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
DOI:
https://doi.org/10.1090/S0002-9947-1973-0337947-9
MathSciNet review:
0337947
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Abstract | References | Similar Articles | Additional Information
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 $I(\bar B/B)$.
- Cahit Arf, Une interprĂ©tation algĂ©brique de la suite des ordres de multiplicitĂ© dâune branche algĂ©brique, Proc. London Math. Soc. (2) 50 (1948), 256â287 (French). MR 31785, DOI https://doi.org/10.1112/plms/s2-50.4.256
- Patrick Du Val, Note on Cahit Arfâs âUne interprĂ©tation algĂ©brique de la suite des ordres de multiplicitĂ© dâune branche algĂ©brique.â, Proc. London Math. Soc. (2) 50 (1948), 288â294. MR 31786, DOI https://doi.org/10.1112/plms/s2-50.4.288
- Sherwood Ebey, The classification of singular points of algebraic curves, Trans. Amer. Math. Soc. 118 (1965), 454â471. MR 176983, DOI https://doi.org/10.1090/S0002-9947-1965-0176983-8
- Nathan Jacobson, Lectures in abstract algebra. Vol. II. Linear algebra, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953. MR 0053905
- Joseph Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649â685. MR 282969, DOI https://doi.org/10.2307/2373463 K. Mount and O. E. Villamayor, Taylor series and higher derivations, Departmento de Matematicas Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires, Serie N$^{o}$. 18, Buenos Aires, 1969.
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
- Yoshikazu Nakai, High order derivations. I, Osaka Math. J. 7 (1970), 1â27. MR 263804
- D. G. Northcott, The neighbourhoods of a local ring, J. London Math. Soc. 30 (1955), 360â375. MR 71110, DOI https://doi.org/10.1112/jlms/s1-30.3.360
- Robert J. Walker, Algebraic Curves, Princeton Mathematical Series, vol. 13, Princeton University Press, Princeton, N. J., 1950. MR 0033083
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
- Oscar Zariski, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math. 87 (1965), 507â536. MR 177985, DOI https://doi.org/10.2307/2373019
- Oscar Zariski, Studies in equisingularity. II. Equisingularity in codimension $1$ (and characteristic zero), Amer. J. Math. 87 (1965), 972â1006. MR 191898, DOI https://doi.org/10.2307/2373257
- Oscar Zariski, Studies in equisingularity. III. Saturation of local rings and equisingularity, Amer. J. Math. 90 (1968), 961â1023. MR 237493, DOI https://doi.org/10.2307/2373492
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Additional Information
Keywords:
Taylor series,
blow-up,
Arf ring,
multiplicity sequence,
decomposition of module of higher differentials
Article copyright:
© Copyright 1973
American Mathematical Society