1 Introduction
1.1 Motivations and overviews
One of the fundamental results in the theory of affine groups states that the category of affine abelian groups is an abelian category, where epimorphisms are faithfully flat morphisms and monomorphism are closed immersions [Reference Demazure and Gabriel10, Chapter III, Section 3, no. 7, Paragraph 7.4, Corollaire, p. 355]. A purely algebraic proof of this theorem was given by Takeuchi in [Reference Takeuchi36, Corollary 4.16], which asserts, in algebraic terms, that the category of commutative and cocommutative Hopf algebras over a field
$\Bbbk $
is an abelian category, hereby generalizing a well-known result from Grothendieck, who proved the same result under the additional condition of finite dimensionality. The main ingredient in Takeuchi’s proof is a Galois-type one-to-one correspondence between all sub-Hopf algebras and all normal Hopf ideals of a given commutative Hopf algebra. This bijection associates any sub-Hopf algebra with the ideal generated by the kernel of its counit. It is noteworthy that injectivity follows from the fact that any Hopf algebra is faithfully flat over its arbitrary sub-Hopf algebras [Reference Takeuchi36, Theorem 3.1]. The same correspondence can be found in [Reference Abe1, Section 4, p. 201], although with a slightly different proof. This correspondence does not rely on commutativity or cocommutativity of the Hopf algebras, and allowed, for example, to an extension of Takeuchi’s result showing that the category of cocommutative (but not necessarily commutative) Hopf algebras is semi-abelian [Reference Gran, Sterck and Vercruysse22].
The aim of this research is to extend the aforementioned correspondence to the “multi-object” setting, that is, from (affine) groups to (affine) groupoids. In Hopf algebraic terms, groupoids can be described as weak Hopf algebras or Hopf algebroids. We will therefore first provide a Galois correspondence between (certain classes of) Hopf ideals and sub-Hopf algebroids of general Hopf algebroids (in fact, even bialgebroids), and specializing to the commutative case, we obtain results for affine groupoids. Similar to the classical case explained above, the motivation of this work is that it could lead ultimately to a better understanding of the exactness properties of the category of (abelian) affine
$\Bbbk $
-groupoids. Furthermore, our results have applications on a wide variety of topics, since the interest in Hopf algebroids is widespread in different branches of mathematics: from algebraic topology, algebraic geometry, and differential geometry (see [Reference Deligne, Cartier, Katz, Manin, Illusie, Laumon and Ribet9, Reference El Kaoutit and Saracco16, Reference Ravenel31]), to the study of linear differential equations (see [Reference El Kaoutit and Gómez-Torrecillas14, Reference El Kaoutit and Saracco17]), noncommutative differential calculus [Reference Ghobadi18], and in the study of the fundamental groupoid of quivers [Reference Ghobadi19], to mention only a few.
1.2 Description of the main results
Recall that a left Hopf algebroid over
$\Bbbk $
(in the sense of Schauenburg [Reference Schauenburg34]) is a pair
$(A, {\mathcal H})$
of
$\Bbbk $
-algebras together with a family of structure maps that make of
${\mathcal H}$
an
$A\otimes _{}A^{o}$
-ring and an A-coring in a compatible way (see Definition 2.1) and whose Hopf–Galois map (described in equation (3.16)) is bijective. The so-called translation map
$\gamma :{\mathcal H}\to {\mathcal H}\times _{A^o}{\mathcal H}$
is obtained from the inverse of the Hopf–Galois map, and it is explicitly defined in equation (3.17). For a ring B over which a Hopf algebroid
$(A, {\mathcal H})$
coacts, we will denote the subring of coinvariant elements by
${B}^{\textsf {co}{{\mathcal H}}}$
, and for any subset
${\mathcal K}$
of
${\mathcal H}$
, the symbol
${\mathcal K}^{+}$
stands for the intersection of the kernel of the counit of
${\mathcal H}$
with
${\mathcal K}$
. Finally, recall that a functor
${\mathcal L} \colon {\mathcal C} \to {\mathcal D}$
admitting a right adjoint
${\mathcal R}$
is said to be comonadic if the comparison functor
${\mathcal K} \colon {\mathcal C} \to {\mathcal D}^{{\mathcal L}{\mathcal R}}$
to the Eilenberg–Moore category of coalgebras for the comonad
${\mathcal L}{\mathcal R}$
is an equivalence.
With these notations at hand, our first main result in the general context of noncommutative Hopf algebroids is the following. The functor
${\mathcal H} \otimes _{B} -$
in the statement below denotes the extension of scalars functor
${_B\textsf {Mod}}\to {_{{\mathcal H}}\textsf {Mod}}$
.
Theorem A (Theorem 3.15)
Let
$(A,{\mathcal H})$
be a left Hopf algebroid over
$\Bbbk $
such that
${{}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}}$
is A-flat. We have a well-defined inclusion-preserving bijective correspondence
When specialized to the commutative setting, it induces our second main theorem.
Theorem B (Theorem 4.26)
Let
$(A,{\mathcal H})$
be a commutative Hopf algebroid such that
${}_{s} {{{\mathcal H}}}$
is A-flat. Then we have a well-defined inclusion-preserving bijective correspondence
Given a commutative
$\Bbbk $
-algebra A, we denote by
$\mathscr {G}_A$
the associated
$\Bbbk $
-functor, that is, the functor from all commutative
$\Bbbk $
-algebras to sets that sends any algebra C the set of all algebra maps from A to C. In this way, a commutative Hopf algebroid
$(A,{\mathcal H})$
gives rise to the pair of
$\Bbbk $
-functors
$(\mathscr {G}_A, \mathscr {G}_{{\mathcal H}})$
which form a presheaf of groupoids (i.e., an affine
$\Bbbk $
-groupoid scheme). If, as a working terminology, we say that a subgroupoid
$\left (\mathscr {G}_A,\mathscr {G}_{\frac {{\mathcal H}}{I}}\right )$
of a given groupoid
$\big (\mathscr {G}_A,\mathscr {G}_{\mathcal H}\big )$
is pure whenever the extension
${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}} \subseteq {\mathcal H}$
is pure, then Theorem B can be rephrased as follows.
Proposition C Let
$(A,{\mathcal H})$
be a commutative Hopf algebroid such that
${}_{s} {{{\mathcal H}}}$
is A-flat. Then we have an inclusion-preserving bijective correspondence
A concrete application of the above to the Hopf algebroid of functions on a finite groupoid (i.e., with a finite set of arrows) is detailed in Example 4.27, where we show that there is a bijective correspondence between normal subgroupoids of a finite groupoid and pure sub-Hopf algebroids of the associated Hopf algebroid of functions.
As a consequence of our main theorem, we have the following remarkable result.
Proposition D (Corollaries 4.32 and 4.33)
Let
$\Bbbk $
be an algebraically closed field, and let
$(A,{\mathcal H})$
be a commutative Hopf algebroid over
$\Bbbk $
such that
${}_{s} {{{\mathcal H}}}$
is A-flat.
-
(a) If
$\big (\mathscr {G}_A(\Bbbk ),\mathscr {G}_{{\mathcal H}/I}(\Bbbk )\big )$
is a normal subgroupoid of
$\big (\mathscr {G}_A(\Bbbk ),\mathscr {G}_{\mathcal H}(\Bbbk )\big )$
such that
${\mathcal H}$
is faithfully flat over
${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$
, then
$$ \begin{align*}\mathscr{G}_{\mathcal H}(\Bbbk) / \mathscr{G}_{{\mathcal H}/I}(\Bbbk) \ \cong \ \mathscr{G}_{{{\mathcal H}}^{\textsf{co}{({{\mathcal H}}/{I})}}}(\Bbbk).\end{align*} $$
-
(b) If
$(A,{\mathcal K})$
is a sub-Hopf algebroid of the commutative Hopf algebroid
$(A,{\mathcal H})$
such that
${\mathcal H}$
is faithfully flat over
${\mathcal K}$
, then
$$ \begin{align*}\mathscr{G}_{\mathcal H}(\Bbbk)/\mathscr{G}_{{\mathcal H}/{\mathcal H}{\mathcal K}^+}(\Bbbk) \ \cong \ \mathscr{G}_{{\mathcal K}}(\Bbbk).\end{align*} $$
This result generalizes corresponding known results for affine algebraic groups (compare, for example, with [Reference Demazure and Gabriel10, Chapter III, Section 3, n°7, Théorème 7.2, p. 353, and Paragraph 7.3, Corollaire, p. 354]).
2 Preliminaries, notation, and first results
We work over a ground field
$\Bbbk $
. All vector spaces, algebras, and coalgebras will be over
$\Bbbk $
. The unadorned tensor product
$\otimes _{}$
stands for
$\otimes _{\Bbbk }$
. By a ring, we always mean a ring with identity element and modules over rings or algebras are always assumed to be unital. All over the paper, we assume a certain familiarity of the reader with the language of monoidal categories and of (co)monoids therein (see, for example, [Reference MacLane28, Chapter VII]).
We begin by collecting some facts about bimodules, (co)rings, and bialgebroids that will be needed in the sequel. The aim is that of keeping the exposition self-contained. Many results and definitions we will present herein hold in a more general context and under less restrictive hypotheses, but we preferred to limit ourselves to the essentials.
Given a (preferably, noncommutative)
$\Bbbk $
-algebra A, the category of A-bimodules forms a nonstrict monoidal category
$\left ({}_{A}{\textsf {Mod}}_{A},\otimes _{A},A,\mathfrak {a},\mathfrak {l},\mathfrak {r}\right )$
. Nevertheless, all over the paper, we will behave as if the structural natural isomorphisms
$$ \begin{gather*} \mathfrak{a}_{M,N,P} \colon (M\otimes_{A}N)\otimes_{A}P \to M\otimes_{A}(N\otimes_{A}P), \ (m\otimes_{A}n)\otimes_{A}p \to m\otimes_{A}(n\otimes_{A}p), \\ \mathfrak{l}_M \colon A \otimes_{A} M \to M, \quad a\otimes_{A}m \mapsto a\cdot m, \qquad \text{and} \\ \mathfrak{r}_M \colon M\otimes_{A}A \to M, \quad m \otimes_{A} a \mapsto m \cdot a \end{gather*} $$
were “the identities,” that is, as if
${}_{A}{\textsf {Mod}}_{A}$
was a strict monoidal category. If A is a noncommutative algebra, then we denote by
$A^{\mathrm {o}}$
its opposite algebra. In this case, an element
$a \in A$
may be denoted by
$a^{\mathrm {o}}$
when it is helpful to stress that it is viewed as an element in
$A^{\mathrm {o}}$
. We freely use the canonical isomorphism between the category of left A-module
${}_{A}\textsf {Mod}$
and that of right
$A^{\mathrm {o}}$
-modules
$\textsf {Mod}_{A^{\mathrm {o}}}$
.
2.1 The enveloping algebra and bimodules
Let A be an algebra, we denote by
the enveloping algebra of A, and we identify the category
${}_{A^{\mathrm {e}}}\textsf {Mod}$
of left
$A^{\mathrm {e}}$
-modules with the category
${}_{A}{\textsf { Mod}}_{A}$
of A-bimodules. Giving a morphism of algebras
$A^{\mathrm {e}} \to R$
is equivalent to providing two commuting algebra maps
$s\colon A \to R$
and
$t\colon A^{\mathrm {o}} \to R$
, called the source and the target, respectively. In particular, the identity of
$A^{\mathrm {e}}$
gives rise to
$s\colon A \to A^{\mathrm {e}}, a \mapsto a \otimes 1^{\mathrm {o}}$
and
$t\colon A^{\mathrm {o}} \to A^{\mathrm {e}}, a^{\mathrm {o}} \mapsto 1 \otimes a^{\mathrm {o}}$
.
Given two
$A^{\mathrm {e}}$
-bimodules M and N, there are several A-bimodule structures underlying M and N. Namely,
$$ \begin{align*}(a \otimes b^{\mathrm{o}}) \cdot m \cdot (c \otimes d^{\mathrm{o}}) = s(a)t(b^{\mathrm{o}}) m s(c)t(d^{\mathrm{o}})\end{align*} $$
for all
$m \in M$
,
$a,b,c,d\in A$
. This leads to several ways of considering the tensor product over A between these underlying A-bimodules. For the sake of clarity, we will adopt the following notations. Given an
$A^{\mathrm {e}}$
-bimodule M, the A-action by elements of the form
$a\otimes _{}1^{\mathrm {o}}$
will be denoted by
${}_{s} {M}$
or
${M}{}_{s} $
and the
$A^{\mathrm {o}}$
-action of the element of the form
$1\otimes _{}a^{\mathrm {o}}$
by
${}_{t} {M}$
or
${M}{}_{t}$
, depending on which side we are putting those elements. We still have other actions by the elements
$t(a)^{\mathrm {o}}$
, or
$s(a)^{\mathrm {o}}$
to produce other A or
$A^{\mathrm {o}}$
actions. In this case, we use the notation
$t^{\mathrm {o}}$
and
$s^{\mathrm {o}}$
located in the corresponding side on which we are declaring the action. Summing up, we denote
from which we can switch the given actions and obtain new two-sided actions involving A and
$A^{\mathrm {o}}$
:
and so on. For example,
${M}{}_{t^{\mathrm {o}}} = {}_{t} {M}$
. For the sake of simplicity, we may also often resort to the following variation of the previous conventions: for
$a,b \in A$
,
$m \in M$
,
2.2 Pure extensions of rings
Purity conditions will play a crucial role in establishing our main theorems. Therefore, we devote this subsection to collect a few results in details in this regard, for the convenience of the reader.
Given a ring R, we recall from [Reference Bourbaki4, Chapter I, Section 2, Example 24, p. 66] that a morphism of left (resp. right) R-modules
$f\colon M \to N$
is called pure (or universally injective) if and only if, for every right (resp. left) R-module P, the morphism of abelian groups
$P \otimes _{R} f$
(resp.
$f \otimes _{R} P$
) is injective. In particular, f itself is injective, by taking
$P = R$
.
Proposition 2.1 (and its corollaries) should be well known: it states that a ring extension
$f \colon {\mathcal A} \to {\mathcal B}$
is pure as a morphism of right
${\mathcal A}$
-modules if and only if the extension-of-scalars functor
${\mathcal B} \otimes _{{\mathcal A}} -$
is faithful. They will be of great help in what follows.
Proposition 2.1 Let R be a ring, and let
$f \colon {\mathcal A} \to {\mathcal B}$
be a morphism of R-rings. The following are equivalent.
-
(1)
$f \colon {\mathcal A} \to {\mathcal B}$
is a pure morphism of right
${\mathcal A}$
-modules. -
(2) For every left
${\mathcal A}$
-module M, the morphism
$\varrho _M\colon M \to {\mathcal B} \otimes _{{\mathcal A}} M$
,
$m \mapsto 1_{\mathcal B} \otimes _{{\mathcal A}} m$
, is injective for all
$m \in M$
. -
(3) For every morphism of left
${\mathcal A}$
-modules
$g \colon M \to N$
,
${\mathcal B} \otimes _{{\mathcal A}} g = 0$
implies
$g = 0$
. -
(4) If
$M \xrightarrow {g} N \xrightarrow {h} P$
are morphisms of left
${\mathcal A}$
-modules such that
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}} M \xrightarrow{{\mathcal B} \otimes_{{\mathcal A}} g} {\mathcal B} \otimes_{{\mathcal A}} N \xrightarrow{{\mathcal B} \otimes_{{\mathcal A}} h} {\mathcal B} \otimes_{{\mathcal A}} P\end{align*} $$
is an exact sequence of
${\mathcal B}$
-modules, then
$M \xrightarrow {g} N \xrightarrow {h} P$
is an exact sequence of
${\mathcal A}$
-modules. -
(5) If
$g \colon M \to N$
is a morphism of left
${\mathcal A}$
-modules such that
${\mathcal B} \otimes _{{\mathcal A}} g \colon {\mathcal B} \otimes _{{\mathcal A}} M \to {\mathcal B} \otimes _{{\mathcal A}} N$
is injective, then g itself is injective.
In addition, any of the foregoing entails that
-
(6) for every left ideal
$\mathfrak {m}$
in
${\mathcal A}$
, we have
$\mathfrak {m} = f^{-1}({\mathcal B}\mathfrak {m})$
,
and hence, in particular, that
-
(7) for every maximal left ideal
$\mathfrak {m}$
in
${\mathcal A}$
, there exists a maximal left ideal
$\mathfrak {n}$
in
${\mathcal B}$
such that
$\mathfrak {m} = f^{-1}(\mathfrak {n})$
.
Proof The equivalence between (1) and (2) is the definition of purity. The equivalence between (3) and (4) is the fact that an additive covariant functor between abelian categories is faithful if and only if it reflects exact sequences (see [Reference Popescu30, Section 3.1, Exercise 4]). The implication from (2) to (3) follows by commutativity of the diagram
To prove that (4) implies (5), consider the morphisms
$0 \to M \xrightarrow {g} N$
and apply the functor
${\mathcal B} \otimes _{{\mathcal A}} -$
. The implication from (5) to (2) follows by considering the morphism
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}}\varrho_M \colon {\mathcal B} \otimes_{{\mathcal A}} M \to {\mathcal B} \otimes_{{\mathcal A}} {\mathcal B} \otimes_{{\mathcal A}} M, \qquad b \otimes_{{\mathcal A}} m \mapsto b \otimes_{{\mathcal A}} 1_{\mathcal B} \otimes_{{\mathcal A}} m,\end{align*} $$
and observing that it admits the retraction
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}} {\mathcal B} \otimes_{{\mathcal A}} M \to {\mathcal B} \otimes_{{\mathcal A}} M, \qquad b \otimes_{{\mathcal A}} b' \otimes_{{\mathcal A}} m \mapsto bb' \otimes_{{\mathcal A}} m,\end{align*} $$
whence it is injective. Finally, let us show that (2)
$\Rightarrow $
(6)
$\Rightarrow $
(7). To prove that (2) implies (6), let
$\mathfrak {m}$
be a left ideal in
${\mathcal A}$
. Since
${\mathcal A}/\mathfrak {m}$
is a left
${\mathcal A}$
-module, (2) entails that
$$ \begin{align*}{\mathcal A}/\mathfrak{m} \xrightarrow{f \otimes_{{\mathcal A}} {\mathcal A}/\mathfrak{m}} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal A}/\mathfrak{m} \cong {\mathcal B} /{\mathcal B}\mathfrak{m}\end{align*} $$
is injective and therefore
$\mathfrak {m} = f^{-1}({\mathcal B}\mathfrak {m})$
. Now, to show that (6) implies (7), observe that, by taking
$\mathfrak {m} = 0$
in (6), we know that f is injective, and hence we may assume that
${\mathcal A} \subseteq {\mathcal B}$
. Suppose that
$\mathfrak {m}$
is a maximal ideal in
${\mathcal A}$
, and let
$\mathfrak {n} \subset {\mathcal B}$
be a maximal ideal containing
${\mathcal B} \mathfrak {m}$
. Then
$\mathfrak {m} = f^{-1}({\mathcal B}\mathfrak {m}) \subseteq f^{-1}(\mathfrak {n})$
and so, since
$1 \notin \mathfrak {n}$
,
$\mathfrak {m} = f^{-1}(\mathfrak {n})$
by maximality of
$\mathfrak {m}$
.
Proposition 2.1 is the analog of [Reference Bourbaki4, Chapter I, Section 3, n°5, Proposition 9] and [Reference Waterhouse39, Section 13.1, Theorem] for pure morphisms.
Corollary 2.2 (of Proposition 2.1)
Let R be a ring, and let
$f \colon {\mathcal A} \to {\mathcal B}$
be a morphism of R-rings. If f is pure as a morphism of right
${\mathcal A}$
-modules, then
$(M,\varrho _M)$
is the equalizer of the pair
in the category of left
${\mathcal A}$
-modules, for every left
${\mathcal A}$
-module M. In particular, we have the equalizer
Proof Consider the parallel arrows
It is clear that the function
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}} \varrho_M \colon {\mathcal B} \otimes_{{\mathcal A}} M \to {\mathcal B} \otimes_{{\mathcal A}} {\mathcal B} \otimes_{{\mathcal A}} M, \qquad b \otimes_{{\mathcal A}} m \mapsto b \otimes_{{\mathcal A}} 1_{\mathcal B} \otimes_{{\mathcal A}} m\end{align*} $$
lands in the equalizer of
${\mathcal B} \otimes _{{\mathcal A}} \lambda _{\mathcal B} \otimes _{{\mathcal A}} M$
and
${\mathcal B} \otimes _{{\mathcal A}} \varrho _{\mathcal B} \otimes _{{\mathcal A}} M$
, and it is injective, since it admits
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}} {\mathcal B} \otimes_{{\mathcal A}} M \to {\mathcal B} \otimes_{{\mathcal A}} M, \qquad b \otimes_{{\mathcal A}} b' \otimes_{{\mathcal A}} m \mapsto bb' \otimes_{{\mathcal A}} m,\end{align*} $$
as a retraction. On the other hand, any
$\sum _i b_i \otimes _{{\mathcal A}} b_i' \otimes _{{\mathcal A}} m_i$
in the equalizer satisfies
$\sum _i b_i \otimes _{{\mathcal A}} b_i' \otimes _{{\mathcal A}} m_i = \sum _i b_i b_i' \otimes _{{\mathcal A}} 1_{\mathcal B} \otimes _{{\mathcal A}} m_i$
, and hence it is in the image of
${\mathcal B} \otimes _{{\mathcal A}} \varrho _M$
. Therefore,
$({\mathcal B} \otimes _{{\mathcal A}} M,{\mathcal B} \otimes _{{\mathcal A}} \varrho _M)$
is the equalizer of
${\mathcal B} \otimes _{{\mathcal A}} \lambda _{\mathcal B} \otimes _{{\mathcal A}} M$
and
${\mathcal B} \otimes _{{\mathcal A}} \varrho _{\mathcal B} \otimes _{{\mathcal A}} M$
, whence
$(M,\varrho _M)$
is the equalizer of
$\lambda _{\mathcal B} \otimes _{{\mathcal A}} M$
and
$\varrho _{\mathcal B} \otimes _{{\mathcal A}} M$
by purity (statement (4) in Proposition 2.1).
Corollary 2.3 (of Proposition 2.1)
Let R be a ring, and let
$f \colon {\mathcal A} \to {\mathcal B}$
be a morphism of R-rings. If f is a pure morphism of right R-modules and
${\mathcal B}_R$
is flat, then
${\mathcal A}_R$
is flat.
Proof Take a monomorphism of left R-modules
$M \xrightarrow {g} N$
. By flatness of
${\mathcal B}$
on R, we have that
${\mathcal B} \otimes _{R} M \xrightarrow {{\mathcal B} \otimes _{R} g} {\mathcal B} \otimes _{R} N$
is injective and so
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}} {\mathcal A} \otimes_{R} M \xrightarrow{{\mathcal B} \otimes_{{\mathcal A}} {\mathcal A} \otimes_{R} g} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal A} \otimes_{R} N\end{align*} $$
is injective, too. By purity (statement (5) in Proposition 2.1),
${\mathcal A} \otimes _{R} M \xrightarrow {{\mathcal A} \otimes _{R} g} {\mathcal A} \otimes _{R} N$
is injective.
Corollary 2.4 (of Proposition 2.1)
Let R be any ring, and let
${\mathcal A} \xrightarrow {g} {\mathcal T} \xrightarrow {h} {\mathcal B}$
be morphisms of R-rings. Set
. If
$f \colon {\mathcal A} \to {\mathcal B}$
is a pure morphism of right
${\mathcal A}$
-modules, then the morphism of right R-modules g is pure as well. In particular, f is pure as a morphism of right R-modules.
Proof Let P be any left R-module. Consider the morphism
$$ \begin{align*}{\mathcal B} \xrightarrow{\cong} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal A} \xrightarrow{{\mathcal B} \otimes_{{\mathcal A}} f} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal B}.\end{align*} $$
Since it is split injective, with retraction induced by the multiplication of
${\mathcal B}$
, the morphism
$$ \begin{align*}{\mathcal B} \otimes_{R} P \xrightarrow{\cong} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal A} \otimes_{R} P \xrightarrow{{\mathcal B} \otimes_{{\mathcal A}} f \otimes_{R} P} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal B} \otimes_{R} P\end{align*} $$
is (split) injective as well, and hence
$$ \begin{align*}{\mathcal B} \otimes_{{\mathcal A}} {\mathcal A} \otimes_{R} P \xrightarrow{{\mathcal B} \otimes_{{\mathcal A}} f \otimes_{R} P} {\mathcal B} \otimes_{{\mathcal A}} {\mathcal B} \otimes_{R} P\end{align*} $$
is injective. Since
$$ \begin{align*} {\mathcal B} \otimes_{{\mathcal A}} f \otimes_{R} P = \left({\mathcal B} \otimes_{{\mathcal A}} h \otimes_{R} P\right) \circ \left({\mathcal B} \otimes_{{\mathcal A}} g \otimes_{R} P\right), \end{align*} $$
then
${\mathcal B} \otimes _{{\mathcal A}} g \otimes _{R} P$
is injective, too. Being
${\mathcal B}_{\mathcal A}$
pure, the morphism
$g \otimes _{R} P$
is injective by statement (5) in Proposition 2.1. The last statement follows by taking f itself as g and
$\operatorname {Id}_{\mathcal B}$
as h.
Lemma 2.5 Let R be any ring, and let
${\mathcal A} \xrightarrow {g} {\mathcal T} \xrightarrow {h} {\mathcal B}$
be morphisms of R-rings. If
${\mathcal B}_{\mathcal A}$
is faithfully flat, then the morphism of right R-modules g is pure. In particular,
is pure.
Proof Since a faithfully flat extension is pure, the statement follows from Corollary 2.4.
2.3 The Takeuchi–Sweedler crossed product
Let
${}_{A^{\mathrm {e}}}M{}_{A^{\mathrm {e}}}$
and
${}_{A^{\mathrm {e}}}N{}_{A^{\mathrm {e}}}$
be two
$A^{\mathrm {e}}$
-bimodules. We first define the A-bimodule
This will usually be the tensor product over A that we are going to consider more often, unless specified otherwise. Then, inside
$M \otimes _{A} N$
, we consider the subspace
It is easy to see that
$M \times _{A} N$
is an A-subbimodule (left
$A^{\mathrm {e}}$
-submodule) with respect to the actions
$$ \begin{align} a\cdot \left(\sum_i m_i \otimes_{A} n_i\right) = \sum_i s(a)m_i \otimes_{A} n_i \quad \!\text{and}\! \quad \left(\sum_i m_i \otimes_{A} n_i\right)\cdot a = \sum_i m_i \otimes_{A} t(a^{\mathrm{o}})n_i \end{align} $$
for all
$\sum _i m_i \otimes _{A} n_i\in M\times _{A}N$
and
$a\in A$
, but it is also an A-subbimodule (right
$A^{\mathrm {e}}$
-submodule) with respect to the actions
$$ \begin{align} a\bullet \left(\sum_i m_i \otimes_{A} n_i\kern-1pt\right) = \sum_i m_i \otimes_{A} n_it(a^{\mathrm{o}}) \!\!\!\quad \text{and}\!\!\! \quad \left(\sum_i m_i \otimes_{A} n_i\kern-1pt\right)\bullet a = \sum_i m_is(a) \otimes_{A} n_i. \end{align} $$
In particular, it is an
$A^{\mathrm {e}}$
-bimodule itself. There exist categorical ways to describe this Takeuchi–Sweedler product in terms of ends and coends (see, e.g., [Reference Sweedler35, Reference Takeuchi37], where the integral notation dating back to [Reference Yoneda40] was used) or in terms of monoidal products (see, e.g., [Reference Böhm3, Reference Takeuchi38]), but, for the convenience of the unaccustomed reader and for the sake of simplicity, we decided to opt for the more elementary description above. In particular, in
$M \times _{A} N$
, the following relations hold for all
$\sum _i m_i \otimes _{A} n_i \in M\times _{A} N$
and all
$a\in A$
:
$$ \begin{align*} \sum_i t(a^{\mathrm{o}})m_i \otimes_{A}{n}_i \stackrel{(2.2)}{=} \sum_i m_i \otimes_{A} s(a)n_i \quad \text{and} \quad \sum_i m_it(a^{\mathrm{o}}) \otimes_{A} n_i \stackrel{(2.3)}{=} \sum_i m_i \otimes_{A}{n}_is(a). \end{align*} $$
In this way, if
${\mathcal U}$
and
${\mathcal V}$
are two
$A^{\mathrm {e}}$
-rings, then
${\mathcal U} \times _{A} {\mathcal V}$
is also an
$A^{\mathrm {e}}$
-ring, with multiplication
$$ \begin{align*}\left(\sum_iu_i \otimes_{A} v_i\right)\cdot \left(\sum_ju^{\prime}_j \otimes_{A} v^{\prime}_j\right) = \sum_{i,j}u_iu^{\prime}_j\otimes_{A}v_iv^{\prime}_j\end{align*} $$
for all
$u_i,u^{\prime }_j\in {\mathcal U}$
,
$v_i,v^{\prime }_j\in {\mathcal V}$
and
$\Bbbk $
-algebra morphism
$A\otimes _{}A^{o} \to {\mathcal U} \times _{A} {\mathcal V}, a \otimes b^{\mathrm {o}} \mapsto s_{\mathcal U}(a) \otimes _{A} t_{\mathcal V}\big (b^{\mathrm {o}}\big )$
. The
$A^{\mathrm {e}}$
-actions (2.4) and (2.5) are induced by this
$A^{\mathrm {e}}$
-ring structure.
2.4 Left bialgebroids
Next, we recall the definition of a left bialgebroid. It can be considered as a revised version of the notion of a
$\times _A$
-bialgebra as it appears in [Reference Schauenburg33, Definition 4.3]. However, we prefer to mimic [Reference Lu25] as presented in [Reference Brzeziński and Militaru6, Definition 2.2].
Definition 2.1 A left bialgebroid is the datum of:
-
(B1) a pair
$\left (A,{\mathcal H}\right )$
of
$\Bbbk $
-algebras; -
(B2) a
$\Bbbk $
-algebra map
, inducing a source
$s\colon A\to {\mathcal H}$
and a target
$t\colon A^{\mathrm {o}}\to {\mathcal H}$
and making of
${\mathcal H}$
an
$A^{\mathrm {e}}$
-bimodule; -
(B3) an A-coring structure
$\left ({\mathcal H},\Delta ,\varepsilon \right )$
on the A-bimodule
${}_{A^{\mathrm {e}}}{\mathcal H}={}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$
, that is to say,
$$ \begin{align*} \Delta \colon {}_{s} {{\mathcal H}}{}_{t^{\mathrm{o}}} \to {}_{s} {{\mathcal H}}{}_{t^{\mathrm{o}}}\otimes_{A}{}_{s} {{\mathcal H}}{}_{t^{\mathrm{o}}} \qquad \text{and} \qquad \varepsilon\colon {}_{s} {{\mathcal H}}{}_{t^{\mathrm{o}}} \to A \end{align*} $$
A-bilinear maps such that
(2.6)
$$ \begin{align} \left(\Delta \otimes_{A} {\mathcal H}\right)\circ \Delta = \left({\mathcal H} \otimes_{A}\Delta\right) \circ \Delta \quad \text{and} \quad \left(\varepsilon\otimes_{A}{\mathcal H}\right)\circ \Delta = \operatorname{Id}_{\mathcal H} = \left({\mathcal H}\otimes_{A}\varepsilon\right)\circ \Delta; \end{align} $$
subject to the following compatibility conditions:
-
(B4)
$\Delta $
takes values into
${}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}} \times _{A} {}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$
and
$\Delta \colon {}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}} \to {}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}\times _{A}{}_{s} {{\mathcal H}}{}_{t^{\mathrm {o}}}$
is a morphism of
$\Bbbk $
-algebras; -
(B5)
$\varepsilon \Big (xs\big (\varepsilon \left (y\right )\big )\Big ) = \varepsilon \left (xy\right ) = \varepsilon \Big (xt\big (\varepsilon \left (y\right )^{\mathrm {o}}\big )\Big )$
for all
$x,y\in {\mathcal H}$
; -
(B6)
$\varepsilon (1_{\mathcal H})=1_A$
.
A
$\Bbbk $
-linear map
$\varepsilon \colon {\mathcal H} \to A$
which is left
$A^{\mathrm {e}}$
-linear and satisfies (B5) and (B6) is called a left character on the
$A^{\mathrm {e}}$
-ring
${\mathcal H}$
(see [Reference Böhm3, Lemma 2.5 and following]).
A morphism of bialgebroids
$\boldsymbol {\phi }\colon (A,{\mathcal H}) \to (B,{\mathcal K})$
is a pair of algebra maps
$\left ( \phi _{{0}} \colon A \to B,\phi _{{1}} \colon {\mathcal H} \to {\mathcal K}\right )$
such that
$$ \begin{gather*} \phi _{1}\circ s_{\mathcal H} = s_{\mathcal K} \circ \phi _{0}, \qquad \phi _{1}\circ t_{\mathcal H} = t_{\mathcal K} \circ \phi_{0}, \\ \varepsilon_{\mathcal K} \circ \phi _{1} = \phi _{0} \circ \varepsilon_{\mathcal H} , \qquad \Delta_{\mathcal K} \circ \phi _{1} =\chi \circ \left( \phi _{1}\otimes _{A}\phi_{1}\right) \circ \Delta_{\mathcal H}, \end{gather*} $$
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,
$\chi \left (h\otimes _{A}k\right ) =h\otimes _{B}k$
. If
$A = B$
and
$\phi _0 = \operatorname {Id}_A$
, then we say that
$\phi _1 : {\mathcal H} \to {\mathcal K}$
is a morphism of bialgebroids over A.
As a matter of terminology, a bialgebroid
$(A,{\mathcal H})$
as in Definition 2.1 is often referred to as a left bialgebroid
${\mathcal H}$
over A. Since in the following we will mainly deal with bialgebroids over a fixed base A, we may often omit to specify it and simply refer to
$(A,{\mathcal H})$
as the left bialgebroid
${\mathcal H}$
.
Remark 2.6 Let us make explicit some of the relations involved in the definition of a left bialgebroid and some of their consequences. In terms of elements of A and
${\mathcal H}$
, and by resorting to Heyneman–Sweedler Sigma Notation, relation (2.6) becomes
$$ \begin{align*} \sum\kern-1pt x_{11} \otimes_{A} x_{12} \otimes_{A} x_2 = \sum\kern-1pt x_{1}\otimes_{A} x_{21}\otimes_{A} x_{22} \ \textrm{ and } \ \sum\kern-1pt s\big(\varepsilon\left(x_1\right)\big)x_2 = x = \sum\kern-1pt t\big(\varepsilon\left(x_2\right)^{\mathrm{o}}\big)x_1 \end{align*} $$
for all
$x\in {\mathcal H}$
. The A-bilinearity of
$\Delta $
forces
$$ \begin{align} \Delta\left(s\left(a\right)\right) = a\cdot \Delta\left(1_{\mathcal H}\right) = s\left(a\right)\otimes_{A}1_{\mathcal H} \quad \text{and} \quad \Delta\left(t\left(a^{\mathrm{o}}\right)\right) = \Delta\left(1_{\mathcal H}\right)\cdot a = 1_{\mathcal H} \otimes_{A} t\left(a^{\mathrm{o}}\right) \end{align} $$
for all
$a\in A$
, and its multiplicativity forces, as a consequence,
$$ \begin{gather*} \Delta\left(xs\left(a\right)\right) = \Delta\left(x\right)\Delta\left(s\left(a\right)\right) = \left(\sum x_1\otimes_{A}x_2\right)\left(s\left(a\right)\otimes_{A}1_{\mathcal H}\right) = \sum x_1s\left(a\right)\otimes_{A}x_2, \\ \Delta\left(xt\left(a^{\mathrm{o}}\right)\right) = \Delta\left(x\right)\Delta\left(t\left(a^{\mathrm{o}}\right)\right) = \left(\sum x_1\otimes_{A}x_2\right)\left(1_{\mathcal H}\otimes_{A}t\left(a^{\mathrm{o}}\right)\right) = \sum x_1\otimes_{A}x_2t\left(a^{\mathrm{o}}\right), \end{gather*} $$
for all
$x\in {\mathcal H}$
. In particular,
$\Delta :{\mathcal H} \to {\mathcal H} \times _{A} {\mathcal H}$
is a morphism of
$A^{\mathrm {e}}$
-rings.
The following result is extremely well known (it underlies the monoidality of the category of left
${\mathcal H}$
-modules).
Proposition 2.7 Let
$(A,{\mathcal H})$
be a left bialgebroid, and let
$M,N$
be two left
${\mathcal H}$
-modules. If we consider M and N as A-bimodules (left
$A^{\mathrm {e}}$
-modules) via the restriction of scalars along
, then
$M \otimes _{A} N$
is a left
${\mathcal H}\times _{A}{\mathcal H}$
-module with
Moreover, (2.8) is a morphism of
$A^{\mathrm {e}} $
-bimodules if we consider
${\mathcal H} \times _{A} {\mathcal H}$
endowed with the actions (2.4) from the left and (2.5) from the right and we consider
$\operatorname {End}_{\Bbbk }\left (M \otimes _{A} N\right )$
endowed with the left
$A^{\mathrm {e}} $
-action coming from the regular A-bimodule structure on the codomain and the right
$A^{\mathrm {e}} $
-action coming from the regular A-bimodule structure on the domain. Equivalently,
$$ \begin{align*}({\mathcal H} \times_{A} {\mathcal H})\otimes_{A^{\mathrm{e}} } (M \otimes_{A} N) & \to M \otimes_{A} N, \\ \left(\sum_ix_i \otimes_{A} y_i\right) \otimes_{A^{\mathrm{e}} } \left(m \otimes_{A} n\right) & \mapsto \sum_ix_i\cdot m\otimes_{A}y_i\cdot n\end{align*} $$
is a left
$A^{\mathrm {e}} $
-linear action.
Corollary 2.8 If
$M,N$
are left
${\mathcal H}$
-modules, then
$M \otimes _{A} N$
is a left
${\mathcal H}$
-module via restrictions of scalars along
$\Delta $
:
$$ \begin{align} {\mathcal H} \to \operatorname{End}_{\Bbbk}\left(M \otimes_{A} N\right), \qquad h \mapsto \Big[ m \otimes_{A} n \mapsto \sum h_1\cdot m \otimes_{A} h_2 \cdot n\Big], \end{align} $$
and the latter is of
$A^{\mathrm {e}} $
-bimodules. Equivalently,
$$ \begin{align*}{\mathcal H} \otimes_{A^{\mathrm{e}} } (M \otimes_{A} N) \to M \otimes_{A} N, \qquad h \otimes_{A^{\mathrm{e}} } \left(m \otimes_{A} n\right) \mapsto \sum h_1\cdot m\otimes_{A} h_2\cdot n\end{align*} $$
is a left
$A^{\mathrm {e}} $
-linear action. In particular, the category of left
${\mathcal H}$
-modules is monoidal with tensor product
$\otimes _{A}$
and unit object A. The left
${\mathcal H}$
-action on A is given by
for all
$a \in A$
,
$h \in {\mathcal H}$
.
Definition 2.2 Given a left bialgebroid
$(A,{\mathcal H})$
, a left
${\mathcal H}$
-module coring is a comonoid in the monoidal category
$\big ({}_{{\mathcal H}}\textsf {Mod},\otimes _{A},A\big )$
of left
${\mathcal H}$
-modules.
2.5 Module corings and relative Hopf modules
Let
$(A,{\mathcal H})$
be a left bialgebroid, and let
$\left (\overline {{\mathcal H}},\overline {\Delta },\overline {\varepsilon }\right )$
be a left
${\mathcal H}$
-module coring. Being an
${\mathcal H}$
-module,
$\overline {{\mathcal H}}$
has an A-bimodule structure
${}_{s} {\overline {{\mathcal H}}}{}_{t^{\mathrm {o}}}$
given by
for all
$a,b \in A$
,
$u \in \overline {{\mathcal H}}$
, with respect to which it is an A-coring. In this setting, we may consider the category
of relative
$\Big (\overline {{\mathcal H}},{\mathcal H}\Big )$
-Hopf modules: these are left comodules over the comonoid
$\overline {{\mathcal H}}$
in the monoidal category
$\left ({}_{{\mathcal H}}\textsf {Mod},\otimes _{A},A\right )$
. In details, they are left
${\mathcal H}$
-modules
$(M,\mu _M\colon {\mathcal H} \otimes M \to M)$
together with a coassociative and counital left
$\overline {{\mathcal H}}$
-coaction
$$ \begin{align*} \partial_M\colon M \to {\overline{{\mathcal H}}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{M}}, \qquad m \mapsto \sum m_{-1} \otimes_{A} m_0, \end{align*} $$
which is also a morphism of left
${\mathcal H}$
-modules, that is, for all
$h \in {\mathcal H}$
,
$m \in M$
, one has
$$ \begin{align*}\sum h_1 \cdot m_{-1} \otimes_{A} h_2 \cdot m_0 = h \cdot \partial_M(m) = \partial_M(h \cdot m) = \sum (h \cdot m)_{-1} \otimes_{A} (h \cdot m)_0.\end{align*} $$
In the present subsection, we are interested in the category
. A Hopf algebroid analog of Doi’s equivalence theorem [Reference Doi11, Theorem 2.3] will be of key importance in proving Theorem 3.14.
Now, suppose that
$\pi \colon {\mathcal H} \to \overline {{\mathcal H}}$
is a morphism of left
${\mathcal H}$
-module corings. Recall that if C is an A-coring with distinguished group-like element
$g\in C$
and if
$(N,\partial _N)$
is a left C-comodule, then
is the so-called space of (left) coinvariant elements in N. In the standing hypotheses,
$\pi (1_{\mathcal H})$
is a distinguished group-like element in
$\overline {{\mathcal H}}$
and we may consider
$$ \begin{align*} {{\mathcal H}}^{\textsf{co}{\overline{{\mathcal H}}}} = \Big\{x\in {\mathcal H} ~\big\vert~ \sum \pi(x_1) \otimes_{A} x_2 = \pi(1_{\mathcal H}) \otimes_{A} x \Big\} \end{align*} $$
(also denoted by
${{\mathcal H}}^{\textsf {co}{\pi }}$
), which is the space of (left) coinvariant elements in
${\mathcal H}$
under the (left)
$\overline {{\mathcal H}}$
-coaction
${\mathcal H} \to \overline {{\mathcal H}} \otimes _{A} {\mathcal H}$
given by
Remark 2.9 Observe that, in general,
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
is just a right A-submodule (left
$A^{\mathrm {o}}$
-submodule) with respect to the action
$\triangleleft $
from (2.1) and a left A-submodule (right
$A^{\mathrm {o}}$
-submodule) with respect to the action
, because of (2.7):
▲
Lemma 2.10 The space
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
is an
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t, and
$\partial $
from (2.11) is right
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
-linear.
Proof Since
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}\subseteq {\mathcal H}$
and clearly
$1_{{\mathcal H}} \in {{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
, to check that it is a
$\Bbbk $
-subalgebra, we just need to verify that the induced multiplication is well defined. To this aim, observe that
$\overline {{\mathcal H}}\otimes _{A}{\mathcal H}$
has a left
${\mathcal H}$
-module structure given by Corollary 2.8 and it has a natural right
${\mathcal H}$
-module structure given by
$$ \begin{align} {\mathcal H} \to \operatorname{End}_{\Bbbk}\left(\overline{{\mathcal H}}\otimes_{A}{\mathcal H}\right)^{\mathrm{o}}, \qquad x \mapsto \left[u\otimes_{A} y \mapsto u \otimes_{A} yx\right]. \end{align} $$
Thanks to this, we may compute directly that for
$x,y\in {{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}},$
$$ \begin{align*} \sum & ~\pi((xy)_1) \otimes_{A} (xy)_2 = \sum \pi(x_1y_1) \otimes_{A} x_2y_2 = \sum x_1\cdot \pi(y_1) \otimes_{A} x_2y_2 \\ & \stackrel{(2.9)}{=} x \cdot \left(\sum \pi(y_1) \otimes_{A} y_2\right) = x\cdot \left( \pi(1_{\mathcal H}) \otimes_{A} y \right) \stackrel{(2.9)}{=} \sum \pi(x_1) \otimes_{A} x_2y \\ & \stackrel{(2.12)}{=} \left(\sum \pi(x_1) \otimes_{A} x_2\right)y = \pi(1_{\mathcal H}) \otimes_{A} xy, \end{align*} $$
so that
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
is indeed a subalgebra. Moreover, since
$$ \begin{align*} \sum \pi(t(a^{\mathrm{o}})_1) \otimes_{A} t(a^{\mathrm{o}})_2 & = (\pi \otimes_{A}{\mathcal H})(\Delta(t(a^{\mathrm{o}}))) \stackrel{(2.7)}{=} (\pi \otimes_{A}{\mathcal H})(1_{\mathcal H}\otimes_{A}t(a^{\mathrm{o}})) \\ & = \pi(1_{\mathcal H})\otimes_{A}t(a^{\mathrm{o}}), \end{align*} $$
for all
$a\in A$
, t takes values in
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
and so
${{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
is
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t.
To conclude, notice that if
$x \in {\mathcal H}$
and
$b \in {{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
, then
$$ \begin{align*} \partial(xb) & = \sum \pi(x_1b_1) \otimes_{A} x_2b_2 \stackrel{(2.9)}{=} x \cdot \left(\sum \pi(b_1) \otimes_{A} b_2\right) = x\cdot \left( \pi(1_{\mathcal H}) \otimes_{A} b \right) \\ & \stackrel{(2.9)}{=} \sum \pi(x_1) \otimes_{A} x_2b \stackrel{(2.12)}{=} \left(\sum \pi(x_1) \otimes_{A} x_2\right)b = \partial(x)b.\\[-37pt] \end{align*} $$
Set
, for the sake of brevity. Since
${\mathcal H}$
becomes a right B-module by restriction of scalars along
$\iota : B \to {\mathcal H}$
, we can consider the category
${}_{B}\textsf {Mod}$
and the extension of scalars functor
which is left adjoint to the restriction of scalars functor
${_{\iota }\big (-\big )} \colon {}_{{\mathcal H}}\textsf {Mod} \to {}_{B}\textsf {Mod}.$
It turns out that the natural transformation
$$ \begin{align*} {\mathcal H} \otimes_{B} (-) \to \overline{{\mathcal H}} \otimes_{A} \big({\mathcal H} \otimes_{B} -\big), \qquad h \otimes_{B} - \mapsto \sum\pi(h_1) \otimes_{A} \big(h_2 \otimes_{B} -\big) \end{align*} $$
induces a morphism of comonads
$$ \begin{align} \Theta\colon {\mathcal H} \otimes_{B} {{}_{\iota}\big(-\big)} \to \overline{{\mathcal H}} \otimes_{A} -, \qquad h \otimes_{B} (-) \mapsto \sum \pi(h_1) \otimes_{A} h_2\cdot (-) \end{align} $$
as in [Reference Gómez-Torrecillas21, Proposition 2.1] and so, since
is also the category of Eilenberg–Moore objects for the comonad
$\overline {{\mathcal H}} \otimes _{A} -$
over
${}_{{\mathcal H}}\textsf {Mod}$
, the functor F induces a functor
. The following theorem is nothing else than [Reference Gómez-Torrecillas21, Theorem 2.7].
Theorem 2.11 Let
$(A,{\mathcal H})$
be a left bialgebroid with
${}_{s} {{{\mathcal H}}} \ A$
-flat. If
$\overline {{\mathcal H}}$
is a left
${\mathcal H}$
-module coring together with a morphism
$\pi \colon {\mathcal H} \to \overline {{\mathcal H}}$
of left
${\mathcal H}$
-module corings and if
$B = {{\mathcal H}}^{\textsf {co}{\overline {{\mathcal H}}}}$
, then
is an equivalence of categories if and only if:
-
(1)
${\mathcal H} \otimes _{B} - \colon {}_{B}\textsf {Mod} \to {}_{{\mathcal H}}\textsf {Mod}$
is comonadic and -
(2) the canonical morphism of comonads
$\Theta $
is a natural isomorphism.
Lemma 2.12 The natural transformation
$\Theta $
of (2.13) is a natural isomorphism if and only if
$$ \begin{align} \xi\colon {\mathcal H}_{\iota} \otimes_{B} {{}_{\iota}{\mathcal H}} \to {\overline{{\mathcal H}}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}, \qquad x \otimes_{B} y \mapsto \sum \pi(x_1) \otimes_{A} x_2y, \end{align} $$
is an isomorphism.
Proof If
$\Theta $
is a natural isomorphism, then
$\Theta _{{\mathcal H}} = \xi $
is an isomorphism. Conversely, since
$\xi $
is right
${\mathcal H}$
-linear with respect to the regular right
${\mathcal H}$
-module structures, we may conclude that if
$\xi $
is an isomorphism, then
$$ \begin{align*} \xi \otimes_{{\mathcal H}} M \colon \Big({\mathcal H}_{\iota} \otimes_{B} {{}_{\iota}{\mathcal H}}\Big) \otimes_{{\mathcal H}} M & \to \left({\overline{{\mathcal H}}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}\right)\otimes_{{\mathcal H}} M, \\ \left(x \otimes_{B} y\right)\otimes_{{\mathcal H}}m & \mapsto \sum \left(\pi(x_1) \otimes_{A} x_2y \otimes_{{\mathcal H}} m\right), \end{align*} $$
is an isomorphism for every M in
${}_{{\mathcal H}}\textsf {Mod}$
, natural in M. The commutativity of the diagram
allows us to conclude that if
$\xi $
is an isomorphism, then
$\Theta $
is a natural isomorphism. Notice that, due to the fact that only regular module structures are involved, the vertical isomorphisms are, in fact, the canonical isomorphism
$$ \begin{gather*} \Big({\mathcal H}_{\iota} \otimes_{B} {{}_{\iota}{\mathcal H}}\Big) \otimes_{{\mathcal H}} M \cong {\mathcal H}_{\iota} \otimes_{B} \Big({{}_{\iota}{\mathcal H}} \otimes_{{\mathcal H}} M\Big) \cong {\mathcal H}_{\iota} \otimes_{B} {{}_{\iota} M} \qquad \text{and} \\ \left({\overline{{\mathcal H}}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}\right)\otimes_{{\mathcal H}} M \cong {\overline{{\mathcal H}}}{}_{t^{\mathrm{o}}} \otimes_{A} \left({}_{s} {{{\mathcal H}}}\otimes_{{\mathcal H}} M\right) \cong {\overline{{\mathcal H}}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{M}}. \\[-35pt] \end{gather*} $$
In general, even when
${\mathcal K}$
is not an equivalence of categories,
${\mathcal K}$
still admits a right adjoint functor, as in the case for ordinary Hopf algebras.
Proposition 2.13 The construction
for every M in
induces a functor
which is right adjoint to
${\mathcal K}$
. Unit and counit are given by
respectively.
Proof Given a relative Hopf module
$(M,\mu _M,\partial _M)$
, denote by
$\Omega (M)$
its underlying H-module structure
$(M,\mu _M)$
. In view of [Reference Gómez-Torrecillas21, Proposition 2.3],
${\mathcal K}$
admits a right adjoint which is explicitly realized on objects as the equalizer of the parallel pair
which is exactly
${M}^{\textsf {co}{\overline {{\mathcal H}}}}$
.
3 The correspondence for arbitrary Hopf Algebroids
This section contains our first main result, namely, Theorem 3.15.
3.1 The correspondence between left ideals two-sided coideals and right coideal subrings
Henceforth, all bialgebroids will be left ones.
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
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,
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:
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
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
${}_{A^{\mathrm {o}}}\textsf {Mod}$
):
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
$$ \begin{align} \pi \circ \iota = \pi \circ \iota \circ t'\circ \varepsilon' = \pi \circ t \circ \varepsilon \circ \iota. \end{align} $$
In particular,
$$ \begin{align} \left(\pi \otimes_{A} {\mathcal H}\right) \circ \Delta \circ \iota = (\pi \otimes_{A} {\mathcal H}) \circ (t \otimes_{A} {\mathcal H}) \circ \iota. \end{align} $$
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
$$ \begin{align} {\mathcal H} \otimes ({\mathcal H} \otimes_{A} {\mathcal H}) \xrightarrow{\cdot} {\mathcal H} \otimes_{A} {\mathcal H}, \qquad x \otimes \left(\sum_j h_j \otimes_{A} k_j\right) \mapsto \sum_{j} x_1h_j \otimes_{A} x_2k_j, \end{align} $$
and a natural action of
${\mathcal H}$
on
${\mathcal H}/{\mathcal H} B^+ \otimes _{A} {\mathcal H}$
given by
$$ \begin{align} {\mathcal H} \otimes \left(\frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A} {\mathcal H}\right) \xrightarrow{\cdot} \frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A} {\mathcal H}, \quad x \otimes \left(\sum_j \pi(h_j) \otimes_{A} k_j\right) \mapsto \sum_{j} \pi(x_1h_j) \otimes_{A} x_2k_j, \end{align} $$
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
$$ \begin{align} \sum \iota(b_{0}) \otimes_{A} b_1 = \sum \iota(b)_1 \otimes_{A} \iota(b)_2. \end{align} $$
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
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
Proposition 3.7 Let
$(A,{\mathcal H})$
be a left bialgebroid over A. Let B be a right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring via t of
${\mathcal H}$
. In view of Proposition 3.6, we know that
${\mathcal H} B^+$
is a left ideal two-sided coideal in
${\mathcal H}$
. Denote by
$\pi \colon {\mathcal H} \to {\mathcal H}/{\mathcal H} B^+$
the canonical projection. Then the inclusion
$\iota \colon B \to {\mathcal H}$
factors through
${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$
. That is to say, we have an inclusion
${B \subseteq {{\mathcal H}}^{\textsf { co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}}$
, which we denote by
$\eta _B$
, such that
$\iota = j'\circ \eta _B$
where
$j'\colon {{\mathcal H}}^{\textsf { co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}} \to {\mathcal H}$
is the canonical inclusion.
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
${j'\circ \eta _B}$
.
Proposition 3.8 Let
$(A,{\mathcal H})$
be a left bialgebroid over A such that
${}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}$
is A-flat. Let I be a left ideal two-sided coideal in
${\mathcal H}$
. In view of Proposition 3.4, we know that
is a right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t. Then we have an inclusion
${\mathcal H} B^+ \subseteq I$
that we denote by
$\epsilon _I$
.
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
$$ \begin{align} \frac{{\mathcal H}}{{\mathcal H} B^+} \cong {\mathcal H} \otimes_{B} \frac{B}{B^+} \cong {\mathcal H} \otimes_{B} A\cong {\mathcal H} \otimes_{B} {\overline{{\mathcal H}}}^{\textsf{ co}{\overline{{\mathcal H}}}} \xrightarrow{\theta_{\overline{{\mathcal H}}}} \overline{{\mathcal H}} = \frac{{\mathcal H}}{I} \end{align} $$
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
It follows that
${\mathcal H} B^+ \subseteq I$
as claimed.
Theorem 3.9 Let
$(A,{\mathcal H})$
be a left bialgebroid such that
${}_{s} {{{\mathcal H}}}$
is A-flat.
-
(1) If I is a left ideal two-sided coideal in
${\mathcal H}$
and if
, then
$B = {{\mathcal H}}^{\textsf { co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$
. That is,
$\Psi \Phi \Psi (I) = \Psi (I)$
. -
(2) If B is a right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t and if
, then
${I = {\mathcal H} \left ({{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}\right )^+}$
. That is,
$\Phi \Psi \Phi (B) = \Phi (B)$
.
In other words,
$\Phi $
and
$\Psi $
form a monotone Galois connection (or, equivalently, an adjunction) between the two lattices.
Proof Flatness is needed to apply Proposition 3.4. Recall that for any left ideal two-sided coideal I and for any right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t, we have that
${B \subseteq {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}}$
via
$\eta _B$
and
${\mathcal H} \left ({{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}\right )^+ \subseteq I$
via
$\epsilon _I$
, by Propositions 3.7 and 3.8, respectively.
-
(1) Let I be a left ideal two-sided coideal in
${\mathcal H}$
, and let
. Since
${\mathcal H} B^+ \subseteq I$
, we have a surjective morphism of left
${\mathcal H}$
-module corings
${\mathcal H}/{\mathcal H} B^+ \twoheadrightarrow {\mathcal H}/I$
such that commutes and hence
$$ \begin{align*}B \subseteq {{\mathcal H}}^{\textsf{co}{\frac{{\mathcal H}}{{\mathcal H} B^+}}} \subseteq {{\mathcal H}}^{\textsf{co}{\frac{{\mathcal H}}{I}}} = B.\end{align*} $$
-
(2) Let B be a right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t and set
. Since
$B \subseteq {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}} = {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$
, we have that
$$ \begin{align*}I = {\mathcal H} B^+ \subseteq {\mathcal H} \left({{\mathcal H}}^{\textsf{co}{\frac{{\mathcal H}}{I}}}\right)^+ \subseteq I.\\[-37pt] \end{align*} $$
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
$$ \begin{align*}{\mathcal H} B^+ \subseteq I \quad \iff \quad B \subseteq {{\mathcal H}}^{\textsf{co}{\frac{{\mathcal H}}{I}}}.\end{align*} $$
Proof This is simply a restatement of the fact that
$\Phi $
is left adjoint to
$\Psi $
.
3.2 The Hopf algebroid case and the bijective correspondence
In case the bialgebroid
$(A,{\mathcal H})$
is a left Hopf algebroid, one can obtain finer results on the correspondence between left ideal two-sided coideals and right comodule subrings than those in Propositions 3.4 and 3.6, as we are going to show in the present subsection.
Remark 3.11 Let
$(A,{\mathcal H})$
be a left bialgebroid over A. We may consider the tensor product
It becomes an
$A^{\mathrm {o}}$
-bimodule via
$$ \begin{align*} a^{\mathrm{o}}~\overline{\triangleright}~\left(x \otimes_{A^{\mathrm{o}}} y\right) = xs(a) \otimes_{A^{\mathrm{o}}} y \qquad \text{and} \qquad \left(x \otimes_{A^{\mathrm{o}}} y\right)~\overline{\triangleleft}~a^{\mathrm{o}} = x \otimes_{A^{\mathrm{o}}} s(a)y \end{align*} $$
for all
$x,y \in {\mathcal H}$
,
$a \in A$
. Inside
${\mathcal H} \otimes _{A^{\mathrm {o}}} {\mathcal H}$
, we isolate the distinguished subspace
It is an
$A^{\mathrm {o}}$
-subbimodule and a
$\Bbbk $
-algebra with unit
$1 \otimes _{A^{\mathrm {o}}} 1^{\mathrm {o}}$
and multiplication
which acts from the right on
${\mathcal H} \otimes _{A^{\mathrm {o}}} {\mathcal H}$
via
$$ \begin{align} \begin{aligned} \left({\mathcal H} \otimes_{A^{\mathrm{o}}} {\mathcal H}\right)\otimes\left({\mathcal H} \times_{A^{\mathrm{o}}} {\mathcal H}\right) & \to {\mathcal H} \otimes_{A^{\mathrm{o}}} {\mathcal H}, \\ \left(\sum_i x_i \otimes_{A^{\mathrm{o}}} y_i\right)\otimes\left(\sum_j u_j \otimes_{A^{\mathrm{o}}} v_j\right) & \mapsto \sum_{i,j} x_iu_j \otimes_{A^{\mathrm{o}}} v_jy_i. \end{aligned} \end{align} $$
If
${\mathcal H}$
is, in addition, a left Hopf algebroid in the sense of [Reference Schauenburg34, Theorem and Definition 3.5] with canonical map
$$ \begin{align} \beta\colon {{\mathcal H}}{}_{t} \otimes_{A^{\mathrm{o}}} {}_{t} {{\mathcal H}} \to {{\mathcal H}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}, \qquad x \otimes_{A^{\mathrm{o}}} y \mapsto \sum x_1 \otimes_{A} x_2y, \end{align} $$
then the assignment
induces a morphism of
$\Bbbk $
-algebras
$$ \begin{align} \gamma\colon {\mathcal H} \to {\mathcal H} \times_{A^{\mathrm{o}}} {\mathcal H}, \qquad x \mapsto \beta^{-1}(x \otimes_{A} 1_{\mathcal H}). \end{align} $$
The fact that
$\gamma $
lands into
${\mathcal H} \times _{A^{\mathrm {o}}} {\mathcal H}$
is [Reference Schauenburg34, equation (3.3)], and the fact that it is multiplicative and unital is [Reference Schauenburg34, equations (3.4) and (3.5)]. The map
$\gamma $
is referred to as the the (left) translation map. In particular,
$$ \begin{align} \sum (xy)_{+}\otimes_{A} (xy)_{-} = \sum x_{+}y_{+}\otimes_{A} y_{-}x_{-} \end{align} $$
for all
$x,y \in {\mathcal H}$
.▲
Lemma 3.12 Let
$(A,{\mathcal H})$
be a left Hopf algebroid over A such that
${}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}$
is A-flat. Let B be a right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring of
${\mathcal H}$
via t such that for all
$b \in B$
, there exists
$\sum _i b_i \otimes _{A^{\mathrm {o}}} h_i \in B \otimes _{A^{\mathrm {o}}} {\mathcal H}$
such that
$$ \begin{align} \beta^{-1}\left(\iota(b)\otimes_{A}1_{{\mathcal H}}\right) = \sum_i \iota(b_i) \otimes_{A^{\mathrm{o}}} h_i. \end{align} $$
Then the canonical isomorphism
$$ \begin{align*}\beta\colon {{\mathcal H}}{}_{t} \otimes_{A^{\mathrm{o}}} {}_{t} {{\mathcal H}} \to {{\mathcal H}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}, \qquad x \otimes_{A^{\mathrm{o}}} y \mapsto \sum x_1 \otimes_{A} x_2y\end{align*} $$
induces an isomorphism
$$ \begin{align*} \xi\colon {{\mathcal H}_{\iota}} \otimes_{B} {{}_{\iota}{\mathcal H}} \to {\frac{{\mathcal H}}{{\mathcal H} B^+}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}. \end{align*} $$
Proof Consider the composition
$$ \begin{align*}{{\mathcal H}}{}_{t} \otimes_{A^{\mathrm{o}}} {}_{t} {{\mathcal H}} \xrightarrow{\beta} {{\mathcal H}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}} \xrightarrow{\pi \otimes_{A}{\mathcal H}} \frac{{\mathcal H}}{{\mathcal H} B^+} \otimes_{A} {\mathcal H}.\end{align*} $$
It satisfies
$$ \begin{align*} \left(\pi \otimes_{A}{\mathcal H}\right) & ~ \beta\left(x\iota(b) \otimes_{A} y \right) = \sum \pi\left(x_1\iota(b)_1\right) \otimes_{A} x_2\iota(b)_2y \\ & \hspace{2pt}\stackrel{(3.9)}{=} x \cdot \left(\sum \pi\left(\iota(b)_1\right) \otimes_{A} \iota(b)_2y\right) \\ & \stackrel{(3.10)}{=} x \cdot \left(\sum \pi\left(\iota(b_0)\right) \otimes_{A} b_1y\right) \\ & \hspace{2pt}\stackrel{(3.6)}{=} x \cdot \left(\sum \pi\iota\left(t'\left(\varepsilon'(b_0)^{\mathrm{o}}\right)\right) \otimes_{A} b_1y\right) \\ & \stackrel{\phantom{(3.10)}}{=} x \cdot \left(\sum \pi(1_{{\mathcal H}}) \otimes_{A} \bigg(s\Big(\varepsilon\big(\iota(b_0)\big)\Big)\bigg)b_1y\right) \\ & \stackrel{(3.10)}{=} x \cdot \left(\sum \pi(1_{{\mathcal H}}) \otimes_{A} \bigg(s\Big(\varepsilon\big(\iota(b)_1\big)\Big)\bigg)\iota(b)_2y\right) \\ & \hspace{2pt}\stackrel{(3.9)}{=} \sum \pi(x_1) \otimes_{A} x_2\iota(b)y = \left(\pi \otimes_{A}{\mathcal H}\right)\beta\left(x \otimes_{A} \iota(b)y \right) \end{align*} $$
for all
$x,y \in {\mathcal H}$
,
$b \in B$
; thus, it factors through
$$ \begin{align*}\xi\colon {{\mathcal H}_{\iota}} \otimes_{B} {{}_{\iota}{\mathcal H}} \to {\frac{{\mathcal H}}{{\mathcal H} B^+}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}}.\end{align*} $$
In the other direction, consider the composition
$$ \begin{align} {{\mathcal H}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}} \xrightarrow{\beta^{-1}} {{\mathcal H}}{}_{t} \otimes_{A^{\mathrm{o}}} {}_{t} {{\mathcal H}} \xrightarrow{p} {{\mathcal H}_{\iota}} \otimes_{B} {{}_{\iota}{\mathcal H}}, \end{align} $$
where p is the canonical projection. Since
${}_{s} {{{\mathcal H}}}$
is A-flat,
${\mathcal H} B^+ \otimes _{A} {\mathcal H} \subseteq {\mathcal H} \otimes _{A} {\mathcal H}$
, and, in fact,
${\mathcal H} B^+ \otimes _{A} {\mathcal H} = \ker \left (\pi \otimes _{A} {\mathcal H}\right )$
. For all
$\sum _j x_j\iota (b_j) \otimes _{A} y_j \in {\mathcal H} B^+ \otimes _{A} {\mathcal H}$
, we have
$$ \begin{align*} \beta^{-1} ~ & \left(\sum_j x_j\iota(b_j) \otimes_{A} y_j\right) = \sum_j \left(x_j\iota(b_j)\right)_{+} \otimes_{A^{\mathrm{o}}} \left(x_j\iota(b_j)\right)_{-}y_j \\ & \stackrel{(3.18)}{=} \sum_j \left(x_j\right)_{+}\left(\iota(b_j)\right)_{+} \otimes_{A^{\mathrm{o}}} \left(\iota(b_j)\right)_{-}\left(x_j\right)_{-}y_j \\ & \stackrel{(3.15)}{=} \sum_j\left(1_{\mathcal H} \otimes_{A^{\mathrm{o}}} y_j\right)\left(\sum\left(x_j\right)_{+}\left(\iota(b_j)\right)_{+} \otimes_{A^{\mathrm{o}}} \left(\iota(b_j)\right)_{-}\left(x_j\right)_{-}\right) \\ & \stackrel{(3.14)}{=} \sum_j\left(1_{\mathcal H} \otimes_{A^{\mathrm{o}}} y_j\right)\left(\sum\left(x_j\right)_{+} \otimes_{A^{\mathrm{o}}} \left(x_j\right)_{-}\right)\left(\sum\left(\iota(b_j)\right)_{+} \otimes_{A^{\mathrm{o}}} \left(\iota(b_j)\right)_{-}\right) \\ & \stackrel{(3.19)}{=} \sum_j\left(1_{\mathcal H} \otimes_{A^{\mathrm{o}}} y_j\right)\left(\sum\left(x_j\right)_{+} \otimes_{A^{\mathrm{o}}} \left(x_j\right)_{-}\right)\left(\sum_i \iota\left(b_{j,i}\right) \otimes_{A^{\mathrm{o}}} h_{j,i}\right) \\ & \stackrel{\phantom{(3.19)}}{=} \sum_{i,j} \left(x_j\right)_{+}\iota\left(b_{j,i}\right) \otimes_{A^{\mathrm{o}}} h_{j,i}\left(x_j\right)_{-}y_j. \end{align*} $$
Therefore,
$$ \begin{align*} p\beta^{-1}\left(\sum_j x_j\iota(b_j) \otimes_{A} y_j\right) & = \sum_{i,j} \left(x_j\right)_{+}\iota\left(b_{j,i}\right) \otimes_{B} h_{j,i}\left(x_j\right)_{-}y_j \\ & = \sum_{j} \left(x_j\right)_{+} \otimes_{B} \left(\sum_i\iota\left(b_{j,i}\right)h_{j,i}\right)\left(x_j\right)_{-}y_j, \end{align*} $$
and since
$$ \begin{align*}\sum_i\iota\left(b_{j,i}\right)h_{j,i} \stackrel{(3.19)}{=} \sum \iota(b_j)_{+}\iota(b_j)_{-} \stackrel{[34, (3.9)]}{=} s\varepsilon\iota\left(b_j\right) = 0, \end{align*} $$
for all j, we conclude that (3.20) induces
$$ \begin{align*}\widetilde{\xi}\colon {\frac{{\mathcal H}}{{\mathcal H} B^+}}{}_{t^{\mathrm{o}}} \otimes_{A} {}_{s} {{{\mathcal H}}} \to {{\mathcal H}_{\iota}} \otimes_{B} {{}_{\iota}{\mathcal H}},\end{align*} $$
which is inverse to
$\xi $
.
In view of Lemma 3.12, one obtains the following improvement to Proposition 3.7.
Theorem 3.13 Let
$(A,{\mathcal H})$
be a left Hopf algebroid over A such that
${}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}$
is A-flat. Let B be a right
${\mathcal H}$
-comodule
$A^{\mathrm {o}}$
-subring via t of
${\mathcal H}$
such that
${\mathcal H}$
is pure over B on the right and such that
$\gamma (B)\subseteq B \otimes _{A^{\mathrm {o}}}{\mathcal H}$
. Then
$B = {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$
, that is,
$\Psi \Phi (B) = B$
.
Proof On the one hand, purity of
${\mathcal H}$
on B entails that
is an equalizer diagram in
$\textsf {Vect}_{\Bbbk }$
(see Corollary 2.2) and, on the other hand, it entails that
$\iota \otimes _{A^{\mathrm {o}}} {\mathcal H}$
is an injective morphism (see Corollary 2.4), so that the condition
$\gamma (B)\subseteq B \otimes _{A^{\mathrm {o}}}{\mathcal H}$
makes sense. By definition,
${{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$
can be realized as the following equalizer in
$\textsf {Vect}_{\Bbbk }$
:
(see (3.5)). Now, commutativity of the following diagram
together with bijectivity of
$\xi $
(Lemma 3.12) entails that
$\eta _B$
is bijective and hence
$B = {{\mathcal H}}^{\textsf {co}{\frac {{\mathcal H}}{{\mathcal H} B^+}}}$
.
At the same time, one may obtain the following improvement to Proposition 3.8 by taking advantage of Theorem 2.11.
Theorem 3.14 Let
$(A,{\mathcal H})$
be a left Hopf algebroid over A such that
${}_{s} {{{\mathcal H}}} = {}_{{A\otimes _{}1^{\mathrm {o}}}}{\mathcal H}$
is A-flat. Let
$I\subseteq {\mathcal H}$
be a left ideal two-sided coideal such that the extension-of-scalars functor
${\mathcal H} \otimes _{B}-$
is comonadic, where
. Then
$I = {\mathcal H} B^+$
, that is to say,
$\Phi \Psi (I)=I$
.
Proof First of all, observe that the comonadicity of
${\mathcal H} \otimes _{B}-$
entails that
$\iota \colon B \to {\mathcal H}$
is pure as a morphism of right
$A^{\mathrm {o}}$
-modules, in view of [Reference Janelidze and Tholen23, Section 5.3, Theorem] and Corollary 2.4. Thus, let us start by proving that
$\gamma (B)\subseteq B \otimes _{A^{\mathrm {o}}}{\mathcal H}$
(which now makes sense, because
$\iota \otimes _{A^{\mathrm {o}}}{\mathcal H}$
is injective by purity). In a nutshell, the following diagram whose rows are equalizers commutes sequentially:
In more detail, we want show that for any
$b\in B$
,
$\gamma (b)= \sum b_{+}\otimes _{A^{\mathrm {o}}} b_{-}\in B \otimes _{A^{\mathrm {o}}} {\mathcal H}$
. Since
$\iota \colon B \to {\mathcal H}$
is pure, it follows that
$ B \otimes _{A^{\mathrm {o}}}{\mathcal H} = {({\mathcal H} \otimes _{A^{\mathrm {o}}} {\mathcal H})}^{\textsf { co}{\frac {{\mathcal H}}{I}}}$
where
${\mathcal H} \otimes _{A^{\mathrm {o}}} {\mathcal H}$
is considered as a left
${\mathcal H}/I$
-comodule via
$(\pi \otimes _{A}\operatorname {Id}_{{\mathcal H}})\Delta \otimes _{A^{\mathrm {o}}}\operatorname {Id}_{{\mathcal H}}$
. On the other hand, we have that
$\sum b_{1}\otimes _{A} b_{2} \in 1 \otimes _{A} B + {}_{t} {I} \otimes _{A} {\mathcal H}$
. By equation (3.6) of [Reference Schauenburg34], we have that
$$ \begin{align*} \sum b_{+1}\otimes_{A} b_{+2}\otimes_{A^{\mathrm{o}}} b_{-} = \sum b_{1}\otimes_{A} b_{2+} \otimes_{A^{\mathrm{o}}} b_{2-}\in 1 \otimes_{A} \sum b_{+}\otimes_{A^{\mathrm{o}}} b_{-} + {}_{t} {I}\otimes_{A}{\mathcal H} \otimes_{A^{\mathrm{o}}} {\mathcal H}. \end{align*} $$
Hence,
$\gamma (b)\in {({\mathcal H} \otimes _{A^{\mathrm {o}}} {\mathcal H})}^{\textsf {co}{\frac {{\mathcal H}}{I}}}$
.
Now, the additional condition on
$\gamma (B)$
we proved above ensures that
$\xi $
of (2.14) is an isomorphism in view of Lemma 3.12. In particular,
${\mathcal K}$
of Theorem 2.11 is an equivalence of categories. Therefore, the morphism
$\theta _{\overline {{\mathcal H}}}$
in (3.12) from the proof of Proposition 3.8 is an isomorphism, and hence the vertical arrows in (3.13) are all isomorphisms. It follows that
$I = {\mathcal H} B^+$
as claimed.
Summing up, we have the first main result of this work. The functor
${\mathcal H} \otimes _{B} -$
in the statement below denotes the extension of scalars functor