Symmetric powers of complete modules over a two-dimensional regular local ring

Authors:
Daniel Katz and Vijay Kodiyalam

Journal:
Trans. Amer. Math. Soc. **349** (1997), 747-762

MSC (1991):
Primary 13B21, 13B22, 13H05, 13H15

DOI:
https://doi.org/10.1090/S0002-9947-97-01819-9

MathSciNet review:
1401523

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free -module , write for the th symmetric power of , mod torsion. We study the modules , , when is complete (i.e., integrally closed). In particular, we show that , for any minimal reduction and that the ring is Cohen-Macaulay.

**[BV]**W. Bruns and U. Vetter,*Determinantal Rings*, Lecture Notes in Mathematics 1327, Springer-Verlag, Berlin and Heidelberg, 1988. MR**89i:13001****[BR]**D. A. Buchsbaum and D.S. Rim,*A generalized Koszul complex II. Depth and multiplicity*, Trans. Amer. Math. Soc.**111**(1964), 197-224. MR**28:3076****[C1]**S. D. Cutkosky,*A new characterization of rational singularities*, Inv. Math.**102**(1990), 157-177. MR**91h:14045****[C2]**S. D. Cutkosky,*On unique and almost unique factorization of complete ideals II*, Inv. Math.**98**(1989), 59-74. MR**90j:14016B****[H]**C. Huneke,*Complete ideals in two-dimensional regular local rings*, Commutative Algebra, Proc. Microprogram, MSRI publication no. 15 (1989), 417-436. MR**90i:13020****[HS]**C. Huneke and J. Sally,*Birational extensions in dimension two and integrally closed ideals*, J. Algebra**115**(1988), 481-500. MR**89e:13025****[Kap]**I. Kaplansky,*Commutative Rings*, University of Chicago Press, 1974. MR**49:10674****[Ko]**V. Kodiyalam,*Integrally closed modules over two-dimensional regular local rings*, Trans. Amer. Math. Soc.**347**(1995), 3551-3573. MR**95m:13015****[L]**J. Lipman,*Rational singularities with applications to algebraic surfaces and unique factorization*, Inst. Hautes Etudes Sci. Publ. Math.**36**(1969), 195-279. MR**43:1986****[LT]**J. Lipman and B. Teissier,*Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals*, Michigan Math. Journal**28**(1981), 97-116. MR**82f:14004****[R]**D. Rees,*Reduction of modules*, Math. Proc. Camb. Phil. Soc.**101**(1987), 431-449. MR**88a:13001****[Z]**O. Zariski,*Polynomial ideals defined by infinitely near base-points*, Amer. J. Math.**60**(1938), 151-204.**[ZS]**O. Zariski and P. Samuel,*Commutative Algebra*, vol. II, Van Nostrand Reinhold, New York, New York, 1960. MR**22:11006**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
13B21,
13B22,
13H05,
13H15

Retrieve articles in all journals with MSC (1991): 13B21, 13B22, 13H05, 13H15

Additional Information

**Daniel Katz**

Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045

Email:
dlk@math.ukans.edu

**Vijay Kodiyalam**

Affiliation:
Department of Mathematics, University of Kansas, Lawrence, Kansas 66045

Address at time of publication:
Vijay Kodiyalam, Institute of Mathematical Sciences, Tharamani, Madras 600 113, India

Email:
vijay@imsc.ernet.in

DOI:
https://doi.org/10.1090/S0002-9947-97-01819-9

Received by editor(s):
March 28, 1995

Additional Notes:
The first author was partially supported by the General Research Fund at the University of Kansas

Article copyright:
© Copyright 1997
American Mathematical Society