# How to compute in exterior powers of a ring

Consider $H := \mathbb{Z}^n$. I have a finite family of matrices $M_i$ , acting over H and hence acting over $\Lambda^2 (\Lambda^3 H)$. I also have $v_1$ and $v_2$ in $\Lambda^2 (\Lambda^3 H)$. I want to compute the submodule generated by $M_i \cdot v_1 , M_i \cdot v_2$ for all $i$. More precisely I want to know if the quotient by this submodule is torsion-free.

Now, my problem is that apparently, SageMaths does not handle $\Lambda^2 (\Lambda^3 H)$.

Q1) Is there any way to do that in a simple way ?

My solution was to send $\Lambda^2 (\Lambda^3 H)$ in $H^{\otimes 6}$, and compute in this submodules using "Finitely generated modules over a PID", but I have a problem with the size of the vectors. Even for $n = 6$, the vectors in $H^6$ are way to big. I tried to use sparse matrices sparse vectors, because all of the vectors will be full of 0's.

Q2) Is there a simple way to use sparse matrix/vectors and still use the same library ?

I think this should be fairly simple, and I would like to solve this problem at least for n = 6 to 10.

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