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Algebras with total integrals. (English) Zbl 0576.16004
Let \(R\) be a commutative ring with identity, let \(A\) be a Hopf algebra over \(R\), and let \(B\) be an \(R\)-algebra which is a right comodule for \(A\) with respect to an algebra homomorphism \(\rho:B\to B\otimes A\). The elements \(c\) of \(B\) for which \(\rho(c)=c\otimes 1\) form a subalgebra \(C\) of \(B\), and a left \(B\)-module homomorphism \(\beta:B\otimes_ CB\to B\otimes A\) is induced by \(\rho\). A comodule homomorphism \(\phi\) of \(A\) into \(B\) is called an integral, and the integral is said to be total if \(\phi(1_ A)=1_ B\). Since \(A\) is a coalgebra and \(B\) is an algebra, there is a convolution product with respect to which the set \(\operatorname{Hom}(A,B)\) of \(R\)-module homomorphisms becomes an \(R\)-algebra; and \(B\) is said to be cleft if there is an integral which is an invertible element of \(\operatorname{Hom}(A,B)\). The author proves that \(B\) is cleft if, and only if, (i) \(\beta:B\otimes_ CB\to B\otimes A\) is an isomorphism and (ii) there exists a left \(C\)-module, right \(A\)-comodule isomorphism between \(C\otimes A\) and \(B\). When \(A\) is a finitely generated, projective module over \(R\), condition (i) is the definition of an \(A\)-Galois extension and condition (ii) is the definition of a normal basis for \(B\). If \(B\) is cleft, then there exists a total integral of \(A\) into \(B\). Again when \(A\) is a finitely generated, projective \(R\)-module; the existence of a total integral of \(A\) into \(B\) is equivalent to the existence of elements \(b_ 1,...,b_ n\) of \(B\) and \(\sigma_ 1,...,\sigma_ n\) of the ideal of integrals of the dual Hopf algebra \(A^*\) (these are the comodule homomorphisms of \(A\) into \(R\)) such that \(\sum_{i}\sigma_ i(b_ i)=1\), where \(B\) is a left \(A^*\)-module according to the right \(A\)-comodule structure of \(B\).
The author relates each of the conditions: \(B\) is cleft and \(B\) has a total integral, to properties of categories of left \((A,B)\)-Hopf modules and right \((A,B)\)-Hopf modules. He considers further cases in which \(A\) is a commutative Hopf algebra over a field and in which \(A\) is a quotient Hopf algebra of a commutative Hopf algebra \(B\) over a field.
Reviewer: H.F.Kreimer

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16W20 Automorphisms and endomorphisms