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Hilbert polynomials and powers of ideals

Published online by Cambridge University Press:  01 November 2008

JÜRGEN HERZOG ,
TONY J. PUTHENPURAKAL  and
JUGAL K. VERMA
JÜRGEN HERZOG
Affiliation:
Fachbereich Mathematik und Informatik, Universität Duisburg-Essen, Campus Essen, 45117 Essen, Germany e-mail: juergen.herzog@uni-essen.de
TONY J. PUTHENPURAKAL
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India e-mail: tputhen@math.iitb.ac.in, jkv@math.iitb.ac.in
JUGAL K. VERMA
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Mumbai 400076, India e-mail: tputhen@math.iitb.ac.in, jkv@math.iitb.ac.in
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Abstract

The growth of Hilbert coefficients for powers of ideals are studied. For a graded ideal I in the polynomial ring S = K[x1, . . ., xn] and a finitely generated graded S-module M, the Hilbert coefficients ei(M/IkM) are polynomial functions. Given two families of graded ideals (Ik)k≥0 and (Jk)k≥0 with Jk ⊂ Ik for all k with the property that JkJℓ ⊂ Jk+ℓ and IkIℓ ⊂ Ik+ℓ for all k and ℓ, and such that the algebras and are finitely generated, we show the function k ↦ e0(Ik/Jk) is of quasi-polynomial type, say given by the polynomials P0,. . ., Pg−1. If Jk = Jk for all k, for a graded ideal J, then we show that all the Pi have the same degree and the same leading coefficient. As one of the applications it is shown that , if I is a monomial ideal. We also study analogous statements in the local case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 3 , November 2008 , pp. 623 - 642
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Amao, J. O.On a certain Hilbert polynomial. J. London Math. Soc. (2) 14 (1976), no. 1, 13–20.CrossRefGoogle Scholar
[2]Brodmann, M.Asymptotic stability of Ass(M/In M). Proc. Amer. Math. Soc. 74 (1979), no. 1, 16–18.Google Scholar
[3]Brodmann, M. and Sharp, R. Y.Local Cohomology: An Algebraic Introduction with Geometric Applications (Cambridge University Press, 1998).CrossRefGoogle Scholar
[4]Bruns, W. and Herzog, J. Cohen-Macaulay Rings, revised edition. Cambridge Studies in Advanced Mathematics, 39 (Cambridge University Press, 1998).CrossRefGoogle Scholar
[5]Bruns, W. and Ichim, B.On the coefficients of Hilbert quasipolynomials. Proc. Amer. Math. Soc. 135 (2007), no. 5, 1305–1308.CrossRefGoogle Scholar
[6]Conca, A., Herzog, J., Trung, N. V. and Valla, G.Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces. Amer. J. Math. 119 (1997), 859–901.CrossRefGoogle Scholar
[7]Cutkosky, D.Symbolic algebras of monomial primes. J. Reine Angew. Math. 416 (1991), 71–89.Google Scholar
[8]Cutkosky, S. D., Ha, H. T., Srinivasan, H. and Theodorescu, E.Asymptotic behaviour of the length of local cohomology. Canad. J. Math. 57 (2005), no. 6, 1178–1192.CrossRefGoogle Scholar
[9]Cutkosky, S. D., Herzog, J. and Trung, N. V.Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compositio Math. 118 (1999), no. 3, 243–261.CrossRefGoogle Scholar
[10]Goto, S., Nishida, K. and Watanabe, K.Non-Cohen-Macaulay symbolic Rees algebras for space monomial curves and counterexamples to Cowsik's question. Proc. Amer. Math. Soc. 120 (1994), 383–392.Google Scholar
[11]Herzog, J., Hibi, T. and Trung, N. V.Symbolic powers of monomial ideals and vertex cover algebras. Adv. Math. 210 (2007), 304–322.CrossRefGoogle Scholar
[12]Hoang, N. D. and Trung, N. V.Hilbert polynomials of non-standard bigraded algebras. Math. Z. 245 (2003), no. 2, 309–334.CrossRefGoogle Scholar
[13]Huneke, C.On the finite generation of symbolic blow-ups. Math. Z. 179 (1982), 465–472.Google Scholar
[14]Huneke, C. and Swanson, I. Integral closures of ideals, rings and modules. London Math. Society Lecture Note Series 336 (Cambridge University Press, 2006).Google Scholar
[15]Kodiyalam, V.Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Amer. Math. Soc. 128 (2000), no. 2, 407–411.CrossRefGoogle Scholar
[16]McMullen, P.Lattice invariant valuations on rational polytopes. Arch. Math. 31 (1978/79), 509–516.Google Scholar
[17]Miller, E. and Sturmfels, B. Combinatorial commutative algebra. Graduate Texts in Mathematics, 227 (Springer-Verlag, 2005).Google Scholar
[18]Puthenpurakal, T. J.Ratliff-Rush Filtration, regularity and depth of higher associated graded modules: Part I. J. Pure Appl. Alg. 208 (2007), 159–176.CrossRefGoogle Scholar
[19]Rees, D.-transforms of local rings and a theorem on multiplicities of ideals. Proc. Camb. Phil. Soc. (2) 57 (1961), 8–17.Google Scholar
[20]Rees, D.A note on analytically unramified local rings. J. London Math. Soc. 36 (1961), 24–28.CrossRefGoogle Scholar
[21]Rees, D.Amao's theorem and reduction criteria. J. London Math. Soc. (2) 32 (1985), no. 3, 404–410.CrossRefGoogle Scholar
[22]Roberts, P. C.A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian. Proc. Amer. Math. Soc. 94 (1985), no. 4, 589–592.Google Scholar
[23]Shah, K.Coefficient ideals. Trans. Amer. Math. Soc. 327 (1991), no. 1, 373–384.CrossRefGoogle Scholar
[24]Stanley, R.Decompositions of rational convex polytopes. Ann. Discr. Math. 6 (1980), 333–342.CrossRefGoogle Scholar
[25]Trung, N. V.Reduction exponent and degree bound for the defining equations of graded rings. Proc. Amer. Math. Soc. 101 (1987), 229–236.CrossRefGoogle Scholar
[26]Trung, N. V.The Castelnuovo regularity of the Rees algebra and the associated graded ring. Trans. Amer. Math. Soc. 350 (1998), 2813–2832.Google Scholar