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I am reading Gunnar Carlsson and Afra Zomorodian's "The theory of multidimensional persistence" and I'm stuck on what an n-graded vector space V is. Their definition seems to be that V can be written as $\bigoplus_i k(v_i)$ for some family of vectors $v_i\in\mathbb{N}^n$. In particular, k is a field that is $\mathbb{N}^n$ graded as follows: $k_0\cong k$ and $k_v={0}$ for all other v. Thus $k=\bigoplus_v k_v$. The notation $k(v_i)$ means to shift the grading to $k(v_i)=\bigoplus_v k_{v-v_i}$.

What I'm missing is that I don't understand at all what this grading means. What does it mean that $k(1,0)$'s generator lies at $(1,0)$? What does the vector space $k(1,0)\bigoplus k(2,3)$ look like?

My second question is why $\rho(M)$ is an n-graded vector space. Here, M is a finitely generated $\mathbb{N}^n$-graded $k[x_1,...,x_n]$ module and $\rho(m)=k\bigotimes_{k[x_1,...,x_n]}M$ where k is given a $k[x_1,...,x_n]$ module structure s.t. $x_i$ acting on k is merely the zero map for all i.

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