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Length of local cohomology of powers of ideals. (English) Zbl 1409.13036
The main aim of the paper is to investigate the asymptotic behavior of the length of local cohomology modules of powers of an ideal \(I\). More precisely, if \(I\) is an ideal in a \(d\)-dimensional local noetherian ring \((R, \mathfrak{m}, k)\) or a polynomial ring over a field \(k\) and maximal homogeneous ideal \(\mathfrak{m}\), the authors study situations when the length \(\lambda(H_{\mathfrak{m}}^i (R/I^n))\) is finite for \(n \gg 0\) and consider the existence of the limit \[ \lim_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\tag{*} \] for \(i > 0\). The paper is motivated by results of S. D. Cutkosky [Adv. Math. 264, 55–113 (2014; Zbl 1350.13032)] who proved that this limit exists if \(i=0\) and \(R\) is analytically unramified. The most concrete results of the paper are obtained for monomial ideals in a polynomial ring. In this case, it is proved that \(\lambda(H_{\mathfrak{m}}^i (R/I^n))< \infty \) for \(n \gg 0\) and the sequence \(\{\lambda(H_{\mathfrak{m}}^i (R/I^n))\}_{n \geq 0}\) agrees with a quasi-polynomial of degree \(d\) for \(n \gg 0\). Moreover, the generating function of this sequence is rational. Even though the limit (*) might not exist, a better behavior is obtained by replacing the filtration of powers of \(I\) with the filtration of integral closures \(\{\overline{I^n}\}_{n \geq 1}\), in which case \(\lim_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/\overline{I^n}))}{n^d}\) exists and is a rational number. In the more general case when \(I\) is a homogeneous ideal, the authors prove that \[ \limsup_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n)_{\geq -\alpha n})}{n^d} < \infty \] for every \(\alpha \in \mathbb{Z}\) and leave open the question of whether or not \(\limsup_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\) is always finite. The authors also raise questions about situations when the limit (*) is rational and about finding interpretations of it in terms of volumes of geometric bodies associated with the ideal \(I\). The paper also discusses several classes of ideals for which \(\liminf_{n \to \infty}\frac{\lambda(H_{\mathfrak{m}}^i (R/I^n))}{n^d}\) is positive.

MSC:
13D45 Local cohomology and commutative rings
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
14B05 Singularities in algebraic geometry
05E40 Combinatorial aspects of commutative algebra
Software:
Macaulay2; Normaliz
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References:
[1] Bayer, David; Stillman, Michael, A criterion for detecting \(m\)-regularity, Invent. Math., 87, 1, 1-11, (1987) · Zbl 0625.13003
[2] BBLSZ B. Bhatt, M. Blickle, G. Lyubeznik, A. K. Singh, and W. Zhang, \emph Stabilization of the cohomology of thickenings, preprint, arXiv:1605.09492 (2016).
[3] Bre H. Brenner, \emph Irrational Hilbert–Kunz multiplicities, preprint, arXiv:1305.5873 (2013).
[4] Brenner, Holger; Caminata, Alessio, Generalized Hilbert–Kunz function in graded dimension \(2\), Nagoya Math. J., 230, 1-17, (2018)
[5] Brodmann, M., The asymptotic nature of the analytic spread, Math. Proc. Cambridge Philos. Soc., 86, 1, 35-39, (1979) · Zbl 0413.13011
[6] Brodmann, M. P.; Sharp, R. Y., Local cohomology, \rmAn algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 136, xxii+491 pp., (2013), Cambridge University Press, Cambridge · Zbl 1263.13014
[7] Bruns, Winfried; Herzog, J\"urgen, Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics 39, xii+403 pp., (1993), Cambridge University Press, Cambridge · Zbl 0788.13005
[8] Normaliz W. Bruns, B. Ichim, T. R\`‘omer, R. Sieg, and C. S\'’oger: \emph Normaliz. Algorithms for rational cones and affine monoids. Available at https://www.normaliz.uni-osnabrueck.de.
[9] Cowsik, R. C.; Nori, M. V., On the fibres of blowing up, J. Indian Math. Soc. (N.S.), 40, 1-4, 217-222 (1977), (1976) · Zbl 0437.14028
[10] Cutkosky, Steven Dale, Asymptotic multiplicities of graded families of ideals and linear series, Adv. Math., 264, 55-113, (2014) · Zbl 1350.13032
[11] Cutkosky, Steven Dale, Limits in commutative algebra and algebraic geometry. Commutative algebra and noncommutative algebraic geometry. Vol. I, Math. Sci. Res. Inst. Publ. 67, 141-162, (2015), Cambridge Univ. Press, New York · Zbl 1359.13024
[12] Cutkosky, S. Dale; Herzog, J\"urgen; Trung, Ng\^o Vi\^et, Asymptotic behaviour of the Castelnuovo–Mumford regularity, Compositio Math., 118, 3, 243-261, (1999) · Zbl 0974.13015
[13] Cutkosky, Steven Dale; H\`a, Huy T\`ai; Srinivasan, Hema; Theodorescu, Emanoil, Asymptotic behavior of the length of local cohomology, Canad. J. Math., 57, 6, 1178-1192, (2005) · Zbl 1095.13015
[14] DDM H. Dao, A. De Stefani, and L. Ma, \emph Cohomologically full rings, in progress.
[15] DS H. Dao and I. Smirnov, \emph On generalized Hilbert–Kunz function and multiplicity, preprint, arXiv:1305.1833 (2013).
[16] Dao, Hailong; Watanabe, Kei-ichi, Some computations of the generalized Hilbert–Kunz function and multiplicity, Proc. Amer. Math. Soc., 144, 8, 3199-3206, (2016) · Zbl 1346.13008
[17] de Concini, C.; Eisenbud, David; Procesi, C., Young diagrams and determinantal varieties, Invent. Math., 56, 2, 129-165, (1980) · Zbl 0435.14015
[18] GS D. Grayson and M. Stillman, \emph Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2.
[19] JH D. Hernandez and J. Jeffries, \emph Local Okounkov bodies and limits in prime characteristic, preprint, arXiv:1701.02575 (2017).
[20] Herzog, J\"urgen; Hibi, Takayuki, Monomial ideals, Graduate Texts in Mathematics 260, xvi+305 pp., (2011), Springer-Verlag London, Ltd., London · Zbl 1206.13001
[21] Herzog, J\"urgen; Hibi, Takayuki; Trung, Ng\^o Vi\^et, Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math., 210, 1, 304-322, (2007) · Zbl 1112.13006
[22] Herzog, J\"urgen; Puthenpurakal, Tony J.; Verma, Jugal K., Hilbert polynomials and powers of ideals, Math. Proc. Cambridge Philos. Soc., 145, 3, 623-642, (2008) · Zbl 1157.13013
[23] Huneke, Craig; Swanson, Irena, Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series 336, xiv+431 pp., (2006), Cambridge University Press, Cambridge · Zbl 1117.13001
[24] Jeffries, Jack; Monta\~no, Jonathan, The \(j\)-multiplicity of monomial ideals, Math. Res. Lett., 20, 4, 729-744, (2013) · Zbl 1295.13035
[25] Jeffries, Jack; Monta\~no, Jonathan; Varbaro, Matteo, Multiplicities of classical varieties, Proc. Lond. Math. Soc. (3), 110, 4, 1033-1055, (2015) · Zbl 1314.13042
[26] Katz, Daniel; Validashti, Javid, Multiplicities and Rees valuations, Collect. Math., 61, 1, 1-24, (2010) · Zbl 1216.13016
[27] Kodiyalam, Vijay, Asymptotic behaviour of Castelnuovo–Mumford regularity, Proc. Amer. Math. Soc., 128, 2, 407-411, (2000) · Zbl 0929.13004
[28] Kunz, Ernst, Characterizations of regular local rings for characteristic \(p\), Amer. J. Math., 91, 772-784, (1969) · Zbl 0188.33702
[29] LT H. M. Lam and N. V. Trung, \emph Associated primes of powers of edge ideals and ear decompositions of graphs, preprint, arXiv:1506.01483 (2015).
[30] Lyubeznik, Gennady, Finiteness properties of local cohomology modules (an application of \(D\)-modules to commutative algebra), Invent. Math., 113, 1, 41-55, (1993) · Zbl 0795.13004
[31] Linquan1 L. Ma and P. H. Quy, \emph Frobenius actions on local cohomology modules and deformation, to appear in Nagoya Math. J., arXiv:1606.02059 (2016).
[32] McMullen, P., Lattice invariant valuations on rational polytopes, Arch. Math. (Basel), 31, 5, 509-516, (1978/79) · Zbl 0387.52007
[33] Minh, Nguyen Cong; Trung, Ngo Viet, Cohen–Macaulayness of powers of two-dimensional squarefree monomial ideals, J. Algebra, 322, 12, 4219-4227, (2009) · Zbl 1206.13028
[34] PAG R. Lazarsfeld, \emph Positivity in algebraic geometry I and II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vols. 48 and 49. Springer-Verlag, Berlin, 2004.
[35] Raicu, Claudiu, Regularity and cohomology of determinantal thickenings, Proc. Lond. Math. Soc. (3), 116, 2, 248-280, (2018) · Zbl 1397.13023
[36] Sbarra, Enrico, Upper bounds for local cohomology for rings with given Hilbert function, Comm. Algebra, 29, 12, 5383-5409, (2001) · Zbl 1097.13516
[37] Schwede, Karl, \(F\)-injective singularities are Du Bois, Amer. J. Math., 131, 2, 445-473, (2009) · Zbl 1164.14001
[38] Simis, Aron; Vasconcelos, Wolmer V.; Villarreal, Rafael H., On the ideal theory of graphs, J. Algebra, 167, 2, 389-416, (1994) · Zbl 0816.13003
[39] Stanley, Richard P., Decompositions of rational convex polytopes \rm, Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Ann. Discrete Math., 6, 333-342, (1980)
[40] Singh, Anurag K.; Walther, Uli, Local cohomology and pure morphisms, Illinois J. Math., 51, 1, 287-298, (2007) · Zbl 1133.13019
[41] Takayama, Yukihide, Combinatorial characterizations of generalized Cohen–Macaulay monomial ideals, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 48(96), 3, 327-344, (2005) · Zbl 1092.13020
[42] Ulrich, Bernd; Validashti, Javid, Numerical criteria for integral dependence, Math. Proc. Cambridge Philos. Soc., 151, 1, 95-102, (2011) · Zbl 1220.13006
[43] Vasconcelos, Wolmer, Integral closure, \rmRees algebras, multiplicities, algorithms, Springer Monographs in Mathematics, xii+519 pp., (2005), Springer-Verlag, Berlin · Zbl 1082.13006
[44] Woods, Kevin, Presburger arithmetic, rational generating functions, and quasi-polynomials, J. Symb. Log., 80, 2, 433-449, (2015) · Zbl 1353.03074
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