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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Klaus G. Fischer
Journal: Trans. Amer. Math. Soc. 186 (1973), 113-128
MSC: Primary 13H15; Secondary 14H20
MathSciNet review: 0337947
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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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Keywords: Taylor series, blow-up, Arf ring, multiplicity sequence, decomposition of module of higher differentials
Article copyright: © Copyright 1973 American Mathematical Society