×

On the structure of the Sally module and the second normal Hilbert coefficient. (English) Zbl 1436.13033

The authors give a complete structure of the Sally module in the case the second normal Hilbert coefficient attains almost minimal value in an analytically unramified Cohen-Macaulay local ring. They present a complete description of the Hilbert function of the associated graded ring of the normal filtration. The Hilbert coefficients of the normal filtration give important geometric information on the base ring like the pseudo-rationality. The Sally module was introduced by W. V. Vasconcelos [Contemp. Math. 159, 401–422 (1994; Zbl 0803.13012)] and it is useful to connect the Hilbert coefficients to the homological properties of the associated graded module of a Noetherian filtration. This study is related to a longstanding conjecture stated by S. Itoh [J. Algebra 150, No. 1, 101–117 (1992; Zbl 0756.13008)].

MSC:

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Blancafort, Cristina, On Hilbert functions and cohomology, J. Algebra, 192, 1, 439-459 (1997) · Zbl 0884.13008 · doi:10.1006/jabr.1996.6937
[2] Corso, Alberto; Polini, Claudia; Rossi, Maria Evelina, Depth of associated graded rings via Hilbert coefficients of ideals, J. Pure Appl. Algebra, 201, 1-3, 126-141 (2005) · Zbl 1103.13005 · doi:10.1016/j.jpaa.2004.12.045
[3] Corso, Alberto; Polini, Claudia; Rossi, Maria Evelina, Bounds on the normal Hilbert coefficients, Proc. Amer. Math. Soc., 144, 5, 1919-1930 (2016) · Zbl 1338.13010 · doi:10.1090/proc/12858
[4] GMV19 K. Goel, V. Mukundan, and J. K. Verma, On the vanishing of the normal Hilbert coefficients of ideals, preprint, available at arxiv1901.06310, J. Ramanujan Math. Soc., to appear. · Zbl 1452.13020
[5] Hoa, L\^e Tu\^an; Zarzuela, Santiago, Reduction number and \(a\)-invariant of good filtrations, Comm. Algebra, 22, 14, 5635-5656 (1994) · Zbl 0843.13002 · doi:10.1080/00927879408825151
[6] Hong, Jooyoun; Ulrich, Bernd, Specialization and integral closure, J. Lond. Math. Soc. (2), 90, 3, 861-878 (2014) · Zbl 1319.13002 · doi:10.1112/jlms/jdu053
[7] Huckaba, Sam; Huneke, Craig, Normal ideals in regular rings, J. Reine Angew. Math., 510, 63-82 (1999) · Zbl 0923.13005 · doi:10.1515/crll.1999.049
[8] Huckaba, Sam; Marley, Thomas, Hilbert coefficients and the depths of associated graded rings, J. London Math. Soc. (2), 56, 1, 64-76 (1997) · Zbl 0910.13008 · doi:10.1112/S0024610797005206
[9] Huneke, Craig, Hilbert functions and symbolic powers, Michigan Math. J., 34, 2, 293-318 (1987) · Zbl 0628.13012 · doi:10.1307/mmj/1029003560
[10] Itoh, Shiroh, Integral closures of ideals generated by regular sequences, J. Algebra, 117, 2, 390-401 (1988) · Zbl 0653.13003 · doi:10.1016/0021-8693(88)90114-7
[11] Itoh, Shiroh, Coefficients of normal Hilbert polynomials, J. Algebra, 150, 1, 101-117 (1992) · Zbl 0756.13008 · doi:10.1016/S0021-8693(05)80052-3
[12] KM15 M. Kummini and S. K. Masuti, On conjectures of Itoh and of Lipman on the cohomology of normalized blow-ups, communicated, available at arxiv1507.03343, J. Commut. Algebra, to appear. · Zbl 1520.13005
[13] Lipman, Joseph; Teissier, Bernard, Pseudo-rational local rings and a theorem of Brian\c{c}on-Skoda about integral closures of ideals, Michigan Math. J., 28, 1, 97-116 (1981) · Zbl 0464.13005
[14] MOR17 S. K. Masuti, K. Ozeki, and M. E. Rossi, On the normal Sally module and the first Hilbert coefficient, to appear in J. Algebra (2019).
[15] Northcott, D. G., A note on the coefficients of the abstract Hilbert function, J. London Math. Soc., 35, 209-214 (1960) · Zbl 0118.04502 · doi:10.1112/jlms/s1-35.2.209
[16] Okuma, Tomohiro; Watanabe, Kei-ichi; Yoshida, Ken-ichi, Rees algebras and \(p_g\)-ideals in a two-dimensional normal local domain, Proc. Amer. Math. Soc., 145, 1, 39-47 (2017) · Zbl 1357.13011 · doi:10.1090/proc/13235
[17] Okuma, Tomohiro; Watanabe, Kei-ichi; Yoshida, Ken-ichi, Normal reduction numbers for normal surface singularities with application to elliptic singularities of Brieskorn type, Acta Math. Vietnam., 44, 1, 87-100 (2019) · Zbl 1420.13019 · doi:10.1007/s40306-018-00311-4
[18] Ooishi, Akira, \( \Delta \)-genera and sectional genera of commutative rings, Hiroshima Math. J., 17, 2, 361-372 (1987) · Zbl 0639.13016
[19] Ozeki, Kazuho; Rossi, Maria Evelina, The structure of the Sally module of integrally closed ideals, Nagoya Math. J., 227, 49-76 (2017) · Zbl 1411.13025 · doi:10.1017/nmj.2016.47
[20] Phuong, Tran Thi, Normal Sally modules of rank one, J. Algebra, 493, 236-250 (2018) · Zbl 1386.13016 · doi:10.1016/j.jalgebra.2017.09.028
[21] Rees, D., A note on analytically unramified local rings, J. London Math. Soc., 36, 24-28 (1961) · Zbl 0115.26202 · doi:10.1112/jlms/s1-36.1.24
[22] Rees, D., Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2), 24, 3, 467-479 (1981) · Zbl 0492.13012 · doi:10.1112/jlms/s2-24.3.467
[23] Rossi, Maria Evelina; Valla, Giuseppe, Hilbert functions of filtered modules, Lecture Notes of the Unione Matematica Italiana 9, xviii+100 pp. (2010), Springer-Verlag, Berlin; UMI, Bologna · Zbl 1201.13003 · doi:10.1007/978-3-642-14240-6
[24] Ng\^o Vi\textviet{\d{\^e}}t Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc., 101, 2, 229-236 (1987) · Zbl 0641.13016 · doi:10.2307/2045987
[25] Vasconcelos, Wolmer V., Hilbert functions, analytic spread, and Koszul homology. Commutative algebra: syzygies, multiplicities, and birational algebra, South Hadley, MA, 1992, Contemp. Math. 159, 401-422 (1994), Amer. Math. Soc., Providence, RI · Zbl 0803.13012 · doi:10.1090/conm/159/01520
[26] Vaz Pinto, Maria, Hilbert functions and Sally modules, J. Algebra, 192, 2, 504-523 (1997) · Zbl 0878.13008 · doi:10.1006/jabr.1996.6820
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.