Duality and Rational Modules in Hopf Algebras over Commutative Rings

Abstract

Let A be an algebra over a commutative ring R. If R is noetherian and A• is pure in R A , then the categories of rational left A-modules and right A • -comodules are isomorphic. In the Hopf algebra case, we can also strengthen the BlattnerMontgomery duality theorem. Finally, we give sufficient conditions to get the purity of A• in R A .

Duality theorems for crossed products over rings

“…As a corollary, with σ trivial, Theorem 2.21 generalizes Koppinen's version of the Blattner-Montgomery duality theorem [23,Corollary 5.4] to the case of arbitrary Noetherian ground rings (this improves also [5,Theorem 3.2]). Corollary 2.22 extends [25,Corollary 9.4.11] to the case of arbitrary QF ground rings (for an arbitrary right H -crossed product see Corollary 2.11).…”

“…extended by C. Chen and W. Nichols [14] to the case of Dedekind domains. In a joint paper with J. Gómez-Torrecillas and F. Lobillo [5,Theorem 3.2] that result was extended to the case of arbitrary Noetherian ground rings.…”

On coreflexive coalgebras and comodules over commutative rings

“…4. An R-algebra A is said to satisfy the -condition or to be an -algebra, if the class K A of all R-coÿnite A-ideals is a ÿlter and the induced R-pairing (A; A • ) satisÿes the -condition (in case R is Noetherian this is equivalent to the purity of A • ⊂ R A ).…”

“…In a joint work with GÃ omez-Torrecillas and Lobillo [4] on the category of comodules of coalgebras over arbitrary commutative base rings, we presented the so calledcondition. That condition has shown to be a natural assumption in the author's study of duality theorems for Hopf algebras [3].…”

“…R C satisÿes the -condition and Ä P (A) ⊆ C * is dense; 3. (A; C) is an -pairing; 4. C ⊂ R A is pure (in the sense of Cohn); 5.…”