Integral closure of a cubic extension and applications

Author:
Sheng-Li Tan

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2553-2562

MSC (2000):
Primary 13B22, 14F05, 14E20

Published electronically:
February 9, 2001

MathSciNet review:
1838377

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we compute the integral closure of a cubic extension over a Noetherian unique factorization domain. We also present some applications to triple coverings and to rank two reflexive sheaves over an algebraic variety.

**[Ha1]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[Ha2]**Robin Hartshorne,*Algebraic vector bundles on projective spaces: a problem list*, Topology**18**(1979), no. 2, 117–128. MR**544153**, 10.1016/0040-9383(79)90030-2**[Ha3]**Robin Hartshorne,*Stable reflexive sheaves*, Math. Ann.**254**(1980), no. 2, 121–176. MR**597077**, 10.1007/BF01467074**[Kap]**Kaplansky, I.,*Commutative Algebra*, W. A. Benjamin Inc., New York, 1970.**[Lan]**Serge Lang,*Old and new conjectured Diophantine inequalities*, Bull. Amer. Math. Soc. (N.S.)**23**(1990), no. 1, 37–75. MR**1005184**, 10.1090/S0273-0979-1990-15899-9**[Laz]**Robert Lazarsfeld,*A Barth-type theorem for branched coverings of projective space*, Math. Ann.**249**(1980), no. 2, 153–162. MR**578722**, 10.1007/BF01351412**[Mir]**Rick Miranda,*Triple covers in algebraic geometry*, Amer. J. Math.**107**(1985), no. 5, 1123–1158. MR**805807**, 10.2307/2374349**[Qui]**Daniel Quillen,*Projective modules over polynomial rings*, Invent. Math.**36**(1976), 167–171. MR**0427303****[ShS]**Harold N. Shapiro and Gerson H. Sparer,*Minimal bases for cubic fields*, Comm. Pure Appl. Math.**44**(1991), no. 8-9, 1121–1136. MR**1127054**, 10.1002/cpa.3160440821**[Sto]**Gabriel Stolzenberg,*Constructive normalization of an algebraic variety*, Bull. Amer. Math. Soc.**74**(1968), 595–599. MR**0224602**, 10.1090/S0002-9904-1968-12023-3**[Sus]**Suslin, A. A.,*Projective modules over a polynomial ring are free*, Soviet Math. Dokl.**17**(1976), 1160-1164.**[Ta1]**Tan, S.-L.,*Cayley-Bacharach property of an algebraic variety and Fujita's conjecture*, J. of Algebraic Geometry**9**(2) (2000), 201-222. CMP**2000:07****[Ta2]**Tan, S.-L.,*Triple coverings on smooth algebraic varieties*, preprint 1999 (Bar-Ilan University).**[TaV]**Tan, S.-L. and Viehweg, E.,*A note on Cayley-Bacharach property for vector bundles*, in: Complex Analysis and Algebraic Geometry (ed: T. Peternell, F.-O. Schreyer), de Gruyter (1999).**[Vas]**Wolmer V. Vasconcelos,*Computing the integral closure of an affine domain*, Proc. Amer. Math. Soc.**113**(1991), no. 3, 633–638. MR**1055780**, 10.1090/S0002-9939-1991-1055780-6

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Additional Information

**Sheng-Li Tan**

Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200062, People’s Republic of China

Email:
sltan@math.ecnu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-01-05902-0

Keywords:
Cubic extension,
integral closure,
covering,
vector bundle

Received by editor(s):
October 18, 1999

Received by editor(s) in revised form:
January 22, 2000

Published electronically:
February 9, 2001

Additional Notes:
This work is partially supported by the Kort Foundation and the Emmy Noether Research Institute for Mathematics. This research is also supported by NSFOY, the 973 Project Foundation and the Doctoral Program Foundation of EMC

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 2001
American Mathematical Society