On the Lowey length of modules of finite projective dimension

Let $(A,\mathfrak{m})$ be a local Gorenstein local ring and let $M$ be an $A$ module of finite length and finite projective dimension. We prove that the Lowey length of $M$ is greater than or equal to order of $A$. This generalizes a result of Avramov, Buchweitz, Iyengar and Miller

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