Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T16:32:11.624Z Has data issue: false hasContentIssue false
  • English
  • Français

TOWARD HILBERT–KUNZ DENSITY FUNCTIONS IN CHARACTERISTIC 0

Part of: Curves Local rings and semilocal rings Surfaces and higher-dimensional varieties Commutative algebra: Homological methods

Published online by Cambridge University Press:  08 March 2018

VIJAYLAXMI TRIVEDI
VIJAYLAXMI TRIVEDI*
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai-400005, India email vija@math.tifr.res.in
Get access Rights & Permissions [Opens in a new window]

Abstract

For a pair $(R,I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic 0, and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim _{p\rightarrow \infty }e_{HK}(R_{p},I_{p})$ is equivalent, for any fixed $m\geqslant d-1$, to the existence of $\lim _{p\rightarrow \infty }\ell (R_{p}/I_{p}^{[p^{m}]})/p^{md}$. This we get as a consequence of Theorem 1.1: as $p\longrightarrow \infty$, the convergence of the Hilbert–Kunz (HK) density function $f(R_{p},I_{p})$ is equivalent to the convergence of the truncated HK density functions $f_{m}(R_{p},I_{p})$ (in $L^{\infty }$ norm) of the mod $p$reductions$(R_{p},I_{p})$, for any fixed $m\geqslant d-1$. In particular, to define the HK density function $f_{R,I}^{\infty }$ in char 0, it is enough to prove the existence of $\lim _{p\rightarrow \infty }f_{m}(R_{p},I_{p})$, for any fixed $m\geqslant d-1$. This allows us to prove the existence of $e_{HK}^{\infty }(R,I)$ in many new cases, for example, when Proj R is a Segre product of curves.

MSC classification

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 14H60: Vector bundles on curves and their moduli 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 13H15: Multiplicity theory and related topics
Type
Article
Information
Nagoya Mathematical Journal , Volume 235 , September 2019 , pp. 158 - 200
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brenner, H., The rationality of the Hilbert–Kunz multiplicity in graded dimension two , Math. Ann. 334 (2006), 91–110.Google Scholar
Brenner, H., Li, J. and Miller, C., A direct limit for limit Hilbert–Kunz multiplicity for smooth projective curevs , J. Algebra 372 (2012), 488–504.10.1016/j.jalgebra.2012.10.004Google Scholar
Bruns, W., Conic divisor classes over a normal monoid algebra , Contemp. Math. 390 (2005), 63–71; Commutative algebra and algebraic geometry.10.1090/conm/390/07293Google Scholar
Buchweitz, R. and Chen, Q., Hilbert–Kunz functions of cubic curves and surfaces , J. Algebra 197 (1997), 246–267.Google Scholar
Buchweitz, R., Chen, Q. and Pardue, K., Hilbert–Kunz functions, preprint (Algebraic Geometry e-print series).Google Scholar
Conca, A., Hilbert–Kunz function of monomial ideals and binomial hypersurfaces , Manuscripta Math. 90 (1996), 287–300.10.1007/BF02568307Google Scholar
Eto, K., Multiplicity and Hilbert–Kunz multiplicity of monoid rings , Tokyo J. Math. 25 (2002), 241–245.10.3836/tjm/1244208851Google Scholar
Fakhruddin, N. and Trivedi, V., Hilbert–Kunz functions and multiplicities for full flag varieties and elliptic curves , J. Pure Appl. Algebra 181 (2003), 23–52.10.1016/S0022-4049(02)00304-3Google Scholar
Gessel, I. and Monsky, M., The limit as $p\rightarrow \infty$ of the Hilbert–Kunz multiplicity of $\sum \,x_{i}^{d_{i}}$ , preprint, 2010, arXiv:1007.2004.Google Scholar
Grothendieck, A., Berthellot, P. and Illusie, L., SGA6, Théorie des Intersections et Théorème de Riemann–Roch, Lect. Notes in Math. 225 , Springer, Heidelberg, 1971.Google Scholar
Grothendieck, A. and Dieudonné, J. A., Eléments de Géometrie Algébrique IV , Publ. Math. Inst. Hautes Études Sci 28(1968) 5–255.Google Scholar
Hans, C. and Monsky, P., Some surprising Hilbert–Kunz functions , Math. Z. 214 (1993), 119–135.Google Scholar
Hartshorne, R., Algebraic Geometry, Springer, New York.Google Scholar
Langer, A., Semistable sheaves in positive characteristic , Ann. of Math. (2) 159 (2004).Google Scholar
Maruyama, M., Openness of a family of torsion free sheaves , J. Math. Kyoto Univ. 16(3) (1976), 627–637.10.1215/kjm/1250522875Google Scholar
Monsky, P., The Hilbert–Kunz function , Math. Ann. 263 (1983), 43–49.10.1007/BF01457082Google Scholar
Monsky, P., The Hilbert–Kunz functions of a characteistic 2 cubic , J. Algebra 197 (1997), 268–277.Google Scholar
Monsky, P., The Hilbert–Kunz multiplicity of an irreducible trinomial , J. Algebra 304 (2006), 1101–1107.Google Scholar
Paranjape, K. and Ramanan, S., “ On the canonical rings of a curve ”, in Algebraic Geometry and Commutative Algebra, Vol. II\unskip , Kinokunia, Tokyo, 1968, 503–616.Google Scholar
Pardue, K., Thesis, Brandeis University (1993).Google Scholar
Trivedi, V., Semistability and HK multiplicity for curves , J. Algebra 284 (2005), 627–644.10.1016/j.jalgebra.2004.10.016Google Scholar
Trivedi, V., Hilbert–Kunz multiplicity and reduction mod p , Nagoya Math. J. 85 (2007), 123–141.Google Scholar
Trivedi, V., Semistability of syzygy bundles on projective spaces in positive characteristic , Int. J. Math. 21(11) (2010), 1475–1504.Google Scholar
Trivedi, V., Frobenius pull backs of vector bundles in higher dimensions , Proc. Indian Acad. Sci. Math. Sci. 122(4) (2012), 615–628.10.1007/s12044-012-0097-0Google Scholar
Trivedi, V., Arithmetic behaviour of Frobenius semistability of syzygy bundles for plane trinomial curves , Bull. Sci. Math. 141(7) (2017), 703–726.10.1016/j.bulsci.2017.07.008Google Scholar
Trivedi, V., Hilbert–Kunz density function and Hilbert–Kunz multiplicity, Trans. Amer. Math. Soc., to appear, doi:10.1090/tran/7268.Google Scholar
Watanabe, K., Hilbert–Kunz multiplicity of toric rings , Proc. Inst. Nat. Sci. (Nihon Univ.) 35 (2000), 173–177.Google Scholar
Watanabe, K. and Yoshida, K., Hilbert–Kunz multiplicity and an inequality between multiplicity and colength , J. Algebra 230 (2000), 295–317.Google Scholar
Watanabe, K. and Yoshida, K., Hilbert–Kunz multiplicity, McKay correspondence and good ideals in two dimensional rational singularities , Manuscripta Math. 104 (2001), 275–294.Google Scholar