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Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not necessarily finite) with $\lambda_{R_{ 0}}(M_{ n})<\infty$ for all $n.$ Suppose there exists a polynomial $P$ such that $P(n)=\lambda_{R_{ 0}}(M_{ n})$ for all large $n.$

Question Can we say dim $M$=deg $P?$

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    $\begingroup$ No, and Remy van Dobben de Bruyn already explained the examples. The graded $R$-module $M = \oplus_n (R/R_+)(n)^{\oplus P(n)}$ has the specified length function, but it has support equal to $R_+ = \oplus_{n>0} R_n$. So the dimension is $0$. $\endgroup$
    – Jason Starr
    Commented Mar 25, 2016 at 18:09

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