I am reading the chapter of Free Resolutions in Using Algebraic Geometry by Cox, Little, O'Shea. It is my first time dealing with graded modules and therefore I would really apprecite some help with following exercise:
Exercise. Suppose that $M$ is a finitely generated $R$-module and $M_t$ denotes the set of homogeneous elemets of $M$ of degree $t$.
a) Prove that $M_t$ is finite dimensional vector space over the field $k$ and $M_t={0}$ when $t<< 0$. (use surjective $\varphi$ constructed below
b) Let $\psi:M\to M$ be a graded homomorphism of degree $0.$ Prove that $\psi $ is an isomorphism if and only if $\psi:M_t\to M_t$ is onto for every $t$. Conclude that $\psi$ is an isomorphism if and only if it is onto.
Where the mentioined $\varphi$ is constructed as:
Sppose that $M$ is a graded $R$-module generated by homogeneous elements $f_1,\dots,f_m$ of degrees $d_1,\dots,d_m$. Then we get a graded homomorphism $$\varphi:R(-d_1)\oplus\dots\oplus R(-d_m)\to M$$ which sends the standard basis element $e_i$ to $f_i\in M$. Note that $\varphi$ is onto and that since $e_i$ has degree $d_i$, it follows that $\varphi$ has degree $0.$
Question 1 Why is $\varphi$ onto? Is it because each generator $f_i$ has corresponding $e_i$ such that $\varphi(e_i)=f_i$?
Question 2 One part of my other problem is that I am not sure if here $R=k[x_1,\dots,x_n]$, with $k$ a fields. But I think it is. And I have no idea how to do the exercise. How can I prove that something is a vector space using a map? I don't know how to do nor a) part nor b)