# Program to calculate homology of a Koszul complex involving univariate polynomials

Let $$R = \mathbb{Z}[x_1,...,x_6]$$ be a polynomial ring. Then we may form the Koszul complex $$K(x_1,...,x_6)$$ which looks something like:

$$R \xrightarrow{d_6} R^6 \xrightarrow{d_5} R^{15} \xrightarrow{} ... \xrightarrow{d_2} R^6 \xrightarrow{d_1} R$$

where, for example, $$d_1 = (x_1,x_2,...,x_6)$$

$$d_2 = \begin{pmatrix} -x_2 &-x_3 &-x_4 &-x_5 &-x_6 & 0 &0& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ x_1 & 0 & 0 & 0 & 0 &-x_3 &-x_4& -x_5& -x_6& 0& 0& 0 & 0 & 0& 0\\ 0 & x_1 & 0 & 0 & 0 & x_2 & 0 & 0 & 0& -x_4& -x_5& -x_6 & 0 & 0 & 0\\ 0 &0& x_1& 0 & 0 & 0 & x_2 & 0 & 0 & x_3 & 0 & 0 &-x_5 &-x_6 & 0\\ 0 & 0 & 0 & x_1 & 0 & 0 & 0 & x_2 & 0 & 0 & x_3 & 0 & x_4 &0 &-x_6\\ 0 & 0 & 0 & 0 & x_1 & 0 & 0 & 0 & x_2 & 0 & 0 & x_3 & 0 & x_4 & x_5\\ \end{pmatrix}$$

etc. Now, each of the $$x_i$$ can be expressed as a polynomial in $$\mathbb{Z}[t]$$ (what is happening here is that the $$x_i$$ are characters of a group, and in writing them as polynomials in $$\mathbb{Z}[t]$$ we are restricting the characters, but this is not so important).

So let's say for example we have:

$$x_1 = (t-2)(t^2 + 1)$$ $$x_2 = (t-2)(t+2)(t^3 + 4t + 8)$$ $$\dots$$

then by tensoring above complex with $$\mathbb{Z}[t]$$ (i.e. $$- \otimes_R \mathbb{Z}[t]$$) we form what I am calling the Koszul complex in one variable:

$$\mathbb{Z}[t] \xrightarrow{d_6} \mathbb{Z}[t]^6 \xrightarrow{d_5} \mathbb{Z}[t]^{15} \xrightarrow{} ... \mathbb{Z}[t]^6 \xrightarrow{d_1} \mathbb{Z}[t]$$

where the $$d_i$$ are just the $$d_i$$ above, but each of the $$x_i$$ are replaced with their expression/restriction in $$\mathbb{Z}[t]$$.

I have found that sagemath has a KoszulComplex function which gives the Koszul complex in terms of the $$x_i$$, but I don't believe there any any implement for using this to compute homology groups.

I cannot seem to find a prebuilt way to compute the homology groups here, although it seems like this should exist. Finally, I will add that if Euclidean-domainness is an issue, I am happy to replace $$\mathbb{Z}[t]$$ with $$F[t]$$ where $$F = \mathbb{F}_p$$ or $$\mathbb{Q}$$.

I do not know any implementations of "homology of a chain complex" over any ring except $$R = k$$ a field or $$R = \Bbb Z$$, and the latter is much less developed.