# cokernel of a map between modules over polynomial rings

Define a polynomial ring $R$ as $F_{2}\left[x_{1},x_{1}^{-1},....x_{D},x_{D}^{-1}\right]$ where $D$ is the dimension and $\mathbb{F}_{2}$ is a binary field. Let $G$ be a free $R$-module of some labels and has rank $t$. $P$ be a free $R$-module of Pauli operators. $\sigma$ is a map from $G$ to $P$. I want to write a snippet to calculate the cokernel of this map.

Just for example (taken from page 54 of arxiv.1305.6973 or page 41 of arxiv.1607.01387), though this is not essential for the question, I can have two interaction'' terms in terms of 2-dimensional Pauli operators $X$, $Z$ and Identity operator $I$ on 4 sites with two 2-dimensional systems per site as $II(0,0)-IX(0,1)-XI(1,0)-XX(1,1)$ and $ZZ(0,0)-IZ(0,1)-ZI(1,0)-II(1,1)$ where on each site $\left(x,y\right)$ (mentioned in the bracket after the Pauli operators), the first(second) Pauli acts on the first(second) two dimensional system on that site. The map $\sigma$ can be written as

$\sigma=\left(\begin{array}{cc} y+xy & 0 \ x+xy & 0 \ 0 & 1+y \ 0 & 1+x \end{array}\right)$

where for example, $y+xy$ is a polynomial that specifies the action on the first two dimensional system as

$y+x y=0 \hspace{1mm} x^0 y^0+ 1 \hspace{1mm} x^0 y^1+0 \hspace{1mm} x^1 y^0 +1 \hspace{1mm} x^1 y^1$

where the exponents are the coordinates of the sites and coefficients $0$ and $1$ imply whether there is a Pauli acting or not.

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