ON GENERALIZED CODERIVATIONS

Yıl 2012, Cilt: 12 Sayı: 12, 37 - 52, 01.12.2012

Atsushi Nakajima

Öz

Let C (resp. A) be a coalgebra (resp. algebra) over a commutative
ring R and M (resp. N) a C-bicomodule (resp. an A-bimodule). We define a
dual notion of a generalized derivation from A to N in the sense of the paper
On categorical properties of generalized derivations, Sci. Math., 2(3) (1999),
345-352, by A. Nakajima, which we call a generalized coderivation from M
to C. We give some elementary properties of generalized coderivations and
discuss the relations of the set of generalized coderivations gCoder(M, C) between
the set of generalized derivations gDer(C∗, M∗) for their dual algebra
C∗ and module M∗. Using these coderivations, we define a notion of a weakly
coseparable coalgebra which is a dual notion of a weakly separable algebra
defined in the paper of N. Hamaguchi and A. Nakajima, Weakly separable
polynomials (in preparation), and give related examples of coseparable coalgebras.

Anahtar Kelimeler

derivation, generalized derivation, coderivation, generalized coderivation, weakly coseparable coalgebra

Kaynakça

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• F. Guzman, Cointegrations, Relative Cohomology for Comodules, and Cosep- arable Corings, J. Algebra, 126 (1989), 211-224.
• N. Hamaguchi and A. Nakajima, Weakly separable polynomials, in preparation.
• R. G. Larson, Coseparable Hopf algebras, J. Pure Appl. Algebra, 3 (1973), 261-267.
• A. Nakajima, Cosemisimple coalgebras and coseparable coalgebras over coalge- bras, Math. J. Okayama Univ., 21 (1979), 125-140.
• A. Nakajima, Coseparable coalgebras and coextensions of coderivations, Math. J. Okayama Univ., 22 (1980), 145-149.
• A. Nakajima, On categorical properties of generalized derivations, Sci. Math., 2(3) (1999), 345-352.
• M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. Atsushi Nakajima
• 1, Nakaku, Nakai,
• Okayama 703-8205, Japan
• e-mail: [e-mail]