TY - JOUR

T1 - The minimal non-Cohen-Macaulay monomial ideals

AU - Lyubeznik, Gennady

N1 - Copyright:
Copyright 2014 Elsevier B.V., All rights reserved.

PY - 1988/4

Y1 - 1988/4

N2 - 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.

AB - 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.

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U2 - 10.1016/0022-4049(88)90065-5

DO - 10.1016/0022-4049(88)90065-5

M3 - Article

AN - SCOPUS:50849147559

VL - 51

SP - 261

EP - 266

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

SN - 0022-4049

IS - 3

ER -