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In this note some generalization of the Chern character is discussed from thechromatic point of view. We construct a multiplicative G_{n+1}-equivariantnatural transformation \Theta from some height (n+1) cohomology theory E^*(-)to the height n cohomology theory K^*(-)\hat{\otimes}_F L, where K^*(-) isessentially the n-th Morava K-theory. As a corollary, it is shown that theG_n-module K^*(X) can be recovered from the G_{n+1}-module E^*(X). We alsoconstruct a lift of \Theta to a natural transformation between characteristiczero cohomology theories.

Equivariance of generalized Chern characters