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We consider the generation of magnetic activity—dynamo waves—in the astrophysical limit of very large magnetic Reynolds number. We consider kinematic dynamo action for a system consisting of helical flow and large-scale shear. We demonstrate that large-scale dynamo waves persist at high Rm if the helical flow is characterized by a narrow band of spatial scales and the shear is large enough. However, for a wide band of scales the dynamo becomes small scale with a further increase of Rm, with dynamo waves re-emerging only if the shear is then increased. We show that at high Rm, the key effect of the shear is to suppress small-scale dynamo action, allowing large-scale dynamo action to be observed. We conjecture that this supports a general “suppression principle”—large-scale dynamo action can only be observed if there is a mechanism that suppresses the small-scale fluctuations. [ABSTRACT FROM AUTHOR]