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On complete monomial ideals. (English) Zbl 1353.13006
Let $$\mathbf{R}=k[x,y]$$, $$\mathfrak{m}=(x,y)$$, and $$I$$ be a monomial ideal of $$\mathbf{R}$$. Recall that an $$\mathfrak{m}$$-primary ideal $$J$$ is said to be $$\mathfrak{m}$$-full if $$\mathfrak{m}J:a=J$$ for some $$a\in\mathfrak{m}\setminus\mathfrak{m}^2$$. As a first result of the present paper, the authors describe how to find the smallest $$\mathfrak{m}$$-full ideal containing a monomial ideal $$I$$. In the case where $$I$$ is minimally generated by $$n$$ monomials that are listed lexicographically, $$I=(x^{a_1}, x^{a_2}y^{b_{n-1}},\ldots, x^{a_i}y^{b_{n-i+1}},\ldots, x^{a_{n-1}}y^{b_2}, y^{b_1})$$, they characterize when $$I$$ is $$\mathfrak{m}$$-full.
Also, in a different direction, they establish separate necessary or sufficient conditions for normality, expressed by systems of linear inequalities $$Q(p_1,\ldots,p_n)\leqslant 0$$, where $$P_i=(a_i, b_{n-i+1})$$. (Recall that an ideal $$I$$ is said to be normal if for all positive integer $$n$$, $$I^n$$ is integrally closed)
Finally, they study the Rees algebras $$\mathbf{R}[It]$$ emphasizing when they are cohen-Macaulay and obtaining their defining equations.

##### MSC:
 13B22 Integral closure of commutative rings and ideals 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Macaulay2
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##### References:
 [1] J.P. Brennan, L.A. Dupont and R.H. Villarreal, Duality, $$a$$-invariants and canonical modules of rings arising from linear optimization problems , Bull. Math. Soc. Sci. Math. Roum. 51 (2008), 279-305. · Zbl 1199.13026 [2] W. Bruns and J. Gubeladze, Polytopes, rings, and $$K$$-theory , Springer Mono. Math., Springer, Dordrecht, 2009. [3] A. Conca, E. De Negri, A.V. Jayanthan and M.E. Rossi, Graded rings associated with contracted ideals , J. Algebra 284 (2005), 593-626. · Zbl 1098.13007 [4] A. Conca, E. De Negri and M.E. Rossi, Contracted ideals and the Gröbner fan of the rational normal curve , Alg. Numer. Theor. 1 (2007), 239-268. · Zbl 1163.13002 [5] A. Corso, L. Ghezzi, C. Polini and B. Ulrich, Cohen-Macaulayness of special fiber rings , Comm. Alg. 31 (2003), 3713-3734. · Zbl 1057.13007 [6] V. Crispin Quiñonez, Integral closure and related operations on monomial ideals , Ph.D. thesis, Stockholm University, 2006. · Zbl 1116.13006 [7] V. Crispin Quiñonez, Integral closure and other operations on monomial ideals , J. Comm. Alg. 2 (2010), 359-386. · Zbl 1238.13005 [8] D. Delfino, A. Taylor, W.V. Vasconcelos, R.H. Villarreal and N. Weininger, Monomial ideals and the computation of multiplicities , in Commutative ring theory and applications , Lect. Notes Pure Appl. Math. 231 , Dekker, New York, 2003. · Zbl 1066.13004 [9] L.A. Dupont and R.H. Villarreal, Edge ideals of clique clutters of comparability graphs and the normality of monomial ideals , Math. Scand. 106 (2010), 88-98. · Zbl 1195.13008 [10] C. Escobar, R.H. Villarreal and Y. Yoshino, Torsion freeness and normality of blowup rings of monomial ideals , in Commutative algebra , Lect. Notes Pure Appl. Math. 244 , Chapman & Hall/CRC, Boca Raton, FL, 2006. · Zbl 1097.13002 [11] I. Gitler, C. Valencia and R.H. Villarreal, A note on Rees algebras and the MFMC property , Beitr. Alg. Geom. 48 (2007), 141-150. · Zbl 1114.13007 [12] S. Goto and Y. Shimoda, Rees algebras of Cohen-Macaulay local rings , in Commutative algebra , Lect. Notes Pure Appl. Math. 68 , Marcel Dekker, New York, 1982. · Zbl 0482.13011 [13] D. Grayson and M. Stillman, Macaulay$$2$$, A software system for research in algebraic geometry , available at http://www.math.uiuc.edu/Macaulay2/. [14] C. Huneke, Complete ideals in two-dimensional regular local rings , in Commutative algebra , Math. Sci. Res. Inst. Publ. 15 , New York, Springer, 1989. · Zbl 0732.13007 [15] C. Huneke and I. Swanson, Integral closure of ideals, rings, and modules , Lond. Math. Soc. Lect. Note 336 , Cambridge University Press, Cambridge, 2006. · Zbl 1117.13001 [16] C. Huneke and B. Ulrich, Residual intersections , J. reine angew. Math. 390 (1988), 1-20. [17] M. Lejeune-Jalabert, Linear systems with infinitely near base conditions and complete ideals in dimension two , in Singularity theory , World Science Publishing, River Edge, NJ, 1995. · Zbl 0948.14004 [18] J. Lipman, On complete ideals in regular local rings , in Algebraic geometry and commutative algebra , Vol. I, Kinokuniya, Tokyo, 1988. · Zbl 0693.13011 [19] J. Lipman and B. Teissier, Pseudo-rational local rings and a theorem of Briançon-Skoda about integral closures of ideals , Michigan Math. J. 28 (1981), 97-112. · Zbl 0464.13005 [20] S. Morey and B. Ulrich, Rees algebras of ideals with low codimension , Proc. Amer. Math. Soc. 124 (1996), 3653-3661. · Zbl 0882.13003 [21] D. Rees, Hilbert functions and pseudorational local rings of dimension two , J. Lond. Math. Soc. 24 (1981), 467-479. · Zbl 0492.13012 [22] A. Schrijver, Theory of linear and integer programming , John Wiley & Sons, New York, 1986. · Zbl 0665.90063 [23] —-, Combinatorial optimization , Alg. Comb. 24 , Springer-Verlag, Berlin, 2003. [24] A. Simis and W.V. Vasconcelos, The syzygies of the conormal module , Amer. J. Math. 103 (1981), 203-224. · Zbl 0467.13009 [25] N.V. Trung, Reduction exponent and degree bound for the defining equations of graded rings , Proc. Amer. Math. Soc. 101 (1987), 229-236. · Zbl 0641.13016 [26] W.V. Vasconcelos, Integral closure , Springer Mono. Math., New York, 2005. · Zbl 1082.13006 [27] J. Watanabe, $$\m$$-full ideals , Nagoya Math. J. 106 (1987), 101-111. · Zbl 0623.13012 [28] O. Zariski, Polynomial ideals defined by infinitely near base points , Amer. J. Math. 60 (1938), 151-204. · Zbl 0018.20101 [29] O. Zariski and P. Samuel, Commutative algebra II, 1960 edition reprint, Grad. Texts Math. 29 , Springer-Verlag, New York, 1975. · Zbl 0322.13001
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