The value semigroups of prime divisors of the second kind in -dimensional regular local rings

Author:
Sunsook Noh

Journal:
Trans. Amer. Math. Soc. **336** (1993), 607-619

MSC:
Primary 13H05

DOI:
https://doi.org/10.1090/S0002-9947-1993-1080735-1

MathSciNet review:
1080735

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, it is shown that the value semigroup of a prime divisor of the second kind on a -dimensional regular local ring is symmetric. Further, a necessary and sufficient condition for two prime divisors of the second kind on a -dimensional regular local ring to have the same value semigroup is obtained.

**[A]**S. S. Abhyankar,*On the valuations centered in a local domain*, Amer. J. Math.**78**(1956), 321-348. MR**0082477 (18:556b)****[AM]**M. F. Atiyah and I. G. MacDonald,*Introduction to commutative algebra*, Addison-Wesley, 1969. MR**0242802 (39:4129)****[Az]**A. Azevedo, Lecture on plane algebroid curve at Purdue University, 1980**[Hos]**M. A. Hoskin,*Zero-dimensional valuation ideals associated with plane curves branches*, Proc. London Math. Soc. (3)**6**(1956), 70-99. MR**0074905 (17:665b)****[H]**C. Huneke,*Complete ideals in two-dimensional regular local rings*, Microprogram in Commutative Algebra, MSRI, Springer-Verlag, 1987. MR**1015525 (90i:13020)****[HS]**C. Huneke and J. D. Sally,*Birational extensions in dimension two and integrally closed ideals*, J. Algebra**115**(1988), 481-500. MR**943272 (89e:13025)****[K1]**E. Kunz,*The value-semigroup of a one dimensional Gorenstein ring*, Proc. Amer. Math. Soc.**25**(1970), 748-751. MR**0265353 (42:263)****[K2]**-,*Introduction to commutative algebra and algebraic geometry*, Birkhäuser, 1985.**[L1]**J. Lipman,*Rational singularities, with applications to algebraic surfaces and unique factorization*, Inst. Hautes Études Sci. Publ. Math.**36**(1969), 195-279. MR**0276239 (43:1986)****[L2]**-,*On complete ideals in regular local rings*, Algebraic Geometry and Commutative Algebra in Honor of Masayoshi Nagata, Vol. I, Academic Press, 1988, pp. 203-231. MR**977761 (90g:14003)****[L3]**-, Handwritten notes, 1966.**[M]**S. McAdam,*Asymptotic prime divisors*, Lecture Notes in Math., vol. 1023, Springer-Verlag, 1983. MR**722609 (85f:13018)****[Ma]**H. Matsumura,*Commutative ring theory*, Cambridge Univ. Press, 1986. MR**879273 (88h:13001)****[N]**M. Nagata,*Local rings*, Interscience, New York, 1962. MR**0155856 (27:5790)****[No]**S. Noh,*Sequence of valuation ideals of prime divisors of the second kind in*-*dimensional regular local rings*, J. Algebra (to appear). MR**1223666 (94d:13003)****[R]**D. Rees,*Lectures on the asymptotic theory of ideals*, London Math. Soc. Lecture Note Series, vol. 113, Cambridge Univ. Press, 1988. MR**988639 (90d:13012)****[S]**J. Sally,*One-fibered ideals*, Microprogram in Commutative Algebra, MSRI, Springer-Verlag, 1987. MR**1015533 (90h:13003)****[Z]**O. Zariski,*Polynomial ideals defined by infinitely near base points*, Amer. J. Math.**60**91938), 151-204. MR**1507308****[ZS]**O. Zariski and P. Samuel,*Commutative algebra*, Vol. 2, Van Nostrand Princeton, N.J., 1960. MR**0120249 (22:11006)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
13H05

Retrieve articles in all journals with MSC: 13H05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1080735-1

Article copyright:
© Copyright 1993
American Mathematical Society