Lifting Idempotents from $\mathbb{F}_pA_5$ to $\mathbb{Z}_{(p)}A_5$

Supplementary code for the paper "On a question by Roggenkamp about group algebras", which was joint work with Dmitriy Rumynin.

Let $\mathbb{F}_p$ be the prime field of characteristic $p$, and let $\mathbb{Z}_{(p)}$ denote the localisation of the integers at the prime $p$. Let $A_5$ be the alternating group on $5$ letters. We have $|A_5| = 60 = 2^2\cdot3\cdot5$.

Lifting an idempotent $\overline{e} \in \mathbb{F}_pA_5$ to $\mathbb{Z}_{(p)}A_5$ amounts to finding an element $e \in \mathbb{Z}_{(p)}A_5$ such that:

• $e \equiv \overline{e} \pmod{p}$
• $e^2 = e$

In this repository, primitive idempotents (equivalently, indecomposable projective modules) are lifted from $\mathbb{F}_pA_5$ to $\mathbb{Z}_{(p)}A_5$ for the "modular primes" $p = 2$, $3$, and $5$.

For each of these primes, there are three indecomposable projective modules to lift. The process of lifting them is as follows:

• Identify primitive idempotents in $\mathbb{F}_pA_5$,
• Lift these primitive idempotents to the group ring over the $p$-adic integers $\widehat{\mathbb{Z}_p}A_5$.
• Check that all coefficients in the lift to $\widehat{\mathbb{Z}_p}A_5$ have $p$-adic expansions with repeating digits, i.e., the coefficients lie in $\mathbb{Z}_{(p)} \subset \mathbb{Q}$.

Details can be found in the following notebooks:

• idempotent-lift-F_2A_5.ipynb
• idempotent-lift-F_3A_5.ipynb
• idempotent-lift-F_5A_5.ipynb

All three notebooks follow a similar structure. In fact, the $p = 3$ case was completed first, and the others are copies of it with the relevant modifications. This approach results in heavy code repetition, but in return, each notebook is fully self-contained. This was deemed to be worth it.