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Powers of Ideals: Betti Numbers, Cohomology and Regularity

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Commutative Algebra

Abstract

The aim of this paper is to provide an approach to some advances over the last decade concerning homological invariants of powers of a graded ideal that derive from finiteness properties of the Rees algebra of the ideal.

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References

  1. Bagheri, A., Chardin, M., Ha, H.T.: Asymptotic linearity of multigraded betti numbers. arXiv:1110.0383 (2012)

    Google Scholar 

  2. Berlekamp, D.: Regularity defect stabilization of powers of an ideal. Math. Res. Lett. 19(1), 109–119 (2012)

    MathSciNet  Google Scholar 

  3. Brodmann, M.: Asymptotic stability of Ass(M/InM). Proc. Amer. Math. Soc. 74(1), 16–18 (1979)

    MathSciNet  Google Scholar 

  4. Brodmann, M.: The asymptotic nature of the analytic spread. Math. Proc. Cambridge Philos. Soc. 86(1), 35–39 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brodmann, M.: Asymptotic depth and connectedness in projective schemes. Proc. Amer. Math. Soc. 108(3), 573–581 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chardin, M.: Some results and questions on Castelnuovo-Mumford regularity. Syzygies and Hilbert Functions, vol. 140. Lecture Notes in Pure Applied Mathematics, vol. 254. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  7. Chardin, M.: Powers of ideals and the cohomology of stalks and fibers of morphisms. Algebra Number Theory. To appear

    Google Scholar 

  8. Chardin, M., Jouanolou, J.-P., Rahimi, A.: The eventual stability of depth, associated primes and cohomology of a graded module. J. Commut. Algebra. To appear

    Google Scholar 

  9. Chandler, K.A.: Regularity of the powers of an ideal. Comm. Algebra 25(12), 3773–3776 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Conca, A.: Regularity jumps for powers of ideals. Commutative Algebra with a Focus on Geometric and Homological Aspects. In: Lecture Notes in Pure Applied Mathematics, vol. 244. Chapman & Hall/CRC, Boca Raton (2006)

    Google Scholar 

  11. Cutkosky, D.: Irrational asymptotic behaviour of Castelnuovo-Mumford regularity. J. Reine Angew. Math. 522, 93–103 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cutkosky, D., Herzog, J., Trung, N.V.: Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compositio Math. 118(3), 243–261 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Cutkosky, D., Ein, L., Lazarsfeld, R.: Positivity and complexity of ideal sheaves. Math. Ann. 321(2), 213–234 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eisenbud, D., Harris, J.: Powers of ideals and fibers of morphisms. Math. Res. Lett. 17(2), 267–273 (2010)

    MathSciNet  MATH  Google Scholar 

  15. Eisenbud, D., Ulrich, B.: Notes on regularity stabilization. Proc. Amer. Math. Soc. 140(4), 1221–1232 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eisenbud, D., Huneke, C., Ulrich, B.: The regularity of Tor and graded Betti numbers. Amer. J. Math. 128(3), 573–605 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Geramita, A.V., Gimigliano, A., Pitteloud, Y.: Graded Betti numbers of some embedded rational n-folds. Math. Ann. 301(2), 363–380 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hà, H.T.: Asymptotic linearity of regularity and a*-invariant of powers of ideals. Math. Res. Lett. 18(01), 1–9 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Herzog, J., Simis, A., Vasconcelos, W.V.: Koszul homology and blowing-up rings. In: Commutative algebra (Trento, 1981). In: Lecture Notes in Pure and Applied Mathematics, vol. 84. pp. 79–169. Dekker, New York (1983)

    Google Scholar 

  20. Hoa, L.T., Hyry, E.: On local cohomology and Hilbert function of powers of ideals. Manuscripta Math. 112(1), 77–92 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. Johnson, M., Ulrich, B.: Artin–Nagata properties and Cohen-Macaulay associated graded rings. Compositio Math. 108, 7–29 (1996)

    MathSciNet  Google Scholar 

  22. Kodiyalam, V.: Homological invariants of powers of an ideal. Proc. Amer. Math. Soc. 118(3), 757–764 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kodiyalam, V.: Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Amer. Math. Soc. 128(2), 407–411 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. McAdam, S.: Asymptotic prime divisors. In: Lecture Notes in Mathematics, vol. 1023. Springer, Berlin (1983)

    Google Scholar 

  25. McAdam, S., Eakin, P.: The asymptotic ass. J. Algebra 61, 71–81 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Niu, W.: A vanishing theorem and asymptotic regularity of powers of ideal sheaves. arXiv:1009.0314

    Google Scholar 

  27. Niu, W.: Some results on asymptotic regularity of ideal sheaves. arXiv:1106.2585

    Google Scholar 

  28. Ratliff, L.J. Jr.: On prime divisors of I n, n large. Michigan Math. J. 23, 337–352 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Römer, T.: Homological properties of bigraded algebras. Illinois J. Math. 45(4), 1361–1376 (2001)

    MathSciNet  MATH  Google Scholar 

  30. Swanson, I.: Powers of ideals. Primary decompositions, Artin–Rees lemma and regularity. Math. Ann. 307(2), 299–313 (1997)

    Google Scholar 

  31. Trung, T.N.: Regularity index of Hilbert functions of powers of ideals. Proc. Amer. Math. Soc. 137(7), 2169–2174 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Trung, N.V., Wang, H.: On the asymptotic linearity of Castelnuovo–Mumford regularity. J. Pure Appl. Algebra 201(1–3), 42–48 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Whieldon, G.: Stabilization of Betti Tables. arXiv:1106.2355

    Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu, CNRS & UPMC, 4 place Jussieu, F-75005, Paris, France

    Marc Chardin

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  1. Marc Chardin
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Correspondence to Marc Chardin .

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Editors and Affiliations

  1. Cornell University, 547 Malott Hall, Ithaca, 14853, New York, USA

    Irena Peeva

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Chardin, M. (2013). Powers of Ideals: Betti Numbers, Cohomology and Regularity. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_9

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