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Morita contexts and Galois theory for weak Hopf comodulelike algebras. (English) Zbl 1231.16027

Let \(A\) be an algebra over a commutative ring with \(1_A\), and \(G\) a group with identity \(e\). A weak \(G\)-\(A\)-coring \(\mathcal C\) is a family \((\mathcal C_\alpha\mid\alpha\in G)\) of \(A\)-bimodules with \(A\)-bimodule maps \(\Delta=\{\Delta_{\alpha,\beta}\colon\mathcal C_{\alpha,\beta}\to\mathcal C_\alpha\otimes_AA\otimes_A\mathcal C_\beta\}\) and \(\varepsilon\colon\mathcal C_e\to A\) such that, for \(\alpha,\beta,\gamma\in G\), \[ (\mathcal C_\alpha\otimes_AA\otimes_A\Delta_{\beta,\gamma})\Delta_{\alpha, \beta\gamma}=(\Delta_{\alpha,\beta}\otimes_AA\otimes_A\mathcal C_\gamma)\Delta_{\alpha\beta,\gamma} \] and \[ (\mathcal C_\alpha\otimes_AA\otimes_A\varepsilon)\Delta_{\alpha,e}=1_A\mathcal C_\alpha 1_A=(\varepsilon\otimes_AA\otimes_A\mathcal C_\alpha)\Delta_{e,\alpha}. \] If \(\mathcal C\) is both right and left unital, then \(\mathcal C\) is a \(G\)-\(A\)-coring. If \((\mathcal C,x)\) contains a family of group like elements \(x=(x_\alpha\mid\alpha\in G)\in A\mathcal CA\), then it is a Galois weak group coring in case \(can\colon(A\otimes_{A^{co\mathcal C}}A)\langle G\rangle\to A\mathcal CA\) is an isomorphism of group corings.
Then the authors show some functors between the category of right weak \(\mathcal C\)-comodules and the category of right \(G\)-graded weak \(\mathcal C\)-comodules. Let \(R=\bigoplus R_\alpha\), \(\alpha\in G\), be a \(G\)-graded weak \(A\)-ring (that is, the left dual graded ring \(^*\mathcal C\) of \(\mathcal C\)). Then there is a functor from the category of right weak \(G\)-\(\mathcal C\)-comodules to the category of right weak \(R\)-modules which is an isomorphism if \(\mathcal C\) is left unital and every \(\mathcal C_\alpha\) is finitely generated and projective as a left \(A\)-module. Moreover, the Morita contexts associated with \(\mathcal C\) are constructed. Let \((\mathcal C,x)\) be a weak \(G\)-\(A\)-coring with a family of group like elements \(x=(x_\alpha\mid\alpha\in G)\in A\mathcal CA\) and \(R\) a left dual ring of \(\mathcal C\), \(Q=\{q=(q_\alpha\mid\alpha\in G)\in\Pi(AR_\alpha A)\mid c_{(1,\beta^{-1})}q_\alpha c_{(2,\alpha^{-1})}=q_{\alpha\beta}(c)x_{\beta^{-1})}\) for any \(\alpha,\beta\in G\), \(c\in\mathcal C_{(\alpha\beta)^{-1}}\}\) and \(T=A^{co\mathcal C}\). Then \((T,ARA,A,Q,\tau,\mu)\) is a Morita context with \(\tau\) and \(\mu\) defined by \(\tau\colon A\otimes_{ARA}Q\to T\) by \(\tau(a\otimes_Rq)=\sum_\alpha q_\alpha(x_{\alpha^{-1}}a)\), and \(\mu\colon Q\otimes_TA\to ARA\) by \(\mu(q\otimes_Ta)=qa\). Equivalent conditions are given for a surjective \(\mu\).
Furthermore, the Galois theory for weak group corings is given. Theorem. \((\mathcal C,x)\) is a Galois weak group coring if and only if \((A\mathcal CA,x)\) is a cofree group coring with a fixed group like family \(x_\alpha=\gamma_\alpha(x_e)\) and \((A\mathcal C_eA,x_e)\) is a Galois coring. – By applying the Galois theory to the weak Hopf comodulelike algebras, some useful results are also obtained.

MSC:

16T15 Coalgebras and comodules; corings
16W50 Graded rings and modules (associative rings and algebras)
16D90 Module categories in associative algebras
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
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References:

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