Cohen-Macaulay local rings with $e_2=e_1-e+1$

Abstract

In this paper we study Cohen-Macaulay local rings of dimension $d$, multiplicity $e$ and second Hilbert coefficient $e_2$ in the case $e_2=e_1-e+1$. Let $h=\mu (\mathfrak{m})-d$. If $e_2\ne 0$ then in our case we can prove that type $A\ge e-h-1$. If type $A=e-h-1$ then we show that the associated graded ring $G(A)$ is Cohen-Macaulay. In the next case when type $A=e-h$ we determine all possible Hilbert series of $A$. In this case we show that the Hilbert Series of $A$ completely determines depth $G(A)$.

arXiv admin note: text overlap with arXiv:1907.11502