CASTELNOUVO-MUMFORD REGULARITY OF GRADED MODULES HAVING A LINEAR FREE PRESENTATION

  • Ahn, Jeaman (Department of Mathematics Education, Kongju National University)
  • Received : 2009.09.25
  • Accepted : 2009.11.24
  • Published : 2009.12.30

Abstract

In this paper we investigate the upper bound on the Castelnuovo-Mumford regularity of a graded module with linear free presentation. Let M be a finitely generated graded module over a polynomial ring R with zero dimensional support. We prove that if M is generated by elements of degree $d{\geq}0$ with a linear free presentation $$\bigoplus^p{R}(-d-1)\longrightarrow^{\phi}\bigoplus^q{R}(-d){\longrightarrow}M{\longrightarrow}0$$, then the Castelnuovo-Mumford regularity of M is at most d+q-1. As an important application, we can prove vector bundle technique, which was used in [11], [13], [17] as a tool for obtaining several remarkable results.

Keywords

References

  1. A. Bertram, L. Ein, and R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer. Math. Soc. 4 (1991), no. 3, 587-602. https://doi.org/10.1090/S0894-0347-1991-1092845-5
  2. D. Bayer and D. Mumford, 1993. "hat can be computed in algebraic geometry-" pp. 1-48 in Computational algebraic geometry and commutative algebra(Cortona, 1991), Sympos. Math. 34, Cambridge Univ. Press, Cambridge.
  3. G. Caviglia and E. Sbarra, Characteristic-free bounds for the Castelnuovo-Mumford regularity, Compos. Math. 141 (2005), no. 6, 1365-1373. https://doi.org/10.1112/S0010437X05001600
  4. D. Eisenbud, Commutative Algebra with a view Toward Algebraic Geometry, no. 150, Springer-Velag New York, (1995).
  5. Eisenbud, D., 2005. The geometry of syzygies: A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, 229. Springer-Verlag, New York.
  6. D. Eisenbud and S. Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), 89-133. https://doi.org/10.1016/0021-8693(84)90092-9
  7. D. Eisenbud, M. Green, K. Hulek, and S. Popescu, Restriction linear syzygies: algebra and geometry, Compositio Math. 141 (2005), 1460-1478. https://doi.org/10.1112/S0010437X05001776
  8. M. Giusti, Some effectivity problems in polynomial ideal theory, EUROSAM 84, Lecture Notes in Computer Science 204 (1984), Springer-Verlag, 159-171.
  9. A. Galligo, Theoreme de division et stabilite en geometrie analytique locale, Ann. Inst. Fourier (Grenoble) 29 (1979), 107-184.
  10. D. Giaimo, On the Castelnuovo-Mumford regularity of connected curves, Trans. Amer. Math. Soc. 358 (2006), no. 1, 267-284 https://doi.org/10.1090/S0002-9947-05-03671-8
  11. L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo and the equationsndefining projective varieties, Inv. Math. 72 (1983), 491-506. https://doi.org/10.1007/BF01398398
  12. S. Kwak, Generic projections, the equations defining projective varieties and Castelnuovo regularity, Math. Z. 234, (2000), no. 3, 413-434. https://doi.org/10.1007/PL00004809
  13. S. Kwak and E. Park, Some effects of property Np on the higher normality and defining equations of nonlinearly normal varieties, J. Reine Angew. Math. 582 (2005), 87–105.
  14. R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces, Duke Math. J. 55 (1987), no. 2, 423-429. https://doi.org/10.1215/S0012-7094-87-05523-2
  15. E. Mayr and A. Meyer, The complexity of the word problems for commutative semigroups and polynomial ideals, Adv. in Math. 46 (1982), no. 3, 305-329. https://doi.org/10.1016/0001-8708(82)90048-2
  16. D. Mumford, Lectures on Curves on an Algebraic Surface, Annals of Math. Studies 59, Princeton University Press, Princeton, NJ.
  17. A. Noma, A bound on the Castelnuovo-Mumford regularity for curves, Math. Ann. 322 (2002), 69-74. https://doi.org/10.1007/s002080100265
  18. I. Peeva and B. Sturmfels, Syzygies of codimension 2 lattice ideals, Math. Z. 229 (1998), no. 1, 163-194. https://doi.org/10.1007/PL00004645
  19. J. Stuckrad and W. Vogel, Castelnuovo's regularity and multiplicity, Math. Ann. 281 (1998), no. 3, 355-368. https://doi.org/10.1007/BF01457149