On associated graded rings having almost maximal depth
We generalize a recent result of Rossi and Valla, and independently Wang, about the depth of $G(m)$ where $m$ is the maximal ideal of a $d$-dimensional Cohen-Macaulay local ring $R$ having embedding dimesion $e_0(m)+d-2$. The generalization removes the restriction on the embedding dimension and replaces it with the condition that $\lambda(m^3/Jm^2)\leq 1$ where $J$ is a $d$-generated minimal reduction of $m$. The main theorem also applies to $m$-primary ideals $I$ satisfying $J\cap I^2=JI$ and $\lambda(I^3/JI^2)\leq 1$, where $J$ is a $d$-generated reduction of $I$. An example of such an $I$ in a $5$-dimensional regular local ring is included as a nontrivial illustration of the theorem.