Suppose we want to compute the direct limit of the direct system $$\mathbb{Z}^{14} \overset{A}{\longrightarrow}\mathbb{Z}^{14} \overset{A}{\longrightarrow}\mathbb{Z}^{14} \overset{A}{\longrightarrow}\cdots$$ where $A$ is a $14 \times 14$ square matrix whose characteristic polynomial is given by $P(x) = (x-2)(x-1)^3(x+1)^6(x^2-2x+2)(x^2-2)$. Then what is the direct limit?
My guess is that, the direct limit is $\mathbb{Z}^9 \oplus \mathbb{Z}[\frac{1}{2}]^5$. Note that $\det(A)= -8$.
I already know that the direct limit is a subset of $\mathbb{Z}[\frac{1}{8}]^{14} = \mathbb{Z}[\frac{1}{2}]^{14}$.
I also know that if the matrix $A$ is diagonalizable such that all its eigenvalues are integers, then the direct limit is just $\bigoplus_i \mathbb{Z}[\frac{1}{m_i}]$, where $m_i := \lambda_i \in\mathbb{Z}$ is the $i$th eigenvalue of $A$. Further, if $A$ is nonsingular with non-integral eigenvalues, together with the condition that its determinant divides all but the leading coefficient of its characteristic polynomial, then the direct limit is just $\mathbb{Z}[\frac{1}{d}]^n$, where $n$ is the dimension of $A$ and $d := \det(A)$.
So, my problem is, what if it's a combination of both scenarios, like the one I am asking now?
