The reduction number of stretched ideals

The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen-Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals in the case where the reduction number attains almost minimal value in a Cohen-Macaulay local ring $(A,\mathfrak{m})$. As an application, we present complete descriptions of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small reduction number.