ON GENERALIZED CODERIVATIONS

Abstract

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.