Forms of the rings R[X] and R[X, Y]
Glasgow Mathematical Journal
Let R be a ring and let S = Spec/?. Let us consider the Etalefini topology on S  . By a form of a given S-scheme T we mean any affine 5-scheme W that is locally (in the etalefini topology) isomorphic to T. We shall consider forms of the J?-schemes T= SpecR [X] and In the case where R = k is a field, the above definition gives the classical definition of forms of A:-algebras . The problem of determining the forms of k[X] is easy. If A is a A>algebra such that, for some separable extension
... eparable extension K of k, there exists a /iT-isomorphism between K®A and K[X], then A and k[X] are isomorphic as ^-algebras. The following result is due to Safarevic . If A: is a field, then there are no nontrivial forms of the affine plane. It means that, if A is a ^-algebra such that, for some separable extension K of k, there exists a ^-isomorphism between K®A and K [X, Y], then A and k [X, Y] are fc-isomorphic. The main results of this paper are the following theorems. THEOREM 1. Let R be a noetherian local ring. Then any form ofSpecR[X] is trivial.