MathDB

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

Problem Statement