Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Formulas for the multiplicity of graded algebras

Author: Yu Xie
Journal: Trans. Amer. Math. Soc. 364 (2012), 4085-4106
MSC (2010): Primary 13H15, 13A30; Secondary 14J70, 14B05
Published electronically: March 28, 2012
MathSciNet review: 2912446
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $ R$ by means of local $ j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $ A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $ A$ in terms of that of $ B$ and of local $ j$-multiplicities of hyperplane sections along $ {\rm Proj}\,(B)$. Our formulas can be used to find the multiplicity of special fiber rings and to obtain the degree of dual varieties for any hypersurface. In particular, it gives a generalization of Teissier's Plücker formula to hypersurfaces with non-isolated singularities. Our work generalizes results by Simis, Ulrich and Vasconcelos on homogeneous embeddings of graded algebras.

References [Enhancements On Off] (What's this?)

  • 1. R. Achilles and M. Manaresi, Multiplicity for ideals of maximal analytic spread and intersection theory, J. Math. Kyoto Univ. 33 (1993), 1029-1046. MR 1251213 (94m:13030)
  • 2. R. Achilles and M. Manaresi, Multiplicities of a bigraded ring and intersection theory, Math. Ann. 309 (1997), 573-591. MR 1483824 (99a:14005)
  • 3. R. Achilles and M. Manaresi, Generalized Samuel multiplicities and applications, Rend. Sem. Mat. Univ. Pol. Torino. 64 (2006), 345-372. MR 2295444 (2007m:13031)
  • 4. W. Fulton, Intersection theory, Springer-Verlag, Berlin, 1998. MR 1644323 (99d:14003)
  • 5. H. Flenner and M. Manaresi, A numerical characterization of reduction ideals, Math. Z. 238 (2001), 205-214. MR 1860742 (2002h:13034)
  • 6. H. Flenner, B. Ulrich and W. Vogel, On limits of joins of maximal dimension, Math. Ann. 308 (1997), 291-318. MR 1464904 (98h:13013)
  • 7. R. Hartshorne, C. Huneke and B. Ulrich, Residual intersections of licci ideals are glicci, preprint.
  • 8. C. Huneke, Strongly Cohen-Macaulay schemes and residual intersections, Trans. Amer. Math. Soc. 277 (1983), 739-763. MR 694386 (84m:13023)
  • 9. C. Huneke and B. Ulrich, Residual intersections, J. Reine Angew. Math. 390 (1988), 1-20. MR 953673 (89j:13024)
  • 10. C. Huneke and B. Ulrich, Generic residual intersections, Commutative algebra (Salvador, 1988), Lecture Notes in Math. 1430, Springer, Berlin, 1990, 47-60. MR 1068323 (91h:13013)
  • 11. S. L. Kleiman, A generalized Teissier-Plücker formula, Contemp. Math. 162 (1994), 249-260. MR 1272702 (95c:14073)
  • 12. M. Kline, Mathematical thought from ancient to modern times, Oxford Univ. Press, 1972. MR 0472307 (57:12010)
  • 13. A. Kustin and B. Ulrich, A family of complexes associated to an almost alternating map; with applications to residual intersection, Memoirs AMS 95 (1992), no 461. MR 1091668 (92i:13012)
  • 14. D.G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50 (1953), 145-158. MR 0059889 (15:596a)
  • 15. P. Samuel, La notion de multiplicité en algèbre et en géométrie algébrique, J. Math. Pures Appl. 30 (1951), 159-274. MR 0048103 (13:980c)
  • 16. J-P. Serre, Algèbre locale. Multiplicités, Springer-Verlag, Berlin-New York, 1965. MR 0201468 (34:1352)
  • 17. A. Simis, B. Ulrich and W. V. Vasconcelos, Codimension, multiplicity and integral extensions, Math. Proc. Camb. Phil. Soc. 130 (2001), 237-257. MR 1806775 (2002c:13017)
  • 18. B. Teissier, Sur diverses conditions numériques d'équisingularité des familles de courbes, Centre de Math. École Polytechnique, 1975.
  • 19. A. Thorup, Generalized Plücker formulas, Recent progress in intersection theory (Bologna, 1997), Trends Math. Birkhäuser Boston, Boston, MA, 2000, 299-327. MR 1849300 (2002h:14007)
  • 20. B. Ulrich, Artin-Nagata properties and reductions of ideals, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math. 159, Amer. Math. Soc. Providence, RI, 1994, 373-400. MR 1266194 (95a:13017)
  • 21. J. Validashti, Multiplicities of graded algebras, Ph.D. thesis, Purdue University, 2007.
  • 22. W.V. Vasconcelos, The reduction number of an algebra, Compositio Math. 104 (1996), 189-197. MR 1421399 (98f:13001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 13H15, 13A30, 14J70, 14B05

Retrieve articles in all journals with MSC (2010): 13H15, 13A30, 14J70, 14B05

Additional Information

Yu Xie
Affiliation: Department of Mathematics, The University of Notre Dame, South Bend, Indiana 46556

Keywords: $j$-multiplicity, associated graded ring, special fiber ring, dual variety, Plücker formula
Received by editor(s): October 7, 2009
Received by editor(s) in revised form: July 26, 2010
Published electronically: March 28, 2012
Additional Notes: This paper is based on the author’s Ph.D. thesis, written under the direction of Professor Bernd Ulrich. The author sincerely thanks Professor Ulrich for suggesting the problem and for advice and many helpful discussions.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society