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 = mu(m) - d. If e(2) &NOTEQUexpressionL; 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. (C) 2022 Elsevier Inc. All rights reserved.