On Hilbert coefficients and sequentially Cohen-Macaulay rings

Ozeki, Kazuho; Truong, Hoang; Yen, Hoang

doi:10.1090/proc/15883

On Hilbert coefficients and sequentially Cohen-Macaulay rings
HTML articles powered by AMS MathViewer

by Kazuho Ozeki, Hoang Le Truong and Hoang Ngoc Yen PDF

Proc. Amer. Math. Soc. 150 (2022), 2367-2383 Request permission

Abstract:

In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of its Hilbert coefficients of non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients of non-parameter ideals.

References

Nguyen Tu Cuong and Doan Trung Cuong, On sequentially Cohen-Macaulay modules, Kodai Math. J. 30 (2007), no. 3, 409–428. MR 2372128
Nguyen Tu Cuong, Shiro Goto, and Hoang Le Truong, The equality $I^2=\mathfrak {q}I$ in sequentially Cohen-Macaulay rings, J. Algebra 379 (2013), 50–79. MR 3019245, DOI 10.1016/j.jalgebra.2012.12.007
Nguyen Tu Cuong, Shiro Goto, and Hoang Le Truong, Hilbert coefficients and sequentially Cohen-Macaulay modules, J. Pure Appl. Algebra 217 (2013), no. 3, 470–480. MR 2974226, DOI 10.1016/j.jpaa.2012.06.026
Alberto Corso, Craig Huneke, and Wolmer V. Vasconcelos, On the integral closure of ideals, Manuscripta Math. 95 (1998), no. 3, 331–347. MR 1612078, DOI 10.1007/s002290050033
Nguyen Tu Cuong and Pham Hung Quy, A splitting theorem for local cohomology and its applications, J. Algebra 331 (2011), 512–522. MR 2774673, DOI 10.1016/j.jalgebra.2010.09.017
Nguyen Tu Cuong, Pham Hung Quy, and Hoang Le Truong, On the index of reducibility in Noetherian modules, J. Pure Appl. Algebra 219 (2015), no. 10, 4510–4520. MR 3346503, DOI 10.1016/j.jpaa.2015.02.029
Nguyen Tu Cuong and Hoang Le Truong, Asymptotic behavior of parameter ideals in generalized Cohen-Macaulay modules, J. Algebra 320 (2008), no. 1, 158–168. MR 2417983, DOI 10.1016/j.jalgebra.2007.12.008
L. Ghezzi, S. Goto, J. Hong, K. Ozeki, T. T. Phuong, and W. V. Vasconcelos, Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals, J. Lond. Math. Soc. (2) 81 (2010), no. 3, 679–695. MR 2650791, DOI 10.1112/jlms/jdq008
W. Gröbner, Ein irreduzibilitätskriterium für Primärmideale in kommutativen ringen, Monatsh. Math. 55 (1951), 138–145., DOI 10.1007/BF01486922
Shiro Goto and Hideto Sakurai, The equality $I^2=QI$ in Buchsbaum rings, Rend. Sem. Mat. Univ. Padova 110 (2003), 25–56. MR 2033000
Shiro Goto and Naoyoshi Suzuki, Index of reducibility of parameter ideals in a local ring, J. Algebra 87 (1984), no. 1, 53–88. MR 736769, DOI 10.1016/0021-8693(84)90160-1
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
Emmy Noether, Idealtheorie in Ringbereichen, Math. Ann. 83 (1921), no. 1-2, 24–66 (German). MR 1511996, DOI 10.1007/BF01464225
D. G. Northcott, On irreducible ideals in local rings, J. London Math. Soc. 32 (1957), 82–88. MR 83477, DOI 10.1112/jlms/s1-32.1.82
P. Schenzel, On the dimension filtration and Cohen-Macaulay filtered modules, Van Oystaeyen, Freddy (ed.), Commutative algebra and algebraic geometry, New York: Marcel Dekker. Lect. Notes Pure Appl. Math. 206 (1999), 245–264.
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Hoang Le Truong, Index of reducibility of distinguished parameter ideals and sequentially Cohen-Macaulay modules, Proc. Amer. Math. Soc. 141 (2013), no. 6, 1971–1978. MR 3034424, DOI 10.1090/S0002-9939-2013-11515-7
Hoang Le Truong, Index of reducibility of parameter ideals and Cohen-Macaulay rings, J. Algebra 415 (2014), 35–49. MR 3229506, DOI 10.1016/j.jalgebra.2014.05.029
Hoang Le Truong, Chern coefficients and Cohen-Macaulay rings, J. Algebra 490 (2017), 316–329. MR 3690336, DOI 10.1016/j.jalgebra.2017.07.012
Hoang Le Truong, The eventual index of reducibility of parameter ideals and the sequentially Cohen-Macaulay property, Arch. Math. (Basel) 112 (2019), no. 5, 475–488. MR 3943468, DOI 10.1007/s00013-018-1284-8
Nguyen Thi Thanh Tam and Hoang Le Truong, A note on Chern coefficients and Cohen-Macaulay rings, Ark. Mat. 58 (2020), no. 1, 197–212. MR 4094646, DOI 10.4310/arkiv.2020.v58.n1.a12
Wolmer V. Vasconcelos, The Chern coefficients of local rings, Michigan Math. J. 57 (2008), 725–743. Special volume in honor of Melvin Hochster. MR 2492478, DOI 10.1307/mmj/1220879434