On Ratliff-Rush closure of modules
DOI:
https://doi.org/10.7146/math.scand.a-119672Abstract
In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.
References
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Buchsbaum, D. A. and Rim, D. S., A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224. https://doi.org/10.2307/1994241
Gaffney, T., Integral closure of modules and Whitney equisingularity, Invent. Math. 107 (1992), no. 2, 301–322. https://doi.org/10.1007/BF01231892
Gaffney, T., Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), no. 2, 209–220. https://doi.org/10.1007/s002220050022
Goto, S., Buchsbaum rings of maximal embedding dimension, J. Algebra 76 (1982), no. 2, 383–399. https://doi.org/10.1016/0021-8693(82)90221-6
Goto, S. and Matsuoka, N., The Rees algebras of ideals in two-dimensional regular local rings, in “The Proceedings of the 27th Symposium on Commutative Algebra, Toyama, 2005”, 2006, pp. 81–89.
Goto, S., Matsuoka, N., Taniguchi, N., and Yoshida, K., The almost Gorenstein Rees algebras over two-dimensional regular local rings, J. Pure Appl. Algebra 220 (2016), no. 10, 3425–3436. https://doi.org/10.1016/j.jpaa.2016.04.007
Hayasaka, F. and Hyry, E., A note on the Buchsbaum-Rim multiplicity of a parameter module, Proc. Amer. Math. Soc. 138 (2010), no. 2, 545–551. https://doi.org/10.1090/S0002-9939-09-10119-3
Heinzer, W., Johnston, B., Lantz, D., and Shah, K., Coefficient ideals in and blowups of a commutative Noetherian domain, J. Algebra 162 (1993), no. 2, 355–391. https://doi.org/10.1006/jabr.1993.1261
Huneke, C. and Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006.
Katz, D. and Kodiyalam, V., Symmetric powers of complete modules over a two-dimensional regular local ring, Trans. Amer. Math. Soc. 349 (1997), no. 2, 747–762. https://doi.org/10.1090/S0002-9947-97-01819-9
Kodiyalam, V., Integrally closed modules over two-dimensional regular local rings, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3551–3573. https://doi.org/10.2307/2155023
Liu, J.-C., Ratliff-Rush closures and coefficient modules, J. Algebra 201 (1998), no. 2, 584–603. https://doi.org/10.1006/jabr.1997.7300
Matsuoka, N., Ratliff-Rush closure of certain two-dimensional monomial ideals and Buchsbaumness of their Rees algebras, in “The Proceedings of the 26th Symposium on Commutative Algebra, Kurashiki, 2004”, 2005, pp. 19–28.
McAdam, S., Asymptotic prime divisors, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0071575
Ratliff, Jr., L. J. and Rush, D. E., Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), no. 6, 929–934. https://doi.org/10.1512/iumj.1978.27.27062
Rossi, H., Picard variety of an isolated singular point, Rice Univ. Stud. 54 (1968), no. 4, 63–73.
Simis, A., Ulrich, B., and Vasconcelos, W. V., Rees algebras of modules, Proc. London Math. Soc. (3) 87 (2003), no. 3, 610–646. https://doi.org/10.1112/S0024611502014144
Stückrad, J. and Vogel, W., Toward a theory of Buchsbaum singularities, Amer. J. Math. 100 (1978), no. 4, 727–746. https://doi.org/10.2307/2373908
Villamayor U., O., On flattening of coherent sheaves and of projective morphisms, J. Algebra 295 (2006), no. 1, 119–140. https://doi.org/10.1016/j.jalgebra.2005.01.022
Brennan, J., Ulrich, B., and Vasconcelos, W. V., The Buchsbaum-Rim polynomial of a module, J. Algebra 241 (2001), no. 1, 379–392. https://doi.org/10.1006/jabr.2001.8764
Buchsbaum, D. A. and Rim, D. S., A generalized Koszul complex. II. Depth and multiplicity, Trans. Amer. Math. Soc. 111 (1964), 197–224. https://doi.org/10.2307/1994241
Gaffney, T., Integral closure of modules and Whitney equisingularity, Invent. Math. 107 (1992), no. 2, 301–322. https://doi.org/10.1007/BF01231892
Gaffney, T., Multiplicities and equisingularity of ICIS germs, Invent. Math. 123 (1996), no. 2, 209–220. https://doi.org/10.1007/s002220050022
Goto, S., Buchsbaum rings of maximal embedding dimension, J. Algebra 76 (1982), no. 2, 383–399. https://doi.org/10.1016/0021-8693(82)90221-6
Goto, S. and Matsuoka, N., The Rees algebras of ideals in two-dimensional regular local rings, in “The Proceedings of the 27th Symposium on Commutative Algebra, Toyama, 2005”, 2006, pp. 81–89.
Goto, S., Matsuoka, N., Taniguchi, N., and Yoshida, K., The almost Gorenstein Rees algebras over two-dimensional regular local rings, J. Pure Appl. Algebra 220 (2016), no. 10, 3425–3436. https://doi.org/10.1016/j.jpaa.2016.04.007
Hayasaka, F. and Hyry, E., A note on the Buchsbaum-Rim multiplicity of a parameter module, Proc. Amer. Math. Soc. 138 (2010), no. 2, 545–551. https://doi.org/10.1090/S0002-9939-09-10119-3
Heinzer, W., Johnston, B., Lantz, D., and Shah, K., Coefficient ideals in and blowups of a commutative Noetherian domain, J. Algebra 162 (1993), no. 2, 355–391. https://doi.org/10.1006/jabr.1993.1261
Huneke, C. and Swanson, I., Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, vol. 336, Cambridge University Press, Cambridge, 2006.
Katz, D. and Kodiyalam, V., Symmetric powers of complete modules over a two-dimensional regular local ring, Trans. Amer. Math. Soc. 349 (1997), no. 2, 747–762. https://doi.org/10.1090/S0002-9947-97-01819-9
Kodiyalam, V., Integrally closed modules over two-dimensional regular local rings, Trans. Amer. Math. Soc. 347 (1995), no. 9, 3551–3573. https://doi.org/10.2307/2155023
Liu, J.-C., Ratliff-Rush closures and coefficient modules, J. Algebra 201 (1998), no. 2, 584–603. https://doi.org/10.1006/jabr.1997.7300
Matsuoka, N., Ratliff-Rush closure of certain two-dimensional monomial ideals and Buchsbaumness of their Rees algebras, in “The Proceedings of the 26th Symposium on Commutative Algebra, Kurashiki, 2004”, 2005, pp. 19–28.
McAdam, S., Asymptotic prime divisors, Lecture Notes in Mathematics, vol. 1023, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0071575
Ratliff, Jr., L. J. and Rush, D. E., Two notes on reductions of ideals, Indiana Univ. Math. J. 27 (1978), no. 6, 929–934. https://doi.org/10.1512/iumj.1978.27.27062
Rossi, H., Picard variety of an isolated singular point, Rice Univ. Stud. 54 (1968), no. 4, 63–73.
Simis, A., Ulrich, B., and Vasconcelos, W. V., Rees algebras of modules, Proc. London Math. Soc. (3) 87 (2003), no. 3, 610–646. https://doi.org/10.1112/S0024611502014144
Stückrad, J. and Vogel, W., Toward a theory of Buchsbaum singularities, Amer. J. Math. 100 (1978), no. 4, 727–746. https://doi.org/10.2307/2373908
Villamayor U., O., On flattening of coherent sheaves and of projective morphisms, J. Algebra 295 (2006), no. 1, 119–140. https://doi.org/10.1016/j.jalgebra.2005.01.022
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Published
2020-05-06
How to Cite
Endo, N. (2020). On Ratliff-Rush closure of modules. MATHEMATICA SCANDINAVICA, 126(2), 170–188. https://doi.org/10.7146/math.scand.a-119672
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