The module decomposition of $I(\bar A/A)$

Let A and B be scalar rings with B an A-algebra. The B-algebra ${D^n}(B/A) = I(B/A)/{I^n}(B/A)$ is universal for n-truncated A-Taylor series on B. In this paper, we consider the $\bar A$ module decomposition of ${D^n}(\bar A/A)$ with a view to classifying the singularity A which is assumed to be the complete local ring of a point on an algebraic curve at a one-branch singularity. We assume that $A/M = k < A$ and that k is algebraically closed with no assumption on the characteristic.

We show that ${D^n}(\bar A/A) = I(\bar A/A)$ for n large and that the decomposition of $I(\bar A/A)$ as a module over the P.I.D. $\bar A$ is completely determined by the multiplicity sequence of A. The decomposition is displayed and a length formula for $I(\bar A/A)$ developed. If B is another such ring, where $\bar B = \bar A = k[[t]]$, we show that $I(\bar A/A) \cong I(\bar B/B)$ as $k[[t]]$ modules if and only if the multiplicity sequence of A is equal to the multiplicity sequence of B. If $A < B < \bar A$, then $I(\bar A/A) \cong I(\bar B/B)$ as $\bar A = \bar B$ modules if and only if the Arf closure of A and B coincide. This is equivalent to the existence of an algebra isomorphism between $I(\bar A/A)$ and $I(\bar B/B)$.