Let $(R, \mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated $R$-module and $\underline{x}=x_{1}, x _{2}, \dots, x_{r}\in \mathfrak{m}$ an $M$-regular sequence with $r\geq 1$. If $\underline{x}$ is also $R$-regular, is it true that $\mathrm{pd}_{R/\underline{x}}(M/\underline{x}M)=\mathrm{pd}_R(M)$? (Here, $\mathrm{pd}$ means projective dimension.)
I know that if $r=1$, then the above result is true. Suppose by induction that the above equality holds for any $M$-regular sequence of length $r-1$. Let $M'=M/\underline{y}M$ and $R'=R/\underline{y}R$, where $\underline{y}=x_{1}, x _{2}, \dots, x_{r-1}$. Then by base step of induction, $\mathrm{pd}_{R'/x_rR}(M'/x_rM')=\mathrm{pd}_{R'}(M')$.
By induction, $\mathrm{pd}_{R}(M)=\mathrm{pd}_{R'}(M')$. Hence, the result follows from the following equality: $\mathrm{pd}_{R/\underline{x}R}(M/\underline{x}M)=\mathrm{pd}_{R'/x_rR}(M'/x_rM')$.