A special case of positivity (II)
Author:
S. P. Dutta
Journal:
Proc. Amer. Math. Soc. 133 (2005), 1891-1896
MSC (2000):
Primary 13H05, 14C17; Secondary 13D15, 14H15
DOI:
https://doi.org/10.1090/S0002-9939-05-07929-3
Published electronically:
February 24, 2005
MathSciNet review:
2137852
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this note we prove the following special case of Serre's conjecture on Intersection Multiplicity: Let be a regular local ring and let
,
be two prime ideals such that
,
and dimension of
. Then
; here
denotes the Hilbert-Samuel multiplicity for any finitely generated module
with respect to
.
- [B] P. Berthelot, Altérations de variétés algébriques [d'aprés A.J. De Jong], Seminaire Bourbaki 815 (1996). MR 1472543 (98m:14021)
- [D1] S. P. Dutta, Generalized Intersection Multiplicity of Modules, Trans. Amer. Math. Soc. 276 (1983), 657-669. MR 0688968 (84i:13024)
- [D2] -, Frobenius and Multiplicities, J. Algebra 85 No. 2 (1983), 424-448. MR 0725094 (85f:13022)
- [D3] -, A Special Case of Positivity, Proc. Amer. Math. Soc. 103 No. 2 (1988). MR 0943042 (89e:13028)
- [D-H-M] S. P. Dutta, M. Hochster, J. E. Mclaughlin, Modules of Finite Projective Dimension with Negative Intersection Multiplicities, Invent. Math. 79 (1985), 253-291. MR 0778127 (86h:13023)
- [Fo] H.-B. Foxby, The MacRae Invariant, Commutative Algebra (Durham 1981), London Math. Soc. Lecture Note Series 72 (1982), 121-128. MR 0693631 (85e:13033)
- [Fu] W. Fulton, Intersection Theory, Springer-Verlag, Berlin, 1984. MR 0732620 (85k:14004)
- [G-S1] H. Gillet and C. Soulé, K the'orie et nullité des multiplicités d'intersection, C. R. Acad. Sci., Paris Ser. I 300 (1985), 71-74. MR 0777736 (86k:13027)
- [G-S2] -, Intersection Theory using Adams operation, Invent. Math. No. 2, 90 (1987), 243-277. MR 0910201 (89d:14005)
- [H] M. Hochster, Nonnegativity of Intersection Multiplicities in Ramified Regular Local Rings following Gabber/De Jong/Berthelot, Preprint.
- [K-R] K. Kurano and P. Roberts, The Positivity of Intersection Multiplicities and Symbolic Power of Prime Ideals, Compositio Math. 122 (2000), 165-182. MR 1775417 (2001g:13031)
- [L] M. Levine, Localization on Singular Varieties, Invent. Math. 91 (1988), 423-464. MR 0928491 (89c:14015a)
- [P-S1] C. Peskine and L. Szpiro, Sygyzies et multiplicités, C. R. Acad. Sc. Paris, t 278 (27 mai 1974), 1421-1424. MR 0349659 (50:2152)
- [R1] P. Roberts, The Vanishing of Intersection Multiplicities of Perfect Complexes, Bull. Amer. Math. Soc. 13 (1985), 127-130. MR 0799793 (87c:13030)
- [R2] -, Multiplicities and Chern Classes in Local Algebra, Cambridge University Press, 1998. MR 1686450 (2001a:13029)
- [S] J. P. Serre, Algèbre locale, multiplicités, Lecture Notes in Math., vol. 11, 3rd ed., Springer-Verlag Berlin/Heidelberg/New York, 1975. MR 0201468 (34:1352)
- [T] B. R. Tennison, Intersection Multiplicities and Tangent cones, Math. Proc. Cambridge Phil. Soc. 85 (1979), 33-42. MR 0510397 (80d:14005)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13H05, 14C17, 13D15, 14H15
Retrieve articles in all journals with MSC (2000): 13H05, 14C17, 13D15, 14H15
Additional Information
S. P. Dutta
Affiliation:
Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email:
dutta@math.uiuc.edu
DOI:
https://doi.org/10.1090/S0002-9939-05-07929-3
Keywords:
Regular local ring,
Hilbert multiplicity,
intersection multiplicity,
blow-up,
Chow-group,
Riemann-Roch Theorem
Received by editor(s):
September 12, 2003
Received by editor(s) in revised form:
February 17, 2004
Published electronically:
February 24, 2005
Additional Notes:
This research was partially supported by NSF grant DMS 99-70263.
Communicated by:
Bernd Ulrich
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.