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Abstract
In this note, we study the global dimension of coalgebras and discuss the class of coalgebras of global dimension less or equal to 1. The coalgebras in this class, which contains all the cosemisimple coalgebras, are called hereditary coalgebras. If C is a finite dimensional coalgebra, then C is hereditary if and only if C (the convolution algebra of C) is a hereditary algebra. Any direct sum of hereditary coalgebras is hereditary too. This gives us many examples of infinite dimensional hereditary coalgebras. A coalgebra is left hereditary if and only if it is right hereditary. Moreover, there do not exist hereditary Hopf algebras of finite dimension which are not cosemisimple.
*This paper is the result of the research carried out by the author during his stay on sabattical at the University of Almeria, supported by DGICYT
†Supported by a grant from NATO and PB91-706 from DGICYT
‡This author wishes to thank the Department of Algebra and Analysis of the University of Almeria for its hospitality
*This paper is the result of the research carried out by the author during his stay on sabattical at the University of Almeria, supported by DGICYT
†Supported by a grant from NATO and PB91-706 from DGICYT
‡This author wishes to thank the Department of Algebra and Analysis of the University of Almeria for its hospitality
Notes
*This paper is the result of the research carried out by the author during his stay on sabattical at the University of Almeria, supported by DGICYT
†Supported by a grant from NATO and PB91-706 from DGICYT
‡This author wishes to thank the Department of Algebra and Analysis of the University of Almeria for its hospitality