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Analytic spread and non-vanishing of asymptotic depth

Published online by Cambridge University Press:  08 March 2017

CLETO B. MIRANDA–NETO
CLETO B. MIRANDA–NETO*
Affiliation:
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB, Brazil. e-mail: [email protected]
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Abstract

Let S be a polynomial ring over a field K of characteristic zero and let M ⊂ S be an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case where M is a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime ideal P ⊂ S containing M be such that the localisation MP has non-maximal analytic spread. Our technique describes, in fact, a concrete obstruction for P to be an asymptotic prime divisor of M with respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 2 , September 2017 , pp. 289 - 299
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Bär, R. and Brodmann, M. Asymptotic depth of twisted higher direct image sheaves. Proc. Amer. Math. Soc. 137 (2009), 1945–1950.Google Scholar
[2] Brodmann, M. Asymptotic stability of Ass(M/I n M). Proc. Amer. Math. Soc. 74 (1979), 16–18.Google Scholar
[3] Brodmann, M. On the asymptotic nature of the analytic spread. Math. Proc. Camb. Phil. Soc. 86 (1979), 35–39.Google Scholar
[4] Brodmann, M. Asymptotic depth and connectedness in projective schemes. Proc. Amer. Math. Soc. 108 (1990), 573–581.Google Scholar
[5] Brumatti, P. and Simis, A. The module of derivations of a Stanley–Reisner ring. Proc. Amer. Math. Soc. 123 (1995), 1309–1318.Google Scholar
[6] Burch, L. Codimension and analytic spread. Math. Proc. Camb. Phil. Soc. 72 (1972), 369–373.CrossRefGoogle Scholar
[7] Eisenbud, D. and Huneke, C. Cohen–Macaulay Rees algebras and their specialisations. J. Algebra 81 (1983), 202–224.Google Scholar
[8] Ghosh, D. and Puthenpurakal, T. J. Asymptotic prime divisors over complete intersection rings. Math. Proc. Camb. Phil. Soc. 160 (2016), 423–436.Google Scholar
[9] Herzog, J., Simis, A. and Vasconcelos, W. V. Arithmetic of normal Rees algebras. J. Algebra 143 (1991), 269–294.Google Scholar
[10] Herzog, J. and Vladoiu, M. Monomial ideals with primary components given by powers of monomial prime ideals. Electron. J. Combin. 21 (2014), #P1.69.Google Scholar
[11] Hochster, M. Criteria for equality of ordinary and symbolic powers of primes. Math. Z. 133 (1973), 53–65.Google Scholar
[12] Huneke, C. On the associated graded ring of an ideal. Illinois J. Math. 26 (1982), 121–137.Google Scholar
[13] Kaplansky, I. An Introduction to Differential Algebra (Hermann, Paris, 1957).Google Scholar
[14] Martínez–Bernal, J., Morey, S. and Villarreal, R. H. Associated primes of powers of edge ideals. Collect. Math. 63 (2012), 361–374.Google Scholar
[15] McAdam, S. and Eakin, P. The asymptotic Ass. J. Algebra 61 (1979), 71–81.Google Scholar
[16] McAdam, S. Asymptotic prime divisors and analytic spreads. Proc. Amer. Math. Soc. 80 (1980), 555–559.Google Scholar
[17] McAdam, S. Asymptotic prime divisors and going down. Pacific J. Math. 91 (1980), 179–186.Google Scholar
[18] McAdam, S. Asymptotic Prime Divisors. Lecture Notes in Math. 1023 (Springer-Verlag, New York, 1983).Google Scholar
[19] Miranda–Neto, C. B. Tangential idealisers and differential ideals. Collect. Math. 67 (2016), 311–328.CrossRefGoogle Scholar
[20] Northcott, D. G. and Rees, D. Reductions of ideals in local rings. Math. Proc. Camb. Phil. Soc. 50 (1954), 145–158.Google Scholar
[21] Ratliff, L. J. Jr., On prime divisors of I n , n large. Michigan Math. J. 23 (1976), 337–352.Google Scholar
[22] Ratliff, L. J. Jr., Integrally closed ideals and asymptotic prime divisors. Pacific J. Math. 91 (1980), 409–431.Google Scholar
[23] Ratliff, L. J. Jr., Note on asymptotic prime divisors, analytic spreads and the altitude formula. Proc. Amer. Math. Soc. 82 (1981), 1–6.Google Scholar
[24] Ratliff, L. J. Jr., Asymptotic sequences. J. Algebra 85 (1983), 337–360.Google Scholar
[25] Ratliff, L. J. Jr., On asymptotic prime divisors. Pacific J. Math. 111 (1984), 395–413.Google Scholar
[26] Ratliff, L. J. Jr., Five notes on asymptotic prime divisors. Math. Z. 190 (1985), 567–581.Google Scholar
[27] Rees, D. Valuations associated with ideals. Proc. London Math. Soc. 6 (1956), 161–174.Google Scholar
[28] Rees, D. Rings associated with ideals and analytic spreads. Math. Proc. Camb. Phil. Soc. 89 (1981), 423–432.Google Scholar
[29] Swanson, I. and Huneke, C. Integral Closure of Ideals, Rings and Modules (Cambridge University Press, 2006).Google Scholar
[30] Tadesse, Y. Derivations preserving a monomial ideal. Proc. Amer. Math. Soc. 137 (2009), 2935–2942.Google Scholar
[31] Vasconcelos, W. V. Arithmetic of Blowup Algebras. London Math. Soc., Lecture Note Series 195 (Cambridge University Press, 1994).CrossRefGoogle Scholar
[32] Vasconcelos, W. V. Computational Methods in Commutative Algebra and Algebraic Geometry (Springer, Heidelberg, 1998).Google Scholar
[33] Villarreal, R. H. Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 238 (Marcel Dekker, Inc., New York, 2001).Google Scholar