Rank of a map between graded modules

Question

Let $M$ a finitely generated $n$-graded module over the ring of polynomials $k[x_1, .., x_n]$, where $n$-graded means that the indexes of the family are elements of $\mathbb{N}^n$. Let $x^{v-u} : M_u \to M_v$, where $M_u, M_v$ are two homogeneous components, such that if $u=(u_1,.., u_n)$ and $v=(v_1, ..., v_n)$, $u_i \leq v_i $ for every $i$. What exactly is the rank of the map $x^{v-u}$? Any reference?