The minimal non-Cohen-Macaulay monomial ideals

Let I⊂R_n= k[X₁,...,X_n]_(X1,...,Xn) be a radical ideal generated by monomials in X₁,...,X_n (k is a field). One calls I minimal non-Cohen-Macaulay if R_n/I is not Cohen- Macaulay, while R_n/I_s is Cohen-Macaulay for every proper subset S⊂{X₁,...,X_n} where I_s 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.