cokernel of a map between modules over polynomial rings

asked 2018-04-24 21:02:38 -0600

arpit gravatar image

updated 2018-04-28 16:43:09 -0600

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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Comments

There is no page 54 in the searched link, possibly https://arxiv.org/pdf/1607.01387.pdf. This is already a problem, when potential helpers try to understand the question. (There are many pages in the link, and the work to understand it, or at least find the minimal path to the example is in no relation to a code snippet for computing a cokernel of a map.) A module is not a "stabilizer map", to realize a map as a matrix we have to know the concrete field and map. Please bring concreteness in the question, best, reduce it to a programming question.

dan_fulea gravatar imagedan_fulea ( 2018-04-25 05:57:58 -0600 )edit

@dan_fulea, my apologies, I have made the correction to page numbers now. And I will edit the question to make the problem more concrete.

arpit gravatar imagearpit ( 2018-04-25 10:15:53 -0600 )edit