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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.

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.)
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