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    $F[x]\oplus F$ as the cokernel of a $2\times 2$ Smith normal form matrix

    P-QVCO4

    Area
    Algebra
    Topics
    Modules, Smith Normal Form, Structure Theorem
    Status
    draft

    Construct a \(2\times2\) matrix \(A\) over \(F[x]\) whose cokernel is \[ F[x]\oplus F. \]

    Solution

    Choose any linear polynomial \(f(x)\in F[x]\), for example \[ f(x)=x+1. \] Since \[ F[x]/(x+1)\cong F, \] we want a Smith normal form with one nonzero invariant factor \(x+1\) and one zero diagonal entry.

    Take \[ A= \begin{pmatrix} x+1&0\\ 0&0 \end{pmatrix}. \] Then \[ \operatorname{coker}(A) \cong F[x]/(x+1)\oplus F[x]/(0) \cong F\oplus F[x]. \] Thus, after reordering the summands, \[ \operatorname{coker}(A)\cong F[x]\oplus F. \]

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