Around the Eakin-Sathaye theorem

We present a generalisation of the Eakin-Sathaye theorem on reductions to the case of complete, and so of joint, reductions in the sense of Rees. Applications are given. Generalisations are also indicated for work of Lipman, Eakin-Sathaye and Sally-Vasconcelos that fed into or resulted from the original version of the Eakin-Sathaye theorem. Finally, the result is reinterpreted as a “multiplicative” normalisation theorem.