Let $(R,\mathfrak{m},k)$ a Noetherian local ring and $I$ an ideal. Suppose that:
$\mu(I)=\operatorname{grade}(I,R)+1$ and $\operatorname{pd}_R(R/I)=\operatorname{grade}(I,R)$. (Some people says $I$ is an almost complete intersection ideal).
In this case, can I chosse a minimal set of generators $x_1,\dots ,x_d,x_{d+1}$ for $I$ such that $x_1,\dots,x_d$ is a regular sequence?
If yes, some idea?
