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A characterization of projective weak Galois extensions. (English) Zbl 1192.16029

Let \(R\) be a commutative ring with 1, \(A\) an \(R\)-algebra with product \(\mu_A\) and unit \(\eta_A\), \(C\) a coalgebra with coproduct \(\delta_C\) and counit \(\varepsilon_C\). The category of left (resp. right) \(A\)-modules is denoted by \(_A\mathcal M\) (resp. \(\mathcal M_A\)), \(^C\mathcal M\) (resp. \({\mathcal M}^C\)) denotes the category of left (resp. right) \(C\)-comodules.
A right-right weak entwining structure in \(\mathcal C\) is a triple \((A,C,\psi_R)\) where \(\psi_R\colon C\otimes A\to A\otimes C\) is a morphism such that (1) \(\psi_R\circ(C\otimes\mu_A)=(\mu_A\otimes C)\circ(A\otimes\psi_R)\circ(\psi_R\otimes A)\), (2) \(\psi_R\circ(C\otimes\eta_A)=(e^C_{RR}\otimes C)\circ\delta_C\), (3) \((A\otimes\delta_C)\circ\psi_R=(\psi_R\otimes C)\circ(C\otimes\psi_R)\circ(\delta_C\otimes A)\), (4) \((A\otimes\varepsilon_C)\circ\psi_R=\mu_A\circ(e^C_{RR}\otimes A)\), where \(e^C_{RR}=(A\otimes\varepsilon_C)\circ\psi_R\circ(C\otimes\eta_A)\colon C\to A\). Denote the category of right-right weak entwining modules \((M,\varphi_M,\rho_M)\) by \(\mathcal M_A^C(\psi_R)\), where \((M,\varphi_M)\) is a right \(A\)-module, \((M,\rho_M)\) is a right \(C\)-comodule, and \(\rho_M\circ\varphi_M=(\varphi_M\otimes C)\circ(M\otimes\psi_R)\circ(\rho_M\otimes A)\). Similarly, a left-left entwining structure in \(\mathcal C\) is a triple \((A,C,\psi_L)\) and the category of left-left weak entwining modules \(_A^C\mathcal M(\psi_L)\) are defined.
Assume \((A,C,\psi_R)\) is a right-right weak entwining structure in \(\mathcal C\) such that \((A,\mu_A,\rho_A)\in|\mathcal M_A^C(\psi_R)|\), the objects of \(\mathcal M_A^C(\psi_R)\). Let \(\Delta^R_{A\otimes C}\colon A\otimes C\to A\otimes C\) by \(\Delta^R_{A\otimes C}=(\mu_A\otimes C)\circ(A\otimes\psi_R)\circ(A\otimes C\otimes\eta_A)\). Then \(\Delta^R_{A\otimes C}\) is an idempotent morphism and there exists an object \(A\square C\) and, \(i^R_{A\otimes C}\colon A\square C\to A\otimes C\), \(p^R_{A\otimes C}\colon A\otimes C\to A\square C\) such that \(\Delta^R_{A\otimes C}=i^R_{A\otimes C}\circ p^R_{A\otimes C}\) and \(id_{A\square C}=p^R_{A\otimes C}\circ i^R_{A\otimes C}\). Let \(\gamma_{A\square C}=p^R_{A\otimes C}\circ(\mu_A\otimes C)\circ(A\otimes \rho_A)\colon A\otimes C\to A\square C\). Then there exists a unique morphism \(\gamma_A^C\colon A\otimes_{A_C}A\to A\square C\) such that \(\gamma_A^C\circ q^A_C=r_{A\square C}\) where \(q^A_C\) is the coequalizer. Assume \(A\otimes{\underline{\;}}\) preserves coequalizers and \((A,\mu_A,\rho_A)\in|\mathcal M_A^C(\psi_R)|\). If \(\gamma_A^C\) is an isomorphism, then \(A\) is called a right weak \(C\)-Galois extension. This generalizes some concepts of Galois extensions as defined by Caenepeel, De Groot and others. Then the authors obtain a criterion under which the surjectivity of \(\gamma_A^C\) implies the bijectivity.
Theorem. Let \((A,C,\psi_A)\) be a right-right weak entwining structure in \(\mathcal C\). Suppose \((A,\mu_A,\rho_A)\in|\mathcal M_A^C(\psi_R)|\). Consider the following statements: (i) \(\gamma_{A\otimes C}\colon A\otimes A\to A\square C\) splits in \(\mathcal M^C\). (ii) (ii-1) \(\gamma_A^C\colon A\otimes_{A_C}A\to A\square C\) is an isomorphism. (ii-2) \(\bigl(A,\varphi_A=\mu_A\circ(A\otimes i_C^A)\bigr)\) is relative projective in \(\mathcal M_{A_C}\), i.e., \(\varphi_A\colon A\otimes A_C\to A\) splits as \(A_C\)-module morphism. Then (ii) \(\Rightarrow\) (i). If \((A,C,\psi_A,\psi_L)\) is an invertible weak entwining structure in \(\mathcal C\) and the factorization \(s_C^A\colon A\otimes {A_C}\to(A\otimes A)_C\) of the morphism \(A\otimes i^A_C\) through the equalizer \(i_C^{A\otimes A}\) is an isomorphism, then (i) \(\Rightarrow\) (ii). Moreover, some results about weak Galois extensions and morphisms of weak entwining structures are obtained.

MSC:

16T05 Hopf algebras and their applications
16T15 Coalgebras and comodules; corings
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16S35 Twisted and skew group rings, crossed products
13B05 Galois theory and commutative ring extensions
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References:

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