Length one ideal extensions and their associated graded rings
Let $(R,\m)$ be a $d$-dimensional Cohen-Macaulay local ring. Given $\m$-primary ideals $J\subset I$ of $R$ such that $I$ is contained in the integral closure of $J$ and $\lambda(I/J)=1$, we compare $\depth G(J)$ and $\depth G(I)$. For example, if $J$ has reduction number one, $JI=I^2$, and $\mu(J)\leq d+1$, we prove that $\depth G(I)\geq d-1$. If, in addition, $\mu(I)=d+1$, we show that $I$ has reduction number one, and hence $G(I)$ is Cohen-Macaulay. These results, besides leading to statements comparing depths of associated graded rings along a composition series, make visible the possibility of studying powers of an ideal by using reductions that are not minimal reductions.