Bulletin of the London Mathematical Society Advance Access originally published online on March 11, 2009
Bulletin of the London Mathematical Society 2009 41(3):483-494; doi:10.1112/blms/bdp020
| ||||||||||||||||||||||||||||||||||||||||||||||
© 2009 London Mathematical Society
The generating condition for coalgebras
Faculty of Mathematics
University of Bucharest
Str. Academiei 14
RO-70109 Bucharest
Romania
State University of New York – Buffalo
244 Mathematics Building
Buffalo, NY 14260-2900
USA
e-mail@yovanov.net
Received 7 April 2008. Revision received 3 November 2008.
For a ring R, the properties of being (left) self-injective or being a cogenerator for the left R-modules do not imply one another, and the two combined give rise to the important notion of pseudo-Frobenius-rings. For a coalgebra C, (left) self-projectivity implies that C is a generator for right comodules and the coalgebras with this property are called right quasi-co-Frobenius; however, whether the converse implication is true is an open question. We provide an extensive study of this problem. We show that this implication does not hold, by giving a large class of examples of coalgebras having the ‘generating property’. In fact, we show that any coalgebra C can be embedded in a coalgebra C
that generates its right comodules, and, if C is local over an algebraically closed field, then C can be chosen local as well. We also give some general conditions under which the implication ‘C-projective (left) 
C generator for right comodules’ does work, and such conditions are when C is right semiperfect or when C has finite coradical filtration.
2000 Mathematics Subject Classification 16W30 (primary), 16S50, 16D90, 16L30 (secondary).
The author was partially supported by the contract nr. 24/28.09.07 with UEFISCU ‘Groups, quantum groups, corings and representation theory’ of CNCIS, PN II (ID_1002).