By Francis Borceux
ISBN-10: 0521061199
ISBN-13: 9780521061193
Lecture notes in arithmetic No.1348
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Hibi, Squarefree Lexsegment Ideals, Math. Z. 228 (1998), 353-378. [B] M. Barile, Arithmetical ranks of Stanley-Reisner ideals via linear algebra, arXiv: math/0703258v2. [BS] V. Bonanzinga and L. 2 (2008), 275-291. [BH] W. Bruns and J. Herzog, Cohen-Macaulay rings, Revised Edition, Cambridge University Press, (1998). [Ha] M. Hachimori, Decomposition of two dimensional simplicial complexes, To appear in Discrete Mathematics. [H] J. Herzog, Combinatorics and Commutative Algebra, IMUB Lecture Notes, 2 (2006), 58-106.
I + 1} and [n] \ {min{j, + 2}, . . , n} are facets of ∆. Hence ∆ is not pure, since i ≤ n − 4. Case 4. v = xn−2 xn , u = xi xn . We may assume u = xn−2 xn−1 . Set u = xi xj , where i ≤ n − 3 and j ≤ n − 1. We distinguish 3 cases: (1) If i = n − 3 and j = n − 2 then it is possible to determine two facets F1 = {1, . . , n − 4, n − 1, n}, with dimF1 = n − 3 and F2 = {1, . . , n − 4, n − 3}, with dimF2 = n − 4 and ∆ is not pure. (2) If i ≤ n − 3 and j = n − 1 then dim(∆ = link[i − 1]) ≥ 1, since {i, n − 2} ∈ ∆ and is not connected.
V = xi+1 xi+2 , u = xi xn , with i ≤ n − 4. First we assume that u = xi xj with i + 3 ≤ j ≤ n − 1. Then 28 4 VITTORIA BONANZINGA AND LOREDANA SORRENTI I∆ = (xi xj , xi xj+1 , . . , xi xn , xi+1 xi+2 ). Then [n] \ {i, i + 1} and [n] \ {i + 1, j, . . , n} are facets of ∆. Hence ∆ is not pure. Next we assume that u = xi xj with i + 1 ≤ j ≤ i + 2. Then [n] \ {i, i + 1} and [n] \ {j, . . , n} are facets of ∆. Hence ∆ is not pure, since i ≤ n − 4. Finally we assume that u = x xj with ≤ i − 1 and < j ≤ n.
Categorical Algebra and Its Applications by Francis Borceux
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