Let (A, m) be a Cohen-Macaulay local ring, M a Cohen-Macaulay A-module of dimension d ≥ 1 and I a proper ideal of analytic deviation one with respect to M. In this paper we study the Cohen-Macaulayness of associated graded module of a Cohen-Macaulay module. We show that if I is generically a complete intersection of analytic deviation one and reduction number at most one with respect to M then G I (M ) is Cohen-Macaulay. When analytic spread of I with respect to M equals d we prove a similar result when reduction number of an ideal is atmost two.
Let (A, m) be a Cohen-Macaulay local ring of dimension d ≥ 1 and I an ideal in A. Let M be a finitely generated maximal Cohen-MacaulayAmodule. Let I be a locally complete intersection ideal with ht M (I) = d − 1, l M (I) = d and reduction number at most one. We prove that the polynomial n → ℓ(Tor A 1 (M, A/I n+1 )) either has degree d − 1 or F I (M ) is a free F (I)−module.
Let R be an Artinian ring and let Γ E (R) be the compressed zero-divisor graph associated to R. The question of when the clique number ω(Γ E (R)) < ∞ was raised by J. Coykendall, S. Sather-Wagstaff, L. Sheppardson, and S. Spiroff, see [8, Section 5]. They proved that if ℓ(R) ≤ 4 thenthey gave an example of a local ring R where ω(Γ E (R)) = ∞ is possible by using the trivial extension of an Artinian local ring by its dualizing module. The question of what happens when ℓ(R) = 5 was stated as an open problem. We show that if ℓWe first reduce the problem to the case of a local ring (R, m, k). We then enumerate all possible Hilbert functions of R and show that the k-vector space m/m 2 admits a symmetric bilinear form in some cases of the Hilbert function. This allows us to relate the orthogonality in the bilinear space m/m 2 with the structure of zero-divisors in R. For instance, in the case when m 2 is principal and m 3 = 0, we show that R is Gorenstein if and only if the symmetric bilinear form on m/m 2 is non-degenerate. Moreover, in the case when ℓ(R) = 4, our techniques also yield a simpler and shorter proof of the finiteness of ω(Γ E (R)) avoiding, for instance, Cohen structure theorem.
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