Quasi-Frobenius Corings and Quasi-Frobenius Extensions

Abstract

In this article, the notions of a Frobenius pair of functors and Frobenius corings are generalized to an l-QF pair of functors and l-QF corings. We prove that an extension ι:B → A is left quasi-Frobenius if and only if (F ₁,G ₁) is an l-QF pair of functors, where F ₁: _A ℳ →  _B ℳ is the restriction of scalars functors, and G ₁ = A⊗ _{B ⁻} : _B ℳ →  _A ℳ is the induction functor. For an A-coring , we prove that  is an l-QF coring if and only if A → ∗ is an l-QF extension and _A  is a finitely generated projective modules if and only if (G ₂,F ₂) is an l-QF pair of functors, where G ₂ =  ⊗ _{A ⁻} : _A ℳ →  ℳ is the induction functor, F ₂:  ℳ →  _A ℳ is the forgetful functor, the result of Brzezinski is generalized.

Key Words:

• l-QF coring
• l-QF extension
• l-QF pair of functors
• l-quasi-adjoint functor

2000 Mathematics Subject Classification:

• 16W30
• 13B02

ACKNOWLEDGMENT

The author takes this opportunity to express his thanks to the referee for his or her helpful comments.

Notes

Communicated by R. Wisbauer.