2021/02/08 by Yadav, Anoot Kumar, Saloni, Kumari · 1 citation
Mathematics · Medicine · #Commutative Algebra and Its Applications #Cholinesterase and Neurodegenerative Diseases
paper · pdf · doi:10.48550/arxiv.2102.04218
Let (A,\mathfrak m) be a Noetherian local ring of dimension d>0 with infinite residue field and I an \mathfrakm-primary ideal. Let \mathcal I be an I-good filtration. We study an equality of Hilbert coefficients, first given by Elias and Valla, versus passage of Buchsbaum property from the local ring to the blow-up algebras. Suppose e1(\mathcal I)-e1(Q)=2e0(\mathcal I)-2ℓ(A/I1)-ℓ(I1/(I2+Q)) where Q⊆ I, a minimal reduction of \mathcal I, is a standard parameter ideal. Under some mild conditions, we prove that if A is Buchsbaum (generalized Cohen-Macaulay respectively), then the associated graded ring G(\mathcal I) is Buchsbaum (generalized Cohen-Macaulay respectively). Our results settle a question of Corso in general for an I-good filtration. Further, let f0(I)= e1(I)-e0(I)-e1(Q)+ℓ(A/I)+μ(I)-d+1 and e1(I)-e1(Q)=2e0(I)-2ℓ(A/I)-ℓ(I/(I2+Q)). We prove, under mild conditions, that (1) if A is generalized Cohen-Macaulay, then the special fiber ring F_\mathfrakm(I) is generalized Cohen-Macaulay; In addition, if depth of A is positive, then depth of F_\mathfrak m(I) is same as depth of A and (2) if A is Buchsbaum and depth A≥ d-1, then F_\mathfrakm(I) is Buchsbaum and the I-invariant of F_\mathfrakm(I) is same as that of A.