The Structure of Weak Coalgebra-Galois Extensions

ABSTRACT

Weak coalgebra-Galois extensions are studied. A notion of an invertible weak entwining structure is introduced. It is proven that, within an invertible weak entwining structure, the surjectivity of the canonical map implies bijectivity, provided the structure coalgebra C is either coseparable or projective as a C-comodule.

Key Words:

• Coring
• Galois comodule
• Invertible weak entwining structure
• Weak coalgebra-Galois extension

1991 Mathematics Subject Classification:

• 16W30

ACKNOWLEDGMENTS

We thank Gabriella Böhm for comments. Tomasz Brzeziński thanks the Engineering and Physical Sciences Research Council for an Advanced Fellowship.

Notes

¹Note that weak entwining structures we discuss in this article are self-dual entwining structures in the terminology of Brzeziński and Wisbauer (2003). More general weak entwining structures discussed in Brzeziński and Wisbauer (2003) are best described in terms of weak corings (Wisbauer, 2001).

²In a recent paper [J. N. Alonso Álvarez, J. M. Fernández Vilaboa, R. Gonzaléz Rodríguez and A. B. Rodríguez Raposo, Invertible weak entwining structures and weak C-cleft extensions, Preprint (2005)] it is shown that condition (c) is a consequence of conditions (a) and (b).

³Example 4.5(2) was kindly provided by Gabriella Böhm.

Communicated by R. Wisbauer.