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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

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