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

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- by Klaus G. Fischer PDF
- Trans. Amer. Math. Soc.
**186**(1973), 113-128 Request permission

## Abstract:

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 $I(\bar B/B)$.

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## Additional Information

- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**186**(1973), 113-128 - MSC: Primary 13H15; Secondary 14H20
- DOI: https://doi.org/10.1090/S0002-9947-1973-0337947-9
- MathSciNet review: 0337947