Regularity comparison of fiber cone and associated graded ring

Abstract

Let $$(A,\mathfrak {m} )$$ be a Noetherian local ring and I be an $$\mathfrak {m} $$-primary ideal of A. In this article, we show that if A is a 2-dimensional Buchsbaum local ring and $${\text {depth}}(A)>0$$, then $${\text {reg}}(F(I))\le \max \{s^*(\mathcal {F})-2, {\text {reg}}(G(I))\}$$ and the equality case is discussed. We also study the upper bounds for $${\text {reg}}(G(\mathbb {M}))$$ in terms of multiplicity and Ratliff-Rush closure for 1-dimensional good I-filtrations $$\mathbb {M}$$.