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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.)
Software:
Macaulay2
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References:
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