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Unmixedness and arithmetic properties of matroidal ideals. (English) Zbl 1436.13027

Let \(I\) be a monomial ideal of \(R=k[x_1,\dots,x_n]\) which is generated in a single degree and assume that \(G(I)\) is the set of minimal monomial generators of \(I\). The ideal \(I\) is called polymatroidal if the following exchange condition is satisfied: For monomials \(u=x^{a_1}_1 \ldots x^{a_n}_ n\) and \(v = x^{b_1}_1\ldots x^{b_n}_ n\) belonging to \(G(I)\) and for every \(i\) with \(a_i > b_i\), one has \(j\) with \(a_j < b_j\) such that \(x_j(u/x_i)\in G(I)\). A squarefree polymatroidal ideal is a matroidal ideal.
In the paper under review, the authors start by studying the unmixedness of matroidal ideals. It is shown that a matroidal ideal \(I\) generated in degree at least two is unmixed if and only if for every variable \(x_i\), the ideal \((I : x_i)\) is unmixed and \(\mathrm{ht}(I)=\mathrm{ht}(I : x_i)\). Next the authors compute the arithmetical rank of matroidal ideals. More precisely, it is shown that the arithmetical rank of a full-supported matroidal ideal in \(R\) is equal to \(n-d+1\), where \(d\) is the degree of the minimal monomial generators of \(I\). This proves a conjecture posed in [Chiang-Hsieh et al., Commun. Algebra. 38, No. 3, 944–952 (2010; Zbl 1189.13010)].

MSC:

13C40 Linkage, complete intersections and determinantal ideals
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13C13 Other special types of modules and ideals in commutative rings

Citations:

Zbl 1189.13010

Software:

Macaulay2
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Full Text: DOI arXiv

References:

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