We define the multiplicity of an $ R $-module $ M $ of dimension $ d > 0 $ to be $$ \text{mult}(M) \stackrel{\text{df}}{=} \text{LC}(P_{M}) \cdot (d - 1)!, $$ where $ P_{M} $ denotes the Hilbert polynomial of $ M $ and $ \text{LC} $ stands for ‘leading coefficient’. Equivalently, $$ \text{mult}(M) \stackrel{\text{df}}{=} {Q_{M}}(1), $$ where $ {HP_{M}}(z) = \dfrac{{Q_{M}}(z)}{(1 - z)^{d}} $ and $ HP_{M} $ denotes the Hilbert-Poincaré series of $ M $.
I can show that if $ I = \langle f_{1},\ldots,f_{r} \rangle $ with $ \deg(f_{i}) = d_{i} $, and $ (f_{1},\ldots,f_{r}) $ is an $ M $-regular sequence, then $$ \text{mult}(M / I M) = (d_{1} \cdots d_{r}) \cdot \text{mult}(M). $$
Consider $ R = \mathbb{k}[x_{1},\ldots,x_{n}] $ and $ I = \langle f_{1},\ldots,f_{r} \rangle $ with $ f_{i} \in R $ homogeneous and $ \deg(f_{i}) = d_{i} \geq 2 $. Does the reverse implication hold, i.e., does $$ \text{mult}(R / I) = d_{1} \cdots d_{r} \quad \Longrightarrow \quad (f_{1},\ldots,f_{r}) ~ \text{is an $ R $-regular sequence}? $$
