is the original arrow.

3. (CH3) The antipode also satisfies $\sum {\mathcal S}(u_1)u_2=(t\circ \varepsilon )(u)$ and $\sum u_1{\mathcal S}(u_2)= (s\circ \varepsilon )(u)$ , which encode the fact that the composition of a morphism with its inverse on either side gives an identity morphism.

In particular, the inverse of the Hopf–Galois map

$$ \begin{align*}\beta\colon {{\mathcal H}}{}_{t} \otimes_{A} {}_{t} {{\mathcal H}} \to {{\mathcal H}}{}_{t} \otimes_{A} {}_{s} {{{\mathcal H}}}, \qquad u \otimes_{A} v \mapsto \sum u_1 \otimes_{A} u_2v,\end{align*} $$

is provided by

$$ \begin{align*}\beta^{-1}\colon {{\mathcal H}}{}_{t} \otimes_{A} {}_{s} {{{\mathcal H}}} \to {{\mathcal H}}{}_{t} \otimes_{A} {}_{t} {{\mathcal H}}, \qquad u \otimes_{A}v \mapsto \sum u_1 \otimes_{A} {\mathcal S}(u_2)v.\end{align*} $$

Thus, explicitly,

(4.2)

A morphism of commutative Hopf algebroids $\boldsymbol {\phi }\colon (A,{\mathcal H}) \to (B,{\mathcal K})$ is a pair of algebra maps $\left ( \phi _{{0}} \colon A \to B,\phi _{{1}} \colon {\mathcal H} \to {\mathcal K}\right )$ such that

$$ \begin{gather*} \phi _{1}\circ s_{\mathcal H} = s_{\mathcal K} \circ \phi _{0}, \qquad \phi _{1}\circ t_{\mathcal H} = t_{\mathcal K} \circ \phi_{0}, \qquad \varepsilon_{\mathcal K} \circ \phi _{1} = \phi _{0} \circ \varepsilon_{\mathcal H} , \\ \Delta_{\mathcal K} \circ \phi _{1} =\chi \circ \left( \phi _{1}\otimes _{A}\phi_{1}\right) \circ \Delta_{\mathcal H} , \qquad {\mathcal S}_{\mathcal K} \circ \phi _{1} =\phi _{1} \circ {\mathcal S}_{\mathcal H}, \end{gather*} $$

where $\chi \colon {\mathcal K} \otimes _{A} {\mathcal K} \rightarrow {\mathcal K} \otimes _{B} {\mathcal K}$ is the obvious projection induced by $\phi _0$ , that is, $\chi \left (k\otimes _{A}k'\right ) =k\otimes _{B}k'$ . If $A = B$ and $\phi _0 = \operatorname {Id}_A$ , then we say that $\phi _1 : {\mathcal H} \to {\mathcal K}$ is a morphism of Hopf algebroids over A.

Similarly to how commutative Hopf algebras over a field $\Bbbk $ are the same thing as affine $\Bbbk $ -group schemes (that is, representable presheaves of groups on $\mathrm {Aff}_{\Bbbk } = \mathrm {CAlg}_{\Bbbk }^{\mathrm {op}}$ , the opposite of the category of commutative $\Bbbk $ -algebras), commutative Hopf algebroids are the same thing as affine groupoid schemes (that is, representable presheaves of groupoids on $\mathrm {Aff}_{\Bbbk }$ ). Precisely, given a commutative Hopf algebroid $(A, {\mathcal H})$ and a commutative $\Bbbk $ -algebra R, we have a groupoid by reversing the structures of $(A,{\mathcal H})$ . We will often adopt this point of view, and we will denote by $\mathscr {G}_{(A,{\mathcal H})} = \big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$ the groupoid scheme defined by $(A,{\mathcal H})$ , that is to say,

$$ \begin{align*} \mathscr{G}_A(R) = A(R) = \mathrm{CAlg}_{\Bbbk}(A,R) \qquad \text{and} \qquad \mathscr{G}_{\mathcal H}(R) = {\mathcal H}(R) = \mathrm{CAlg}_{\Bbbk}({\mathcal H},R), \end{align*} $$

in order to stress the groupoid structure on the pair $\big (\mathscr {G}_A(R),\mathscr {G}_{\mathcal H}(R)\big )$ . Observe that, under this perspective,

(4.3) $$ \begin{align} (\psi \bullet \varphi)(h) = \sum \varphi\big(h_1\big)\psi\big(h_2\big) \end{align} $$

for all , $h \in {\mathcal H}$ and R in $\mathrm {CAlg}_{\Bbbk }$ .

Remark 4.7 In particular, $\big (\mathscr {G}_A(\Bbbk ),\mathscr {G}_{\mathcal H}(\Bbbk )\big )$ will be called an affine $\Bbbk $ -groupoid, and when $A,{\mathcal H}$ are finitely generated and reduced, then it will be called an affine algebraic $\Bbbk $ -groupoid. Suppose that $\Bbbk $ is algebraically closed. Denote by $\mathrm {Aff}_1:\mathrm {CAlg}_{\Bbbk } \to \mathrm {Set}$ the affine scheme represented by the polynomial algebra $\Bbbk [T]$ and by $\operatorname {Var}_{\Bbbk }$ the category of affine $\Bbbk $ -varieties (i.e., the category whose objects are $\mathrm {CAlg}_{\Bbbk }(R,\Bbbk )$ for R a finitely generated and reduced commutative algebra and whose morphisms are given by precomposition by morphisms in $\mathrm {CAlg}_{\Bbbk }$ ). Then

$$ \begin{gather*} (A,{\mathcal H}) \mapsto \big(\mathscr{G}_A(\Bbbk),\mathscr{G}_{\mathcal H}(\Bbbk)\big) \quad \text{and} \\ \big(\mathscr{G}_A(\Bbbk),\mathscr{G}_{\mathcal H}(\Bbbk)\big) \mapsto \Big(\operatorname{Var}_{\Bbbk}\big(\mathscr{G}_A(\Bbbk),\mathrm{Aff}_1(\Bbbk)\big),\operatorname{Var}_{\Bbbk}\big(\mathscr{G}_{\mathcal H}(\Bbbk),\mathrm{Aff}_1(\Bbbk)\big)\Big) \end{gather*} $$

give an anti-equivalence between the category of affine (i.e., finitely generated, commutative, reduced) Hopf algebroids and the category of affine algebraic $\Bbbk $ -groupoids, where

$$ \begin{align*} \operatorname{Var}_{\Bbbk}\big(\mathscr{G}_A(\Bbbk),\mathrm{Aff}_1(\Bbbk)\big) = \mathrm{CAlg}_{\Bbbk}(\Bbbk[T],A).\\[-34pt] \end{align*} $$

▲

Let $(\mathscr {G}_A,\mathscr {G}_{\mathcal H})$ be an affine groupoid scheme, and let $I\subseteq {\mathcal H}$ be an ideal. Denote by $\pi \colon {\mathcal H} \to {\mathcal H}/I$ the canonical projection and by $\pi ^*\colon \mathrm {CAlg}_{\Bbbk }({\mathcal H}/I,-) \to \mathrm {CAlg}_{\Bbbk }({\mathcal H},-)$ the natural transformation given by precomposition by $\pi $ . We say that $(\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I})$ is a wide closed groupoid subscheme of $(\mathscr {G}_A,\mathscr {G}_{\mathcal H})$ (or, simply, a wide closed subgroupoid or wide affine subgroupoid of $(\mathscr {G}_A,\mathscr {G}_{\mathcal H})$ ) if $(\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I})$ is a groupoid itself and $\pi ^*:\mathscr {G}_{{\mathcal H}/I} \hookrightarrow \mathscr {G}_{\mathcal H}$ induces a morphism of groupoids $(\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I}) \to (\mathscr {G}_A,\mathscr {G}_{\mathcal H})$ . Since the structure maps of the commutative Hopf algebroid ${\mathcal H}/I$ should be induced by the structure maps of ${\mathcal H}$ via $\pi $ , $(\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I})$ is a wide subgroupoid of $(\mathscr {G}_A,\mathscr {G}_{\mathcal H})$ if and only if $(A,{\mathcal H}/I)$ is a commutative Hopf algebroid and $\pi :{\mathcal H} \to {\mathcal H}/I$ is a morphism of commutative Hopf algebroids, if and only if the ideal I satisfies:

1. (HI1) $\varepsilon (I) = 0$ ;

2. (HI2) $\Delta (I) \subseteq \operatorname {Im}({\mathcal H}\otimes _{A}I + I\otimes _{A}{\mathcal H})$ ;

3. (HI3) ${\mathcal S}(I) \subseteq I$ (and so ${\mathcal S}(I) = I$ ).

An ideal $I \subseteq {\mathcal H}$ satisfying (HI1) and (HI2) will be called a wide bi-ideal of $(A,{\mathcal H})$ . When I satisfies (HI3) as well, it will be called a wide Hopf ideal of $(A,{\mathcal H})$ .

In what follows, we are mainly concerned with wide subgroupoids and wide ideals; hence, we will often omit the adjective “wide.” For the sake of completeness, the following proposition can be proved by finding inspiration from the proof of Proposition 3.3.

Now, let $(A,{\mathcal H})$ be a commutative Hopf algebroid and consider the ideal $\langle s-t \rangle $ generated by the vector subspace $\{s(a) - t(a) \mid a \in A\}$ . Since $s(a) - t(a) \in {\mathcal P}_{A}({\mathcal H})$ (the space of primitive elements) for every $a \in A$ , $\langle s-t \rangle $ is a Hopf ideal in ${\mathcal H}$ . Its associated quotient Hopf algebroid will be denoted by . For $(A, {{\mathcal H}}_{(i)})$ , the source equals the target. That is, it is a commutative Hopf A-algebra. Denote by $\eta \colon A \to {{\mathcal H}}_{(i)}$ the unit map (that is, $\eta (a) = s(a) + \langle s-t\rangle = t(a) + \langle s-t\rangle $ for all $a \in A$ ) and by ${\varpi }_{(i)} \colon {\mathcal H} \to {{\mathcal H}}_{(i)}$ the canonical projection sending x to $\overline {x} = x + \langle s-t \rangle $ .

Lemma 4.10 Let $(A,{\mathcal H})$ be a commutative Hopf algebroid. Then the presheaf of groupoids $\big (\mathscr {G}_A, \mathscr {G}_{\mathcal H}^{(i)}\big )$ obtained by considering the isotropy groupoid of $\big (\mathscr {G}_A(R), \mathscr {G}_{\mathcal H}(R)\big )$ for every R in $\mathrm {CAlg}_{\Bbbk }$ is represented by the Hopf A-algebra ${{\mathcal H}}_{(i)}$ . In other words, we have an isomorphism

$$ \begin{align*} \Big(A(R), {\mathcal H}(R)^{{(i)}}\Big) \cong \Big(A(R), {{\mathcal H}}_{(i)}(R)\Big), \end{align*} $$

natural in R, between the isotropy groupoid of $\big (\mathscr {G}_A, \mathscr {G}_{\mathcal H}\big )$ and the groupoid $\big (\mathscr {G}_A,\mathscr {G}_{{{\mathcal H}}_{(i)}}\big )$ .

Proof Let R be a commutative $\Bbbk $ -algebra. The fact that the correspondence

is bijective is almost tautological, and it is a morphism of groupoids because it is induced by a morphism of commutative Hopf algebroids. The naturality in R follows from the associativity of the composition.

Even for Hopf algebroids over noncommutative base algebras, ${\mathcal H}/\langle s-t \rangle $ forms a natural Hopf algebroid structure and is called the isotropy quotient of ${\mathcal H}$ [Reference Ghobadi19]. In particular, Lemma 4.10 was previously discussed in Example 3.13 of [Reference Ghobadi19].

4.2.2 Comodule algebras and normal Hopf ideals

With an eye to the definition of normal Hopf ideals, we give the analog of the adjoint action for groupoids (see Section 4.1.2) in the Hopf algebroid context. To this aim, recall first that a (commutative) right ${\mathcal H}$ -comodule algebra for a commutative Hopf algebroid $(A,{\mathcal H})$ is a (commutative) monoid in the symmetric monoidal category $ \textsf {Comod}{}^{{\mathcal H}}$ (see Remark 3.2). That is, a (commutative) right $\mathcal{H}$ -comodule algebra is a (commutative) A-algebra a ${\mathcal R}$ (with structure map $r\colon A\to {\mathcal R}$ ) which is also a right ${\mathcal H}$ -comodule with coaction $\delta _{{\mathcal R}}\colon {\mathcal R} \to {\mathcal R}\otimes _{A}{\mathcal H}, x \mapsto \sum x_{0} \otimes _{A} x_1$ , satisfying for all $x, y \in {\mathcal R}$ ,

(4.4) $$ \begin{align} \delta_{{\mathcal R}}(xy) = \sum x_{0}y_{0} \otimes_{A} x_{1}y_{1} \qquad \mbox{and} \qquad \delta_{{\mathcal R}}(1_{{\mathcal R}})= 1_{{\mathcal R}}\otimes_{A}1_{{\mathcal H}}. \end{align} $$

In others words, the coaction $\delta _{{\mathcal R}}$ is an A-algebra map, where ${\mathcal R}\otimes _{A}{\mathcal H}$ is seen as an A-algebra via $A \to {\mathcal R} \otimes _{A} {\mathcal H},a \mapsto 1_{{\mathcal R}}\otimes _{A} t(a)$ . A morphism of right ${\mathcal H}$ -comodule algebras is an A-algebra map which is also a right ${\mathcal H}$ -comodule morphism. Left ${\mathcal H}$ -comodule algebras are analogously defined. Henceforth, all comodule algebras will be commutative comodule algebras. A trivial example of a comodule algebra is the base algebra A of a Hopf algebroid $(A, {\mathcal H})$ itself.

Proof If $({\mathcal B},b:A\rightarrow {\mathcal B})$ is an A-algebra with a right ${\mathcal H}$ -comodule algebra structure given by a coaction $\delta _{{\mathcal B}}$ , then

and

turn $\mathscr {N}_{\mathcal B}(R)$ into a left $\mathscr {G}_{(A,{\mathcal H})}(R)$ -set. Conversely, if $\Big (\mathscr {N}_{\mathcal B}(R),\rho _R,\mu _R\Big )$ is a left $\mathscr {G}_{(A,{\mathcal H})}(R)$ -set natural in $\mathrm {CAlg}_{\Bbbk }$ , then

and

make of ${\mathcal B}$ a right ${\mathcal H}$ -comodule algebra.

In the setting of Proposition 4.11, we say that $\mathscr {N}_{\mathcal B}$ is an affine (left) $\mathscr {G}_{(A,{\mathcal H})}$ -set.

Note that, for a right ${\mathcal H}$ -comodule algebra $({\mathcal R},r)$ , the $\Bbbk $ -vector subspace

$$ \begin{align*} {{\mathcal R}}^{\textsf{co}{{\mathcal H}}} = \{x \in {\mathcal R} \mid \delta_{{\mathcal R}}(x) = x \otimes_{A} 1_{{\mathcal H}} \} \end{align*} $$

of ${\mathcal H}$ -coinvariant elements is a $\Bbbk $ -subalgebra of ${\mathcal R}$ (this follows easily from (4.4)).

Dually to the construction of the left translation groupoid, if $({\mathcal R},r)$ is a right ${\mathcal H}$ -comodule algebra, then $({\mathcal R}, {\mathcal R}\otimes _{A}{\mathcal H})$ admits a natural structure of commutative Hopf algebroid such that $(r, 1_{{{\mathcal R}}}\otimes _{A}-)\colon (A,{\mathcal H}) \to ({\mathcal R},{\mathcal R}\otimes _{A}{\mathcal H})$ is a morphism of Hopf algebroids. Namely,

$$ \begin{gather*} s\colon {\mathcal R} \to {\mathcal R} \otimes_{A} {\mathcal H}, \quad x \mapsto x \otimes_{A} 1_{\mathcal H}, \qquad t\colon {\mathcal R} \to {\mathcal R} \otimes_{A} {\mathcal H}, \quad x \mapsto \sum x_0 \otimes_{A} x_1, \\ \Delta\colon {\mathcal R} \otimes_{A} {\mathcal H} \to \big({\mathcal R} \otimes_{A} {\mathcal H}\big) \otimes_{{\mathcal R}} \big({\mathcal R} \otimes_{A} {\mathcal H}\big), \kern-7pt\qquad (x \otimes_{A} h) \mapsto \sum (x \otimes_{A} h_1) \otimes_{{\mathcal R}} (1_{\mathcal R} \otimes_{A} h_2), \\ \varepsilon\colon {\mathcal R} \otimes_{A} {\mathcal H} \to {\mathcal R}, \ x \otimes_{A} h \mapsto x r\big(\varepsilon(h)\big), \qquad \text{and} \\ {\mathcal S} \colon {\mathcal R} \otimes_{A} {\mathcal H} \to {\mathcal R} \otimes_{A} {\mathcal H}, \qquad x \otimes_{A} h \mapsto \sum x_0 \otimes_{A} x_1{\mathcal S}(h). \end{gather*} $$

This is called the left translation Hopf algebroid of $(A, {\mathcal H})$ along r. Of course, if ${}_{s} {{{\mathcal H}}}$ is flat over A, then ${\mathcal R}\otimes _{A}{\mathcal H}$ is flat over ${\mathcal R}$ (see [Reference El Kaoutit and Kowalzig15, Lemma 3.4] for more properties).

Now, dually to the case of groupoids, we introduce the adjoint coaction.

Proposition 4.13 The pair $({{\mathcal H}}_{(i)}, \eta )$ is a right ${\mathcal H}$ -comodule algebra with coaction defined by

(4.5) $$ \begin{align} \delta_{{{\mathcal H}}_{(i)}} \colon {{\mathcal H}}_{(i)} \to {{\mathcal H}}_{(i)} \otimes_{A} {\mathcal H}, \qquad \overline{h} \mapsto \sum \overline{h_2} \otimes_{A} {\mathcal S}(h_1)h_3. \end{align} $$

In particular, $({{\mathcal H}}_{(i)}, {{\mathcal H}}_{(i)}\otimes _{A}{\mathcal H})$ is a commutative Hopf algebroid (it is the left translation Hopf algebroid along $\eta $ ).

Proof The fact that $\delta _{{{\mathcal H}}_{(i)}}$ is well defined can be deduced by using the following map:

$$ \begin{align*} {{\mathcal H}}{}_{t} \otimes_{A} {}_{s} {{\mathcal H}}{}_{t} \to {{{\mathcal H}}_{(i)}}_{\eta} \otimes_{A} {}_{s} {{\mathcal H}}{}_{s}, \qquad u\otimes_{A}v \mapsto \overline{v}\otimes_{A} {\mathcal S}(u), \end{align*} $$

which is well defined and right A-linear. The map $\delta _{{{\mathcal H}}_{(i)}}$ is clearly right A-linear, that is, $\delta _{{{\mathcal H}}_{(i)}}\Big ( \overline {h}\eta (a)\Big ) = \delta _{{{\mathcal H}}_{(i)}}(\overline {h})t(a)$ for every $a \in A$ and $\overline {h} \in {{\mathcal H}}_{(i)}$ . Aside from the coaction being well defined, the rest of the proof showing the coassociativity and counital property of $\delta $ and the comodule algebra structure on ${{\mathcal H}}_{(i)}$ follow exactly as in the case of the right adjoint coaction on ordinary Hopf algebras, by simply using Sweedler’s notation. Therefore, $({{\mathcal H}}_{(i)}, \delta _{{{\mathcal H}}_{(i)}})$ is a right ${\mathcal H}$ -comodule. Since $\delta _{{{\mathcal H}}_{(i)}}$ is manifestly multiplicative and unital, $({{\mathcal H}}_{(i)}, \eta )$ is a right ${\mathcal H}$ -comodule algebra.

Let I be an ideal of ${\mathcal H}$ which contains $\langle s-t \rangle $ . We denote by ${I}_{(i)}$ its image in ${{\mathcal H}}_{(i)}$ : .

Remark 4.14 If ${}_{s} {{{\mathcal H}}}$ is A-flat, then (NI2) is equivalent to requiring that the image of ${I}_{(i)}$ is a right ${\mathcal H}$ -subcomodule of ${{\mathcal H}}_{(i)}$ with respect to the coaction (4.5). Namely, that there exists $\delta _{{I}_{(i)}}\colon {I}_{(i)} \to {I}_{(i)} \otimes _{A} {\mathcal H}$ such that the following diagram commutes:

▲

For commutative Hopf algebroids with source equal to target, that is, for Hopf algebras over a commutative algebra, one recovers the classical definition. Notice further that in this latter case, the associated isotropy Hopf algebra coincides with the Hopf algebroid itself.

The following proposition is the Hopf algebroid analog of Lemma 4.3.

Proof The statement follows from the fact that I is normal if and only if the left-hand side square in the following diagram with exact rows commutes:

if and only if the right-hand side square commutes, because $\operatorname {Im}\left ({I}_{(i)} \otimes _{A} {\mathcal H}\right ) = \ker \left ({\pi }_{(i)} \otimes _{A} {\mathcal H}\right )$ .

Let $\big (\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I}\big )$ be an affine subgroupoid of $\big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$ (i.e., $I \subseteq {\mathcal H}$ is a Hopf ideal). We say that $\big (\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I}\big )$ is a normal affine subgroupoid of $\big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$ if and only if $\big (\mathscr {G}_A(R),\mathscr {G}_{{\mathcal H}/I}(R)\big )$ is a normal subgroupoid of $\big (\mathscr {G}_A(R),\mathscr {G}_{\mathcal H}(R)\big )$ in the sense of Section 4.1.2, for all R in $\mathrm {CAlg}_{\Bbbk }$ . The following proposition shows that normal Hopf ideals are in bijective correspondence with normal affine subgroupoids.

Proof Let I be a normal Hopf ideal of ${\mathcal H}$ . Then, in view of (NI1) and Lemma 4.15, ${\mathcal H}/I \cong {{\mathcal H}}_{(i)}/{I}_{(i)}$ is a commutative Hopf A-algebra with a surjective morphism of commutative Hopf algebroids $(\operatorname {Id}_A,\tilde {\pi }) \colon (A,{\mathcal H}) \to (A, {{\mathcal H}}_{(i)}/{I}_{(i)})$ , where $\tilde {\pi }\colon {\mathcal H} \to {{\mathcal H}}_{(i)}/{I}_{(i)}$ is induced by $\pi \colon {\mathcal H} \to {\mathcal H}/I$ , up to the isomorphism above. We have then the affine subgroupoid $\Big (\mathscr {G}_A,\mathscr {G}_{{{\mathcal H}}_{(i)}/{I}_{(i)}}\Big )$ of $\big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$ . For R in $\mathrm {CAlg}_{\Bbbk }$ , consider $(\operatorname {Id},\tilde {\pi }^*) \colon \Big (\mathscr {G}_A(R),\mathscr {G}_{{{\mathcal H}}_{(i)}/{I}_{(i)}}(R)\Big ) \hookrightarrow \big (\mathscr {G}_A(R),\mathscr {G}_{\mathcal H}(R)\big )$ . Condition (NG0) is always satisfied in our treatment. Condition (NG1) is satisfied because for every $\varphi \in \mathrm {CAlg}_{\Bbbk }\big ({{\mathcal H}}_{(i)}/{I}_{(i)},R\big )$ , we have

$$ \begin{align*} \sigma(\varphi) = \varphi \circ (\tilde{\pi} \circ s) \stackrel{((NI1))}{=} \varphi \circ (\tilde{\pi} \circ t) = \tau(\varphi). \end{align*} $$

Concerning condition (NG2), for every $\psi \in \mathrm {CAlg}_{\Bbbk }({\mathcal H},R)$ and for every $\varphi \in \mathrm {CAlg}_{\Bbbk }\big ({{\mathcal H}}_{(i)}/{I}_{(i)},R\big )$ such that $\varphi \circ (\tilde {\pi }\circ s) = \psi \circ s = \varphi \circ (\tilde {\pi } \circ t)$ (i.e., source and target of $\varphi $ coincide with the source of $\psi $ ), set . Notice that

$$ \begin{align*} \varphi'\big(s(a)-t(a)\big) & \stackrel{\phantom{(4.3)}}{=} \big(\psi \bullet \tilde{\pi}^*(\varphi) \bullet \psi^{-1}\big)\big(s(a)-t(a)\big) \\ & \stackrel{(4.3)}{=} \psi^{-1}\big(s(a)\big)\varphi\Big(\overline{1}\Big)\psi(1) - \psi^{-1}(1)\varphi\Big(\overline{1}\Big)\psi\big(t(a)\big) \\ & \stackrel{\phantom{(4.3)}}{=}\psi\big({\mathcal S} s(a)\big)\varphi\Big(\overline{1}\Big)\psi(1) - \psi({\mathcal S} (1))\varphi\Big(\overline{1}\Big)\psi\big(t(a)\big) \\ & \stackrel{\phantom{(4.3)}}{=}\psi\big(t(a)\big)\varphi\Big(\overline{1}\Big)\psi(1) - \psi(1)\varphi\Big(\overline{1}\Big)\psi\big(t(a)\big) = 0 \end{align*} $$

for all $a \in A$ , whence there exists a unique $\tilde {\varphi }'\colon {{\mathcal H}}_{(i)} \to R$ such that $\tilde {\varphi }' \circ {\varpi }_{(i)} = \varphi '$ . Now, for every $\overline {x} = {\varpi }_{(i)}(x) \in {I}_{(i)}$ , we have that

$$ \begin{align*} &\tilde{\varphi}'\big(\overline{x}\big) = \big(\psi \bullet \tilde{\pi}^*(\varphi) \bullet \psi^{-1}\big)(x) \stackrel{(4.3)}{=} \sum\psi\big({\mathcal S}(x_1)\big)\varphi\big(\overline{x_2} + {I}_{(i)}\big)\psi(x_3) \\ = &\sum\varphi\big(\overline{x_2} + {I}_{(i)}\big)\psi\big({\mathcal S}(x_1)x_3\big) = \big(\varphi \otimes_{A} \psi\big)\big({\pi}_{(i)} \otimes_{A} {\mathcal H}\big)\left(\sum\overline{x_2} \otimes_{A} {\mathcal S}(x_1)x_3\right) \stackrel{(NI2)}{=} 0, \end{align*} $$

whence there exists a unique $\phi \colon {{\mathcal H}}_{(i)}/{I}_{(i)} \to R$ such that $\phi \circ {\pi }_{(i)} = \tilde {\varphi }'$ . Summing up, $\psi \bullet \tilde {\pi }^*(\varphi ) \bullet \psi ^{-1} = \tilde {\pi }^*(\phi )$ for some $\phi \in \mathscr {G}_{{{\mathcal H}}_{(i)}/{I}_{(i)}}(R)$ and so (NG2) is satisfied.

Conversely, take a normal affine subgroupoid $\big (\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I}\big )$ of $\big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$ induced by $\pi \colon {\mathcal H} \to {\mathcal H}/I$ . By definition, I is a Hopf ideal of ${\mathcal H}$ . Moreover, for any commutative $\Bbbk $ -algebra R and for any $\varphi \in \mathrm {CAlg}_{\Bbbk }\big ({\mathcal H}/I,R\big )$ , we have that $\varphi \circ (\pi \circ s) = \varphi \circ (\pi \circ t)$ , because in $\big (\mathscr {G}_A(R),\mathscr {G}_{{\mathcal H}/I}(R)\big )$ the only arrows that we have are loops. By taking $R={\mathcal H}/I$ and $\varphi = \operatorname {Id}_{{\mathcal H}/I}$ , we get $\pi \circ t = \pi \circ s $ , which shows that $\langle s - t \rangle \subseteq I$ and hence (NI1) is satisfied and ${\mathcal H}/I \cong {{\mathcal H}}_{(i)}/{I}_{(i)}$ . We still need to check condition (NI2). For this, recall from Lemma 4.3 that $\big (\mathscr {G}_A,\mathscr {G}_{{\mathcal H}/I}\big )$ is a normal affine subgroupoid of $\big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$ if and only if

commutes for every R in $\mathrm {CAlg}_{\Bbbk }$ , if and only if there exists a right ${\mathcal H}$ -comodule algebra structure $\delta _{{\mathcal H}/I}$ on ${\mathcal H}/I$ such that

commutes (where ${\tilde {\pi }}_{(i)}$ is the composition ${{\mathcal H}}_{(i)} \xrightarrow {{\pi }_{(i)}} {{\mathcal H}}_{(i)}/{I}_{(i)} \cong {\mathcal H}/I$ with kernel ${I}_{(i)}$ ), if and only if $\delta _{{{\mathcal H}}_{(i)}}\big ({I}_{(i)}\big ) \subseteq \ker \big ({\tilde {\pi }}_{(i)} \otimes _{A} {\mathcal H}\big ) = \operatorname {Im}\big ({I}_{(i)} \otimes _{A} {\mathcal H}\big )$ .

4.3.3 The space of invariants

Let $(A, {\mathcal H})$ be a commutative Hopf algebroid, and let $I \subseteq {\mathcal H}$ be a normal Hopf ideal. Denote by $\tilde {\pi } \colon {\mathcal H} \to {{\mathcal H}}_{(i)}/{I}_{(i)}$ the composition of the canonical projection $\pi \colon {\mathcal H} \to {\mathcal H}/I$ with the isomorphism ${\mathcal H}/I \cong {{\mathcal H}}_{(i)}/{I}_{(i)}$ . Since this leads to a morphism of Hopf algebroids, we have an induced ${{\mathcal H}}_{(i)}/{I}_{(i)}$ -comodule structure on ${\mathcal H}$ as in Section 3, given by

(4.6) $$ \begin{align} \delta_{{\mathcal H}} \colon {\mathcal H} \to {\mathcal H}\otimes_{A} {\displaystyle \frac{{{\mathcal H}}_{(i)}}{{I}_{(i)}}}, \qquad h \mapsto \sum h_1\otimes_{A} \left(\overline{h_2} + {I}_{(i)}\right). \end{align} $$

In this way, $({\mathcal H}, s)$ becomes a right ${{\mathcal H}}_{(i)}/{I}_{(i)}$ -comodule algebra. Let us denote its coinvariant subalgebra by