A Counter-example to the Cancellation Problem for the Affine Space A³ in characteristic p

We show that the Cancellation Conjecture does not hold for the affine space A^3_k over any field k of positive characteristic. We prove that an example of T. Asanuma provides a three-dimensional k-algebra A for which A is not isomorphic to k[X_1,X_2,X_3] although A[T] is isomorphic to k[X_1, X_2, X_3, X_4].