Convergent higher derivations on local rings

Abstract

NICKOLAS HEEREMA[June operation and taking inverses, all in the case in which R is quasi-local.Theorem 2 states that the convergent higher derivations form a subsemigroup HC(R, R) of H(R, R) and Theorem 3 states that the subsets H?(R, R), H^(R, R), and Hf, (R, R) of M-convergent, uniformly M-convergent and strongly M-convergent higher derivations (see Definition 3) form subgroups of H(R, R).An example following the proof of Theorem 2 illustrates the fact that D may be convergent and aD may be an automorphism whereas Z>-1 is not convergent.It is readily seen that if D is M-convergent then aD is in Hu the subgroup of those inertial automorphisms a satisfying the condition a(a) -a e M2 for all a in M. Conversely, if a = aD and a is in H1 then D is M-convergent.If F is a t'-ring (unramified) every inertial automorphism is in Hx.If R is an unramified complete regular local ring then the mapping A: D -> aD is a homomorphism of H^(R, R) onto Hx.As a matter of fact A restricted to HtM(R, R) still maps onto Hx.It follows from work of Wishart [7, pp.50, 51] that a ramified taring may have inertial automorphisms represented by D in H^(R, R) but not by D in HtM(R, R).111 deals with the question of necessary and sufficient conditions on a complete local ring R that every convergent higher derivation be uniformly convergent.Theorem 5 asserts that if the residue field k has characteristic p, the condition that k have a finite /-basis is sufficient and if R is regular this condition is necessary.If R is regular and k has characteristic zero (R is a power series ring over k) then every convergent higher derivation is uniformly convergent if and only if k has finite transcendency degree over its prime field.