Partial actions of groups as powerful tools have been introduced during the study of operator algebras by Exel [2]. With the further development many positive results have been proposed [3-6]. S.Caenepeel and the other authors developed a theory of partial actions of Hopf algebras [1] and introduced the notion of a partial entwining structure as a generalization of entwining structure (see [9]).

On other hand, the notion of a Hopf π-coalgebra which generalized that of a Hopf algebra was introduced and played an important role, consequently group entwining structures and group weak entwining structure were carefully studied. Motivated by this fact, we introduce the notion of a partial group comodule and give a Maschke type theorem for them. Because the "coassociativity" of a partial structure is destroyed, the generalization is not trivial and easy.

In this paper, we first recall basic definitions of partial group comodules and give some examples. Then we state a Maschke-type theorem of partial group Hopf modules which generalizes the relevant results of Hopf modules (see [7, 8]), entwined modules, group Hopf modules, etc..

The organization of the paper is as follows: First we introduce the notion of partial group comodules and then give our main result-Maschke type theorem.

Conventions   We work over a commutative ring $k$. We denote by $i$ the unit of the group $\pi$ and use the standard (co)algebra notation, i.e., $\Delta$ is a coproduct, $\varepsilon$ is a counit, $m$ is a product and $1$ is a unit. If $1$ appears more than once in the same expression, then we use different $1^{'}$. The identity map from any $k$-space $V$ to itself is denoted by ${id}_{V}$. Write $a_{\alpha}$ for any element in $A_{\alpha}$ and $[a]$ for an element in $\overline{A}=A/kerf$, where $f$ is a $k$-linear map. For a right $\pi$-$C$-comodule $M$, we write ${\rho}_{\alpha, \beta}(m)=\sum m_{([0], \alpha)}\otimes m_{([1], \beta)}$ for any $\alpha, \beta\in\pi$ and $m\in M_{\alpha\beta}$.

Deflnition 2.1  A $\pi$-coalgebra over $k$ is a family of $C={\{C_{\alpha}\}}_{\alpha\in \pi}$ of $k$-spaces endowed with a family $k$-linear maps $\Delta={\{{\Delta}_{\alpha, \beta}:C_{\alpha\beta}\rightarrow C_{\alpha}\otimes C_{\beta}\}}_{\alpha, \beta\in \pi}$ and a $k$-linear map $\varepsilon:C_{i}\rightarrow k$ such that for any $\alpha, \beta, \gamma\in \pi$,

(1) $({\Delta}_{\alpha, \beta}\otimes {id}_{{C}_{\gamma}}){\Delta}_{\alpha\beta, \gamma}=({id}_{{C}_{\alpha}}\otimes{\Delta}_{\beta, \gamma}){\Delta}_{\alpha, \beta\gamma}.$

(2) $({id}_{{C}_{\alpha}}\otimes\varepsilon){\Delta}_{\alpha, i}=(\varepsilon\otimes {id}_{{C}_{\alpha}}){\Delta}_{i, \alpha}={id}_ {C_{\alpha}}.$

Here we extend the Sweedler notation for comultiplication, we write

$ {\Delta _{\alpha, \beta }}({c_{\alpha \beta }}) = \sum {{c_{\alpha \beta 1\alpha }}} \otimes {c_{\alpha \beta 2\beta }}, \alpha, \beta \in \pi, c \in {C_{\alpha \beta }}. $

Remark  $(C_{i}, {\Delta}_{i, i}, \varepsilon)$ is a coalgebra in the usual sense.

Deflnition 2.2  A Hopf $\pi$-coalgebra(Hopf group coalgebra) is a family of algebras $H={\{H_{\alpha}\}}_{\alpha\in\pi}$ and also a $\pi$-coalgebra ${\{H_{\alpha}, \Delta=\{{\Delta}_{\alpha, \beta}\}, \varepsilon\}}_{\alpha, \beta\in\pi}$ endowed with a family $S={\{S_{{\alpha}^{-1}}:H_{\alpha}\rightarrow H_{{\alpha}^{-1}}\}}_{\alpha\in\pi}$ of $k$-linear maps called an antipode such that for any $\alpha\in \pi.$

(3) $\sum {{S_{{\alpha ^{ - 1}}}}} ({h_{1\alpha }}){h_{2{\alpha ^{ - 1}}}} = \varepsilon (h){1_{{\alpha ^{ - 1}}}},\sum {{h_{1\alpha }}} {S_\alpha }({h_{2{\alpha ^{ - 1}}}}) = \varepsilon (h){1_\alpha }.$

Deflnition 2.3  Let $H$ be a Hopf group coalgebra and ${A=\{A_{\alpha}\}}_{\alpha\in\pi}$ be a family of algebras endowed with a family of $k$-linear maps ${\{{\rho}_{\alpha, \beta}:A_{\alpha\beta}\rightarrow A_{\alpha}\otimes A_{\beta}\}}_{\alpha, \beta\in\pi}$. $A$ is called a right partial group comodule-algebra if the following conditions are satisfied:

(4) ${\rho _{\alpha ,\beta }}(ab) = {\rho _{\alpha ,\beta }}(a){\rho _{\alpha ,\beta }}(b),a,b \in {A_{\alpha \beta }}.$

(5) $({\rho _{\alpha ,\beta }} \otimes i{d_{{H_\gamma }}}){\rho _{\alpha \beta ,\gamma }}(c) = \sum {{c_{([0],\alpha )}}} {1_{([0],\alpha )}} \otimes {c_{([1],\beta \gamma )1\beta }}{1_{([1],\beta )}} \otimes {c_{([1],\beta \gamma )2\gamma }},c \in {A_{\alpha \beta \gamma }}.$

(6) $\sum \varepsilon ({d_{([1],i)}}){d_{([0],\alpha )}} = d,\;\;d \in {A_\alpha }.$

Example 1  Let $H$ ba a Hopf group coalgebra and $e={\{e_{\alpha}\}}_{\alpha\in\pi}$ be a central idempotent such that ${\Delta}_{\alpha, \beta}(e_{\alpha\beta})(e_{\alpha}\otimes 1_{\beta})=e_{\alpha}\otimes e_{\beta}$ and $\varepsilon(e_{i})=1$, then $H$ is a right partial group comodule-algebra.

Deflnition 2.4  Let $H$ ba a Hopf group coalgebra and $A$ be a right partial group comodule-algebra. An $A$-module $M={\{M_{\alpha}\}}_{\alpha\in\pi}$ with a family of $k$-linear maps

$ {\{ {\rho _{\alpha, \beta }}:{M_{\alpha \beta }} \to {M_\alpha } \otimes {H_\beta }\} _{\alpha, \beta \in \pi }} $

is called a partial $(H, A)$-Hopf module if the following conditions are verified for any $m \in {M_\alpha },m' \in {M_{\alpha \beta \gamma }},m'' \in {M_{\alpha \beta }},a \in {A_{\alpha \beta }}$:

(7) $\sum\varepsilon(m_{([1], i)})m_{([0], \alpha)}=m.$

(8) $({\rho}_{\alpha, \beta}\otimes{id}_{H_{\gamma}}){\rho}_{\alpha\beta, \gamma}(m^{'})=\sum m^{'}_{([0], \alpha)}\cdot 1_{([0], \alpha)}\otimes m^{'}_{([1], \beta\gamma)1\beta}1_{([1], \beta)}\otimes m^{'}_{([1], \beta\gamma)2\gamma}.$

(9) ${\rho}_{\alpha, \beta}(m^{''}\cdot a)=\sum m^{''}_{([0], \alpha)}\cdot a_{([0], \alpha)}\otimes m^{''}_{([1], \beta)}a_{([1], \beta)}.$

We define the coinvariants of $M$ as

$ {M^{COH}} = \{ m = {\{ {m_\alpha }\} _{\alpha \in \pi }}\mid {\rho _{\alpha, \beta }}({m_{\alpha \beta }}) = \sum {{m_\alpha }} \cdot {1_{([0], \alpha )}} \otimes {1_{([1], \beta )}}\} $

and denote $M_{A}^{\pi-H}$ the category of partial $(H, A)$-Hopf modules.

Example 2  Let $H$ ba a Hopf group coalgebra and $A$ be a right partial group comodule-algebra. It is easy to prove that $A$ is a partial $(H, A)$-Hopf module with the multiplications as $A$-actions.

Deflnition 2.5  Let $H$ ba a Hopf group coalgebra and $A$ be a right partial group comodule-algebra. A right partial $\pi-H$-comodule map $\theta={\{{\theta}_{\alpha}:{H}_{\alpha}\rightarrow {A}_{\alpha}\}}_{\alpha\in\pi}$ such that $\sum 1_{([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(1_{([1], {\alpha}^{-1})}))=1_{\alpha}$ is called a right total integral of $A$.

Deflnition 2.6  Let $M\in M_{A}^{\pi-H}$ and $\text{tr}={\{\text{tr}_{\alpha}:M_{i}\rightarrow M_{\alpha}\}}_{\alpha\in\pi}$ be a family of $k$-linear maps such that $\text{tr}_{\alpha}(m)=\sum m_{([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})}))$. Then $\text{tr}$ is called a trace map of $M$.

We define $\widehat {\text{tr}}:\prod {{M_\alpha }} \to \prod {{M_\alpha }} ,\widehat {\text{tr}}({({m_\alpha })_{\alpha \in \pi }}) = {({\text{t}}{{\text{r}}_\alpha }\}({m_i}))_{\alpha \in \pi }}$.

In the following parts we always suppose that

$ \sum {{1_{([0], \alpha )}}} a \otimes {1_{([1], \beta )}}b = \sum a {1_{([0], \alpha )}} \otimes b{1_{([1], \beta )}} $

for any $a\in A_{\alpha}$ and $b\in H_{\beta}$.

Proposition 2.7  $m={(m_{\alpha})}_{\alpha\in\pi}$ is a coinvariant of $M$ if and only if $\text{tr}(m)=m$.

Proof  If $m={(m_{\alpha})}_{\alpha\in\pi}\in M^{COH}$, then

$ \widehat {\text{tr}}(m) = \sum {{m_{([0], \alpha )}}} \cdot {\theta _\alpha }({S_\alpha }({m_{([1], {\alpha ^{ - 1}})}})) = \sum {{m_\alpha }} \cdot {1_{([0], \alpha )}}{\theta _\alpha }({S_\alpha }({1_{([1], {\alpha ^{ -1}})}})) = {m_\alpha }. $

Conversely, if $\hat{\text{tr}}(m)=m$, we have

$ \begin{array}{l} \;\;\;\;\;{\rho _{\alpha, \beta }}({\text{t}}{{\text{r}}_{\alpha \beta }}(m))\\ = \sum {m_{([0], \alpha \beta )([0], \alpha )}} \cdot {\theta _\alpha }({S_\alpha }({m_{([1], {{(\alpha \beta )}^{ - 1}})2{\alpha ^{ - 1}}}})) \otimes {m_{([0], \alpha \beta )([1], \beta )}}{S_\beta }({m_{([1], {{(\alpha \beta )}^{ - 1}})1{\beta ^{ - 1}}}})\\ = \sum {{m_{([0], \alpha )}}} \cdot {1_{([0], \alpha )}}{\theta _\alpha }({S_\alpha }({m_{([1], {\alpha ^{ - 1}})2{\beta ^{ - 1}}{\alpha ^{ - 1}}2{\alpha ^{ - 1}}}}))\\ \;\;\;\;\; \otimes {m_{([1], {\alpha ^{ - 1}})1\beta }}{1_{([1], \beta )}}{S_\beta }({m_{([1], {\alpha ^{ - 1}})2{\beta ^{ - 1}}{\alpha ^{ - 1}}1{\beta ^{ - 1}}}})\\ = \sum {{m_{([0], \alpha )}}} \cdot {1_{([0], \alpha )}}{\theta _\alpha }({S_\alpha }({m_{([1], {\alpha ^{ - 1}})}})) \otimes {1_{([1], \beta )}} = \sum {{m_\alpha }} \cdot {1_{([0], \alpha )}} \otimes {1_{([1], \beta )}}. \end{array} $

Given $M\in M_{A}^{\pi-H}$, we define

$ {\omega _\alpha }:{M_i} \otimes {H_\alpha } \to {M_i} \otimes {H_\alpha }, {\omega _\alpha }(m \otimes h) = \sum m \cdot {1_{([0], i)}} \otimes h{1_{([1], \alpha )}}, m \in {M_i}, h \in {H_\alpha }. $

It is trivial to prove that ${\omega}_{\alpha}^{2}={\omega}_{\alpha}$, so we can define

$ \overline {{M_i} \otimes {H_\alpha }} = ({M_i} \otimes {H_\alpha })/ker{\omega _\alpha } $

as a $k$-space.

We claim that $\overline{M_{i}\otimes H}={\{\overline{M_{i}\otimes H_{\alpha}}\}}_{\alpha\in\pi}\in M_{A}^{\pi-H}$ with the actions given by

$ {\rm{[}}m \otimes {h_a}] \cdot {a_\alpha } = \sum {{\rm{[}}m \cdot {a_{(\left[ 0 \right],i)}} \otimes {h_\alpha }{a_{(\left[ 1 \right],\alpha )}}]} $

and the partial coactions given by

$ {\rho _{\alpha ,\beta }}({\rm{[}}m \otimes {h_{a\beta }}]){\rm{ = }}\sum {{\rm{[}}m \cdot {1_{(\left[ 0 \right],i)}} \otimes {h_{a\beta 1\alpha }}{1_{(\left[ 1 \right],\alpha \beta )1\alpha }}]} \otimes {h_{a\beta 2\beta }}{1_{(\left[ 1 \right],\alpha \beta )2\beta .}} $

It is easy to prove that the actions are well-defined. We only have to show the coactions are well-defined. In fact, we have

$ \begin{eqnarray*}&&{\rho}_{\alpha, \beta}(m\otimes h_{\alpha\beta}-m\cdot 1_{([0], i)}\otimes h_{\alpha\beta}a_{([1], \alpha\beta)})\\&=&\sum m\cdot 1_{([0], i)}\otimes h_{\alpha\beta1\alpha}1_{([1], \alpha\beta)1\alpha}\otimes h_{\alpha\beta2\beta}1_{([1], \alpha\beta)2\beta}\\&&-m\cdot 1_{([0], i)}1_{([0], i)}^{'}\otimes h_{\alpha\beta1\alpha}1_{([1], \alpha\beta)1\alpha}1_{([1], \alpha\beta)1\alpha}^{'}\otimes h_{\alpha\beta2\beta}1_{([1], \alpha\beta)2\beta}1_{([1], \alpha\beta)2\beta}^{'}=0.\\&&\sum {[m\otimes h_{\alpha\beta\gamma}]}_{([0], \alpha)}\cdot 1_{([0], \alpha)}\otimes {[m\otimes h_{\alpha\beta\gamma}]}_{([1], \beta\gamma)1\beta}1_{([1], \beta)}\otimes {[m\otimes h_{\alpha\beta\gamma}]}_{([1], \beta\gamma)2\gamma} \\ &=&\sum [m\cdot 1_{([0], i)}^{'}1_{([0], i)}1_{([0], i)}^{''}\otimes h_{\alpha\beta\gamma1\alpha}1_{([1], \alpha\beta\gamma)1\alpha}^{'}1_{([1], \alpha\beta)1\alpha}1_{([1], \alpha)1\alpha}^{''}]\\ &&\otimes h_{\alpha\beta\gamma2\beta\gamma1\beta}1_{([1], \alpha\beta\gamma)2\beta\gamma1\beta}^{'}1_{([1], \alpha\beta)2\beta}\otimes h_{\alpha\beta\gamma2\beta\gamma2\gamma}1_{([1], \alpha\beta\gamma)2\beta\gamma2\gamma}^{'} \\&=&\sum [m\cdot 1_{([0], i)}^{'}1_{([0], i)}\otimes h_{\alpha\beta\gamma1\alpha\beta1\alpha}1_{([1], \alpha\beta\gamma)1\alpha\beta1\alpha}^{'}1_{([1], \alpha\beta)1\alpha} \\ &&\otimes h_{\alpha\beta\gamma1\alpha\beta2\beta}1_{([1], \alpha\beta\gamma)1\alpha\beta2\beta}^{'}1_{([1], \alpha\beta)2\beta}\otimes h_{\alpha\beta\gamma2\gamma}1_{([1], \alpha\beta\gamma)2\gamma}^{'} \\&=&({\rho}_{\alpha, \beta}\otimes {id}_{H_{\gamma}}){\rho}_{\alpha\beta, \gamma}([m\otimes h_{\alpha\beta\gamma}]). \\ &&\sum\varepsilon({[m\otimes h_{\alpha}]}_{([1], i)}){[m\otimes h_{\alpha}]}_{([0], \alpha)} =\sum\varepsilon(h_{\alpha2i} 1_{([1], \alpha)2i})[m\cdot 1_{([0], i)}\otimes h_{\alpha1\alpha} 1_{([1], \alpha)1\alpha})]\\&=&[m\otimes h_{\alpha}]. \\&&{\rho}_{\alpha, \beta}([m\otimes h_{\alpha\beta}]\cdot a_{\alpha\beta})=\sum{\rho}_{\alpha, \beta}([m\cdot a_{\alpha\beta([0], i)}\otimes h_{\alpha\beta}a_{\alpha\beta([1], \alpha\beta)}])\\& =&\sum [m\cdot a_{\alpha\beta([0], i)}1_{([0], i}\otimes h_{\alpha\beta1\alpha}a_{\alpha\beta([1], \alpha\beta)1\alpha}1_{([1], \alpha\beta)1\alpha}]\otimes h_{\alpha\beta2\beta}a_{\alpha\beta([1], \alpha\beta)2\beta}1_{([1], \alpha\beta)2\beta} \\&=&\sum{[m\otimes h_{\alpha\beta}]}_{([0], \alpha)}\cdot a_{\alpha\beta([0], \alpha)}\otimes {[m\otimes h_{\alpha\beta}]}_{([1], \beta)}a_{\alpha\beta([1], \beta)}. \end{eqnarray*} $

We define $\xi={\{{\xi}_{\alpha}:\overline{M_{i}\otimes H_{\alpha}}\rightarrow M_{\alpha}\}}_{\alpha\in\pi}$, where

$ {\xi _\alpha }([m \otimes h]) = \sum {{m_{([0],\alpha )}}} {\theta _\alpha }({S_\alpha }({m_{([1],{\alpha ^{ - 1}})}})h),m \in {M_i},h \in {H_\alpha }. $

Then we claim that ${\xi}_{\alpha}$ is well-defined for any $\alpha\in\pi$. Indeed, for any $m \in {M_i},h \in {H_\alpha }, $

$ \begin{array}{l} \;\;{\xi _\alpha }(m \otimes h - m \cdot {1_{([0],i)}} \otimes h{1_{([1],\alpha )}})\\ {\rm{ = }}\sum {{m_{([0],\alpha )}}} \cdot {\theta _\alpha }({S_\alpha }({m_{([1],{\alpha ^{ - 1}})}})h) - {(m \cdot {1_{([0],i)}})_{([0],\alpha )}}{\theta _\alpha }({S_\alpha }({(m \cdot {1_{([0],i)}})_{([1],{\alpha ^{ - 1}})}})h{1_{([1],\alpha )}})\\ {\rm{ = }}\sum {{m_{([0],\alpha )}}} \cdot {\theta _\alpha }({S_\alpha }({m_{([1],{\alpha ^{ - 1}})}})h) - {m_{([0],\alpha )}} \cdot {1_{([0],i)([0],\alpha )}}{\theta _\alpha }({S_\alpha }({m_{([1],{\alpha ^{ - 1}})}}{1_{([0],i)([1],{\alpha ^{ - 1}})}})h{1_{([1],\alpha )}})\\ {\rm{ = }}\sum {{m_{([0],\alpha )}}} \cdot {\theta _\alpha }({S_\alpha }({m_{([1],{\alpha ^{ - 1}})}})h) - {m_{([0],\alpha )}} \cdot {1_{([0],\alpha )}}{\theta _\alpha }({S_\alpha }({m_{([1],{\alpha ^{ - 1}})}}{1_{([0],i)1{\alpha ^{ - 1}}}})h{1_{([1],i)2\alpha }}) = 0. \end{array} $

Lemma 2.8  For any $\alpha \in \pi ,{\xi _\alpha }^\circ {p_\alpha }^\circ \rho _{i,\alpha }^M = i{d_{{M_\alpha }}}$, where $p_{\alpha}:M_{i}\otimes H_{\alpha}\rightarrow\overline{M_{i}\otimes H_{\alpha}}$ is a canonical map.

Proof  In fact, for any $m\in M_{\alpha}$ we have

$ \begin{eqnarray*}&&{\xi}_{\alpha}\circ p_{\alpha}\circ {\rho}^{M}_{i, \alpha}(m)=\sum m_{([0], i)([0], \alpha)}\cdot{\theta}_{\alpha}(S_{\alpha}(m_{([0], i)([1], {\alpha}^{-1})})m_{([1], \alpha)}) \\&=&\sum m_{([0], \alpha)}\cdot1_{([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(m_{([0], i)1{\alpha}^{-1}} 1_{([1], {\alpha}^{-1})})m_{([1], i)2\alpha})=m. \end{eqnarray*} $

Lemma 2.9  If for any $\alpha \in \pi ,h \in {H_\alpha },a \in {A_\alpha }, $

$ \sum {{a_{([0], \alpha )}}} {\theta _\alpha }({S_\alpha }({a_{([0], i)1{\alpha ^{ - 1}}}})h{a_{([1], i)2\alpha }}) = {\theta _\alpha }(h)a, $

then ${\xi}_{\alpha}$ is a right $A_{\alpha}$-linear map.

Proof  Indeed for any $\alpha \in \pi ,h \in {H_\alpha },a \in {A_\alpha }$ we have

$ \begin{eqnarray*}&&{\xi}_{\alpha}([m\otimes h]\cdot a)= {(m\cdot a_{([0], i)})}_{([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}({(m\cdot a_{([0], i)})}_{([1], {\alpha}^{-1})})h a_{([1], \alpha)}) \\&=&m_{([0], \alpha)}\cdot a_{([0], i)([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} a_{([0], i)([1], {\alpha}^{-1})})h a_{([1], \alpha)}) \\&=&m_{([0], \alpha)}\cdot a_{([0], \alpha)}1_{([0], \alpha)}^{'}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} a_{([0], i)1{\alpha}^{-1}}1_{([1], {\alpha}^{-1})}^{'})h a_{([1], i)2\alpha}) \\&=&m_{([0], \alpha)}\cdot a_{([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} a_{([0], i)1{\alpha}^{-1}})h a_{([1], i)2\alpha}) \\&=&m_{([0], \alpha)}\cdot {\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} )h)a={\xi}_{\alpha}([m\otimes h])\cdot a. \end{eqnarray*} $

Lemma 2.10  Let $H$ ba a Hopf group coalgebra and $A$ be a right partial group comodule-algebra. If the condition in Lemma 2.9 is satisfied, then $\xi=\{{\xi}_{\alpha}\}\in M_{A}^{\pi-H}$.

Proof  It is sufficed to prove that $\xi$ is right partial $\pi-H$-colinear. In fact, for any $\alpha ,\beta \in \pi ,h \in {H_{\alpha \beta }},m \in {M_i}, $ on one hand,

$ \begin{eqnarray*}&&{\rho}_{\alpha, \beta}^{M}{\xi}_{\alpha\beta}([m\otimes h]) \\&=&\sum m_{([0], \alpha\beta)([0], \alpha)}\cdot {{\theta}_{\alpha\beta}(S_{\alpha\beta}(m_{([1], {(\alpha\beta)}^{-1})} )h)}_{([0], \alpha)}\\ && \otimes m_{([0], \alpha\beta)([1], \beta)} {{\theta}_{\alpha\beta}(S_{\alpha\beta}(m_{([1], {(\alpha\beta)}^{-1})} )h)}_{([1], \beta)} \\&=&\sum m_{([0], \alpha\beta)([0], \alpha)}\cdot {\theta}_{\alpha}(S_{\alpha}(m_{([1], {(\alpha\beta)}^{-1})2{\alpha}^{-1}} )h_{1\alpha}) \otimes m_{([0], \alpha\beta)([1], \beta)} S_{\beta}(m_{([1], {(\alpha\beta)}^{-1})1{\beta}^{-1}} )h_{2\beta} \\&=&\sum m_{([0], \alpha)}\cdot 1_{([0], \alpha)} {\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})2{(\alpha\beta)}^{-1}2{\alpha}^{-1}} )h_{1\alpha}) \otimes m_{([1], {\alpha}^{-1})1\beta}1_{([1], \beta)} \\&&S_{\beta}(m_{([1], {\alpha}^{-1})2{(\alpha\beta)}^{-1}1{\beta}^{-1}} )h_{2\beta} =\sum m_{([0], \alpha)}\cdot 1_{([0], \alpha)} {\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} )h_{1\alpha}) \otimes1_{([1], \beta)}h_{2\beta}. \end{eqnarray*} $

On the other hand,

$ \begin{eqnarray*}&&({\xi}_{\alpha}\otimes{id}_{H_{\beta}}){\rho}_{\alpha, \beta}^{\overline{M_{i}\otimes H_{\alpha\beta}}}([m\otimes h])\\&=&\sum {(m\cdot 1_{([0], i)})}_{([0], \alpha)}\cdot {\theta}_{\alpha}(S_{\alpha}({(m\cdot 1_{([0], i)})}_{([1], {\alpha}^{-1})})h_{1\alpha}1_{([1], \alpha\beta)1\alpha}) \otimes h_{2\beta}1_{([1], \alpha\beta)2\beta} \\&=&\sum m_{([0], \alpha)}\cdot 1_{([0], i)([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} 1_{([0], i)([1], {\alpha}^{-1})})h_{1\alpha}1_{([1], \alpha\beta)1\alpha}) \otimes h_{2\beta}1_{([1], \alpha\beta)2\beta} \\&=&\sum m_{([0], \alpha)}\cdot 1_{([0], \alpha)}1_{([0], \alpha)}^{'}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} 1_{([1], \beta)1{\alpha}^{-1}}1_{([1], {\alpha}^{-1})}^{'})h_{1\alpha}1_{([1], \beta)2\alpha\beta1\alpha})\\ &&\otimes h_{2\beta}1_{([1], \beta)2\alpha\beta2\beta} \\&=&\sum m_{([0], \alpha)}\cdot 1_{([0], \alpha)([0], \alpha)}{\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} 1_{([1], i)1{\alpha}^{-1}})h_{1\alpha}1_{([1], i)2\alpha}) \otimes h_{2\beta}1_{([1], \beta)} \\&=&\sum m_{([0], \alpha)}\cdot 1_{([0], \alpha)} {\theta}_{\alpha}(S_{\alpha}(m_{([1], {\alpha}^{-1})} )h_{1\alpha}) \otimes1_{([1], \beta)}h_{2\beta}. \end{eqnarray*} $

Therefore we complete the proof. Now we can give our main result.

Theorem 2.11  Let $H$ ba a Hopf group coalgebra and $A$ be a right partial group comodule-algebra with a total integral $\theta$, and $M, N\in M_{A}^{\pi-H}$. Supposing the condition in Lemma 2.9 is satisfied, if $f_{i}:M_{i}\rightarrow N_{i}$ splits as $A_{i}$-module map, then $f={\{f_{\alpha}:M_{\alpha}\rightarrow N_{\alpha}\}}_{\alpha\in\pi}$ splits as partial $\pi-H$-comodule map.

Proof  Assume that there exits an $A_{i}$-module map $g_{i}:N_{i}\rightarrow M_{i}$ such that $g_{i}f_{i}={id}_{M_{i}}$. We define

$ \bar g = {\{ \overline {{g_\alpha }} :{N_\alpha } \to {M_\alpha }, \overline {{g_\alpha }} (n) = \sum {{\xi _\alpha }} ([{g_i}({n_{([0], i)}}) \otimes {n_{([1], \alpha )}}]), n \in {N_\alpha }\} _{\alpha \in \pi }}. $

First we claim that $\overline {{g_\alpha }} $ is right $A_{\alpha}$-linear. In fact, for any $\alpha \in \pi ,n \in {N_\alpha },a \in {A_\alpha }$,

$ \begin{eqnarray*} &&\overline{g_{\alpha}}(n\cdot a)=\sum {\xi}_{\alpha}([g_{i}(n_{([0], i)}\cdot a_{([0], i)})\otimes n_{([1], \alpha)}a_{([1], \alpha)}])\\ &=&\sum {\xi}_{\alpha}([g_{i}(n_{([0], i)})\cdot a_{([0], i)}\otimes n_{([1], \alpha)}a_{([1], \alpha)}])=\sum {\xi}_{\alpha}([g_{i}(n_{([0], i)})\otimes n_{([1], \alpha)}])\cdot a =\overline{g_{\alpha}}(n)\cdot a. \end{eqnarray*} $

Second we prove that $\overline{g}$ is right partial $\pi-H$-colinear. Indeed, for any $n\in N_{\alpha\beta}$,

$ \begin{eqnarray*}&&(\overline{g_{\alpha}}\otimes{id}_{H_{\beta}}){\rho}_{\alpha, \beta}^{N}(n) =\sum {\xi}_{\alpha}([g_{i}(n_{([0], \alpha)([0], i)})\otimes n_{([0], \alpha)([1], \alpha)}])\otimes n_{([1], \beta)} \\&=&\sum {\xi}_{\alpha}([g_{i}(n_{([0], i)})\cdot 1_{([0], i)}\otimes n_{([1], \alpha\beta)1\alpha}1_{([1], \alpha)}])\otimes n_{([1], \alpha\beta)2\beta}\\&=&\sum{\rho}_{\alpha, \beta}^{M}{\xi}_{\alpha}([g_{i}(n_{([0], i)})\otimes n_{([1], \alpha\beta)}])={\rho}_{\alpha, \beta}^{M}\overline{g_{\alpha\beta}}(n). \end{eqnarray*} $

Finally we have to show $\overline{g}f={id}_{M}$. In fact, for any $\alpha \in \pi ,m \in {M_\alpha }$ we have,

$ \begin{eqnarray*}&&\overline{g_{\alpha}}f_{\alpha}(m)=\sum {\xi}_{\alpha}([g_{i}({f_{\alpha}(m)}_{([0], i)})\otimes {f_{\alpha}(m)}_{([1], \alpha)}])\\&=&\sum {\xi}_{\alpha}([g_{i}(f_{i}(m_{([0], i)}))\otimes m_{([1], \alpha)}])=\sum {\xi}_{\alpha}([m_{([0], i)}\otimes m_{([1], \alpha)}])\\&=&{\xi}_{\alpha}\circ{p}_{\alpha}\circ{\xi}_{i, \alpha}(m)=m. \end{eqnarray*} $

Hence we complete the proof.