Classification of finitely generated multigraded modules over $K[x_1,\ldots,x_n]$? 
Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded
  polynomial ring. Have
  finitely-generated multigraded $R$-modules been classified? Are they
  of the form $R^r\oplus\bigoplus_{i=1}^sR/Rx^{a_i}$ for some (unique?)
  $a_1,\ldots,a_s\in\mathbb{N}^n$? 

I was thinking along the following lines. If an $\mathbb{N}^n$-graded $R$-module $M$ is generated by $v_1,\ldots,v_r$, then we may assume that every $v_i$ is homogenous of degree $a_i$ (otherwise each of these $v_i$ is a further finite combination of homogenous vectors). Let $R^{[a]}$ be the $R$-module $R$ with the grading shifted by $a\!\in\!\mathbb{N}^n$. Thus the map $R^r\!=\!\bigoplus_{i=1}^rR^{[a_i]}\rightarrow M$ that sends $e_i\mapsto v_i$ is a surjective graded morphism, so its kernel $A$ is a graded submodule and $M\cong R^r/A$, i.e. $$\textstyle{A=\bigoplus_{b\in\mathbb{N}^n}(R^r)_b\cap A=\bigoplus_{b\in\mathbb{N}^n}\prod_iKx^{a_i+b}\cap A}.$$ Hence $A$ is generated by $u_1,\ldots,u_s$ where every component of $u_i$ is $\alpha_{ji}x^{a_i+b_j}$ for some $\alpha_{ji}\!\in\!K$. 
Thus our module is isomorphic to the cokernel of the matrix $$\left[\begin{matrix}
\alpha_{11}x^{a_1+b_1} &\alpha_{12}x^{a_1+b_2}&\ldots&\alpha_{1s}x^{a_1+b_s}\\
\alpha_{21}x^{a_2+b_1} &\alpha_{22}x^{a_2+b_2}&\ldots&\alpha_{2s}x^{a_2+b_s}\\
\vdots&\vdots&\vdots&\vdots\\
\alpha_{r1}x^{a_r+b_1} &\alpha_{r2}x^{a_r+b_2}&\ldots&\alpha_{rs}x^{a_r+b_s}\\
\end{matrix}\right].$$
Now we can do row and column operations, without changing the isomorphism type of the module. But I'm not sure how the above can be transformed into
$$\left[\begin{matrix}
x^{c_1} &&&\\
 &x^{c_2}&&\\
&&\ddots&\\
 &&&x^{c_s}\\
\end{matrix}\right].$$
If my conjecture is not valid, I ask for a counterexample. For instance, if $n=2$ and $R=K[x,y]$, are $Coker\left[\begin{smallmatrix}
x^2y &xy\\ & xy^2\\
\end{smallmatrix}\right]$ or $Coker\left[\begin{smallmatrix}
x &y\\
\end{smallmatrix}\right]$  or $Coker\left[\begin{smallmatrix}
x \\y
\end{smallmatrix}\right]$ not of the above form? 

Is there some other classification of
  $\mathbb{N}^n$-graded $K[x_1,\ldots,x_n]$-modules? Maybe they are of the 
  form $\bigoplus_{i=1}^sR^{[a_i]}/\langle x^a; a\!\in\!A_i\rangle$ for some
  (unique?) $a_1,\ldots,a_s\!\in\!\mathbb{N}^n$ and $A_1,\ldots,A_s\!\subseteq\!\mathbb{N}^n$?

By Monomial Ideals (Herzog & Hibi, 2011), Dickson’s lemma 2.1.1, every $A_i$ may be finite. Also, elements of $A_i$ are assumed to be incomparable w.r.t. the componentwise partial order on $\mathbb{N}^n$.
 A: For $n=2$ let $M$ be the module with graded parts
$$M_{i,j}=\begin{cases}
K&\mbox{ if }(i,j)=(1,1),(1,0)\mbox{ or }(0,1)\\
0&\mbox{ otherwise.}
\end{cases}$$
and with $x_1$ and $x_2$ acting as isomorphisms $M_{0,1}\to M_{1,1}$ and $M_{1,0}\to M_{1,1}$ respectively.
Then your conjecture is not true for $M$.
If $n>2$ (possibly also $n=2$) then this is a "wild" problem: classifying such modules is at least as hard as classifying pairs of square matrices up to simultaneous conjugacy.
Edit: This last claim is not too complicated to see for $n=5$:
Let $V=K^d$ and let $A,B$ be $d\times d$ matrices over $K$. Define a multigraded $K[x_1,\dots,x_5]$-module $N=N(A,B)$ with
$$
N_{00000}=V\oplus V,$$
$$N_{10000}=N_{01000}=N_{00100}=N_{00010}=N_{00001}=V,$$
and all other components zero,
so the action of each $x_i$ is determined by a map $V\oplus V\to V$, given by the matrices
$$\begin{align}
X_1&=\begin{pmatrix}I_n&0\end{pmatrix}\\
X_2&=\begin{pmatrix}0&I_n\end{pmatrix}\\
X_3&=\begin{pmatrix}I_n&I_n\end{pmatrix}\\
X_4&=\begin{pmatrix}I_n&A\end{pmatrix}\\
X_5&=\begin{pmatrix}I_n&B\end{pmatrix}\\
\end{align}$$
An easy computation shows that an isomorphism $N(A,B)\cong N(A',B')$ is determined by a single automorphism of $V$, given by a matrix $T$ such that
$T^{-1}AT=A'$ and $T^{-1}BT=B'$, so the classification of this particular class of multigraded modules up to isomorphism is just the classification of pairs of square matrices up to simultaneous conjugacy.
A: Carlsson and Zomorodian (2009) gave a classification theorem for finitely generated multigraded modules over $R=k[X_1, \ldots, X_n]$ in their paper on multiparameter persistence. The motivation for their study is that the category of discrete multiparameter persistence modules is equivalent to the category $\mathcal{C}$ of finitely generated multigraded $R$-modules.
The classification is a little complicated - in particular, there is no discrete complete parametrisation as in the $n=1$ case. I will try to describe their work briefly.
A free object $F \in \mathcal{C}$ has as its basis a multiset $\xi(F) := \{(x_1, \alpha_1), \ldots, (x_m, \alpha_m)\} \subset \mathbb{Z}^n \times \mathbb{N}$, such that $F$ is the direct sum $\bigoplus_{(x, \alpha) \in X} \Sigma^x R^\alpha$ (where $\Sigma^x$ denotes a degree shift by $x$). For arbitrary $M$ in $\mathcal{C}$ the classification proceeds by considering the generating set and relations.

*

*Let $\rho(M)$ denote the multigraded $k$-vector space $k \otimes_R M$, where $k$ has the $R$-module structure such that the $x_i$ act trivially. There is a free object $F$ in $\mathcal{C}$ and an epimorphism $\pi_M: F \to M$ such that $1 \otimes_R \pi : \rho(F) \to \rho(M)$ is an isomorphism of multigraded $k$-vector spaces, and $(F, \pi_M)$ is unique upto isomorphism.

*For every $M \in \mathcal{C}$ we have two invariants $\xi_0(M) := \xi(\rho(M))$ and $\xi_1(M) := \xi(\rho(\ker(\pi_M))$.

*The isomorphism class of $M$ is the isomorphism class of $F/L$ where $L$ is any member of the orbit of $G_F := \operatorname{Aut}(F)$ acting on the set of subobjects $S := \{L \subset F \mid \xi_0(L) = \xi_1(M)\}$.

In general $\operatorname{Aut}(F)$ is algebraic group and $S$ is a quasiprojective variety so the orbit space might not have a nice structure.
