A Counter-example to the Cancellation Problem for the Affine Space A^3 in characteristic p

TL;DR

This paper demonstrates that the Cancellation Conjecture fails for the affine space A^3 over fields of positive characteristic by providing a specific counter-example where A is not isomorphic to the standard polynomial ring despite its extension being isomorphic.

Contribution

The paper presents a counter-example to the Cancellation Conjecture in characteristic p, showing that A^3 does not satisfy cancellation, which was previously unknown.

Findings

Counter-example to the Cancellation Conjecture in characteristic p

A specific three-dimensional algebra A where A[T] is isomorphic to a polynomial ring

A is not isomorphic to the standard polynomial ring in three variables

Abstract

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].