The minimal non-Cohen-Macaulay monomial ideals

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Let I⊂Rn= k[X1,...,Xn](X1,...,Xn) be a radical ideal generated by monomials in X1,...,Xn (k is a field). One calls I minimal non-Cohen-Macaulay if Rn/I is not Cohen- Macaulay, while Rn/Is is Cohen-Macaulay for every proper subset S⊂{X1,...,Xn} where Is is the (suitably defined) restriction of I to S. A complete list of minimal non-Cohen-Macaulay ideals of pure height t would produce a simple Cohen-Macaulayness criterion for ideals of pure height t. This paper gives a complete list of minimal non-Cohen-Macaulay monomial ideals of pure height 2. In particular, it turns out there exists exactly one such ideal for every fixed n≥4. As an immediate application one obtains a new Cohen-Macaulayness criterion for monomial ideals of pure height 2, which is simpler than Reisner's topological criterion.

Original languageEnglish (US)
Pages (from-to)261-266
Number of pages6
JournalJournal of Pure and Applied Algebra
Issue number3
StatePublished - Apr 1988


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