ALIBABA 2021 Problem 20

Let $M=\bigoplus_{i \in \mathbb{Z}} \mathbb{C}e_i$ be an infinite dimensional $\mathbb{C}$ -vector space, and let $\text{End}(M)$ denote the $\mathbb{C}$ -algebra of $\mathbb{C}$ -linear endomorphisms of $M$ . Let $A$ and $B$ be two commuting elements in $\text{End}(M)$ satisfying the following condition: there exist integers $m \le n<0<p \le q$ satisfying $\text{gd}(-m,p)=\text{gcd}(-n,q)=1$ , and such that for every $j \in \mathbb{Z}$ , one has $$Ae_j=\sum_{i=j+m}^{j+n} a_{i,j}e_i, \text{with } a_{i,j} \in \mathbb{C}, a_{j+m,j}a_{j+n,j} \ne 0,$$ $$Be_j=\sum_{i=j+p}^{j+q} b_{i,j}e_i, \text{with } b_{i,j} \in \mathbb{C}, b_{j+p,j}b_{j+q,j} \ne 0.$$ Let $R \subset \text{End}(M)$ be the $\mathbb{C}$ -subalgebra generated by $A$ and $B$ . Note that $R$ is commutative and $M$ can be regarded as an $R$ -module.

(a) Show that $R$ is an integral domain and is isomorphic to $R \cong \mathbb{C}[x,y]/f(x,y)$ , where $f(x,y)$ is a non-zero polynomial such that $f(A,B)=0$ .

(b) Let $K$ be the fractional field of $R$ . Show that $M \otimes_R K$ is a $1$ -dimensional vector space over $K$ .