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Noether normalizations, reductions of ideals, and matroids


Authors: Joseph P. Brennan and Neil Epstein
Journal: Proc. Amer. Math. Soc. 139 (2011), 2671-2680
MSC (2010): Primary 13A30; Secondary 05B35, 13B21, 13H15
DOI: https://doi.org/10.1090/S0002-9939-2011-10719-6
Published electronically: January 24, 2011
MathSciNet review: 2801606
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Abstract: We show that given a finitely generated standard graded algebra of dimension $ d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $ d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $ k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which is a common generalization of a topological space and a matroid.


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  • [BH97] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, revised ed., Cambridge Studies in Advanced Mathematics, no. 39, Cambridge Univ. Press, Cambridge, 1997.
  • [Eps05] Neil Epstein, A tight closure analogue of analytic spread, Math. Proc. Cambridge Philos. Soc. 139 (2005), no. 2, 371-383. MR 2168094 (2006e:13003)
  • [Eps10] -, Reductions and special parts of closures, J. Algebra 323 (2010), no. 8, 2209-2225. MR 2596375
  • [HH02] Jürgen Herzog and Takayuki Hibi, Discrete polymatroids, J. Algebraic Combin. 16 (2002), no. 3, 239-268 (2003). MR 1957102 (2004c:52017)
  • [HHRT97] Manfred Herrmann, Eero Hyry, Jürgen Ribbe, and Zhongming Tang, Reduction numbers and multiplicities of multigraded structures, J. Algebra 197 (1997), no. 2, 311-341. MR 1483767 (98k:13006)
  • [HS06] Craig Huneke and Irena Swanson, Integral closure of ideals, rings, and modules, London Math. Soc. Lecture Note Ser., vol. 336, Cambridge Univ. Press, Cambridge, 2006. MR 2266432 (2008m:13013)
  • [Hyr99] Eero Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2213-2232. MR 1467469 (99i:13005)
  • [KR94] D. Kirby and D. Rees, Multiplicities in graded rings. I. The general theory, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 209-267. MR 1266185 (95b:13002)
  • [Kun85] Ernst Kunz, Introduction to commutative algebra and algebraic geometry, Birkhäuser Boston Inc., Boston, MA, 1985. MR 789602 (86e:14001)
  • [Kun86] Joseph P. S. Kung, A source book in matroid theory, Birkhäuser Boston Inc., Boston, MA, 1986, with a foreword by Gian-Carlo Rota. MR 0890330 (88e:05028)
  • [NR54] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1954), no. 2, 145-158. MR 0059889 (15:596a)
  • [O'C87] Liam O'Carroll, On two theorems concerning reductions in local rings, J. Math. Kyoto Univ. 27 (1987), no. 1, 61-67. MR 878490 (87m:13034)
  • [O'C05] -, Around the Eakin-Sathaye theorem, J. Algebra 291 (2005), no. 1, 259-268. MR 2158522 (2006m:13005)
  • [OH98] Hidefumi Ohsugi and Takayuki Hibi, Normal polytopes arising from finite graphs, J. Algebra 207 (1998), no. 2, 409-426. MR 1644250 (2000a:13010)
  • [Oxl92] James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587 (94d:05033)
  • [Ree84] D. Rees, Generalizations of reductions and mixed multiplicities, J. London Math. Soc. (2) 29 (1984), no. 3, 397-414. MR 754926 (86e:13023)
  • [RS88] D. Rees and Judith D. Sally, General elements and joint reductions, Michigan Math. J. 35 (1988), no. 2, 241-254. MR 959271 (89h:13034)
  • [Sim94] Robert Samuel Simon, Combinatorial properties of ``cleanness'', J. Algebra 167 (1994), no. 2, 361-388. MR 1283293 (96c:13018)
  • [SoP75] J. M. S. Simões Pereira, On matroids on edge sets of graphs with connected subgraphs as circuits. II, Discrete Math. 12 (1975), 55-78. MR 0419275 (54:7298)
  • [SVV98] Aron Simis, Wolmer V. Vasconcelos, and Rafael H. Villarreal, The integral closure of subrings associated to graphs, J. Algebra 199 (1998), no. 1, 281-289. MR 1489364 (99c:13004)
  • [Swa92] Irena Swanson, Joint reductions, tight closure, and the Briançon-Skoda theorem, J. Algebra 147 (1992), no. 1, 128-136. MR 1154678 (93g:13001)
  • [Swa94] -, Joint reductions, tight closure, and the Briançon-Skoda theorem. II, J. Algebra 170 (1994), no. 2, 567-583. MR 1302856 (95m:13003)
  • [Tan99] Zhongming Tang, A note on the Cohen-Macaulayness of multi-Rees rings, Comm. Algebra 27 (1999), no. 12, 5967-5974. MR 1726287 (2000m:13005)
  • [Ver91] J. K. Verma, Joint reductions and Rees algebras, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 2, 335-342. MR 1085400 (92h:13007)
  • [Wag99] David G. Wagner, Algebras related to matroids represented in characteristic zero, European J. Combin. 20 (1999), no. 7, 701-711. MR 1721927 (2001j:13002)
  • [Whi86] Neil White (ed.), Theory of matroids, Encyclopedia of Mathematics and its Applications, vol. 26, Cambridge Univ. Press, Cambridge, 1986. MR 849389 (87k:05054)
  • [Yao02] Yongwei Yao, Primary decomposition: compatibility, independence and linear growth, Proc. Amer. Math. Soc. 130 (2002), no. 6, 1629-1637. MR 1887009 (2003a:13021)

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Additional Information

Joseph P. Brennan
Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, Orlando, Florida 32816
Email: jpbrenna@mail.ucf.edu

Neil Epstein
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
Address at time of publication: Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany
Email: nepstein@uos.de

DOI: https://doi.org/10.1090/S0002-9939-2011-10719-6
Received by editor(s): December 13, 2009
Received by editor(s) in revised form: August 1, 2010
Published electronically: January 24, 2011
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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