Frobenius Extensions of Corings

Abstract

Let  and  be two corings over a ring A and be a morphism of corings. We investigate the situation when the associated induced (“corestriction of scalars”) functor ℳ → ℳ  is a Frobenius functor, and call these morphisms Frobenius extensions of corings. The characterization theorem generalizes notions such as Frobenius corings and is applied to several situations; in particular, provided some (general enough) flatness conditions hold, the notion proves to be dual to that of Frobenius extensions of rings (algebras). Several finiteness theorems are given for each case we consider; these theorems extend existing results from Frobenius extensions of rings or from Frobenius corings, showing that a certain finiteness property almost always occur for many instances of Frobenius functors.

Key Words:

• Corings
• Frobenius functor

2000 Mathematics Subject Classification:

• Primary 16W30
• Secondary 16S90, 16Lxx, 16Nxx, 18E40

ACKNOWLEDGMENTS

The author wishes to thank his Ph.D. adviser C. Năstăsescu for very useful remarks on the subject as well as for his continuous support throughout the past years. He would also like to address special thanks to the referee for very useful comments on the subject and a helpful report.

This paper was partially supported by a contract nr. 24/28.09.07 with UEFISCU “Groups, quantum groups, corings and representation theory” of CNCIS, PNII (ID1002), and by the bilateral project BWS04/04 “New Techniques in Hopf Algebra Theory and Graded Ring Theory” of the Flemish and Romanian governments.

Notes

Communicated by R. Wisbauer.