Cohen-Macaulay local rings with e2 = e1 − e + 1.

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 = μ (m) − d. If e 2 ≠ 0 then in our case we can prove that type (A) ≥ 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 depth G (A) completely determines the Hilbert Series of A. [ABSTRACT FROM AUTHOR]