Dilatation properties of regular local rings

The results and methods of algebraic geometry, when analysed in terms of modern algebra, have revealed on several occasions algebraic principles of surprising generality. Recently it has become apparent that the geometric theory of infinitely near points has, as it were, an abstract form which forms part of the ideal theory of commutative rings, but there are many details which have yet to be worked out. Roughly speaking, one may say that what corresponds to the theory of the sequence of points on a curve branch is now known in some detail, and forms a substantial addition to our knowledge of the properties of one-dimensional local rings†; but the construction of an abstract theory similarly related to the theory of neighbourhoods in n-dimensional projective space can hardly be said to have been started. A number of necessary preliminary steps were taken by the author in (3)—in the process of providing algebraic foundations for certain applications of dilatation theory—and later some applications were made to 2-dimensional problems. However, the present paper should be regarded as an attempt to initiate a dilatation theory of regular local rings to run parallel to the general theory of infinitely near points in n-dimensional space.