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Smash products, separable extensions and a Morita context over Hopf algebroids. (English) Zbl 1302.16029

Let \(k\) be a commutative ring with \(1\). A left bialgebroid \(H_L=(H,L,s_L,t_L,\Delta_L,\varepsilon_L)\) consists of two algebras \(H\) and \(L\) over \(k\) where \(H\) is an \(L\otimes_kL^o\)-ring via the algebra homomorphisms \(s_L\colon L\to H\) and \(t_L\colon L^o\to H\) and \((H,\Delta_L,\varepsilon_L)\) an \(L\)-coring via \(\Delta_L\) and \(\varepsilon_L\). Similarly, a right bialgebroid \(H_R=(H,L,s_R,t_R,\Delta_R,\varepsilon_R)\) is defined. A Hopf algebroid \(H=(H_L,H_R,S)\) with the antipode \(S\colon H\to H\) is defined by G. Böhm [Algebr. Represent. Theory 8, No. 4, 563-599 (2005; Zbl 1137.16037)]. Then, for a left module of \(H_L\), a left \(H_L\)-module algebra, the smash product algebra \(A\#H\) as defined in [L. Kadison and K. Szlachányi, Adv. Math. 179, No. 1, 75-121 (2003; Zbl 1049.16022)], and for a Hopf algebroid \(H\) with bijective \(S\), the authors show an equivalent condition for the existence of non-trivial homomorphisms between a left \(H_L\)-module algebra \(A\) projective as an \(L\)-module and \(A\#H\). Next, a trace function for \(H\) on \(A\) to the invariant subalgebra \(A^{H_L}\) is defined and shown that the left trace function is surjective if and only if \(A\) contains an element of trace one. It is also shown that if the \(L\)-ring \((H,s_L)\) underlying \(H_L\) is left semisimple, then the \(L\)-ring \((A\#H,1_A\#\varepsilon_L)\) is a separable extension of \((A,\eta_A)\). Moreover, for a Frobenius Hopf algebroid \(H\) and a left \(H_L\)-module algebra \(A\), a grouplike character \(\chi\colon A\#H\to A\) is defined and a Morita context connecting \(A\#H\) and \(A^{H_L}\) is constructed. Thus the weak structure theorem is obtained for the category of left \((H,A)\)-Hopf modules over Hopf algebroids.

MSC:

16T05 Hopf algebras and their applications
16T15 Coalgebras and comodules; corings
16S40 Smash products of general Hopf actions
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