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The minimal non-Cohen-Macaulay monomial ideals. (English) Zbl 0655.13020

Let \(R=k[X_ 1,...,X_ m]_{(X_ 1,...,X_ m)}\) and I an ideal of R generated by square free monomials in \(X_ 1,...,X_ m\), S a proper subset of \(\{X_ 1,...,X_ m\}\). The author defines the restriction \(I_ S\) and defines that I is minimal non Cohen-Macaulay if \(R/I\) is not a Cohen-Macaulay ring while \(R/I_ S\) is a Cohen-Macaulay ring for all proper S. He gives a complete list of such ideals of pure height 2. As an application he obtains a Cohen-Macaulayness criterion for monomial ideals of pure \(height\quad 2\) [see also: G. A. Reisner, Adv. Math. 21, 30-49 (1976; Zbl 0345.13017)].
Reviewer: M.Morales

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Citations:

Zbl 0345.13017
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References:

[1] Harshorne, R., Cohomological dimension of algebraic varieties, Ann. of Math., 88, 403-450 (1968) · Zbl 0169.23302
[2] Lyubeznik, G., On the local cohomology modules \(H^iu\) for ideals \(U\) generated by monomials in an \(R\)-sequence, (Greco, S.; Strano, R., Complete Intersections. Complete Intersections, Lecture Notes in Mathematics, 1092 (1984), Springer: Springer Berlin), 214-220
[3] Lyubeznik, G., Set-theoretic intersections and monomial ideals, (Thesis (1984), Columbia University)
[4] Reisner, G. A., Cohen-Macaulay quotients of polynomial rings, Adv. in Math., 21, 30-49 (1976) · Zbl 0345.13017
[5] Schwartau, Ph. W., Liaison addition and monomial ideals, (Thesis (1982), Brandeis University)
[6] Taylor, D., Ideals generated by monomials in an \(R\)-sequence, (Thesis (1960), University of Chicago)
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