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Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$.

Let $M$ be a finitely generated $kG$-module.

We denote the first syzygy of $M$ by $\Omega(M)$, i.e. $\Omega(M):=\text{Ker}(p)$ where $P\stackrel{p}{\rightarrow} M$ is a minimal projective cover of $M$.

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A $kG$-homomorphism $m: M\rightarrow \Omega(M)$ is called almost projective,

if $m$ is a generator of the simple socle of $\text{Hom}_{kG}(M,\Omega(M))/\text{Proj}_{kG}(M,\Omega(M))$ as $\text{End}_{kG}(M)$-module.

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Here, $\text{Proj}_{kG}(M,\Omega(M))$ is the set of all projective $kG$-homomorphisms from $M$ to $\Omega(M)$ (i.e. those that factor through a projective $kG$-module).

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I would like to ask the following question:

Can MAGMA produce / compute such an almost projective $kG$-homomorphism $m$?

Thank you very much for the help.

EDIT (9th April): The motivation is to construct almost split sequences with MAGMA and this is related to the following earlier question:

https://math.stackexchange.com/questions/3317886/can-magma-compute-auslander-reiten-sequences-in-group-algebras

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I cannot answer 100%, but I can tell you what I know is there, and maybe its enough with some tweaking. AR-sequences are not something I've needed to implement in Magma yet, so I've not grappled with this one.

Magma can compute projective covers and a syzygy, first off. Then it can compute $\texttt{AHom(A,B)}$, which is simply $\mathrm{Hom}_{kG}(A,B)$. It can also compute $\texttt{PHom(A,B)}$, but this is only for basic algebras, not general group algebras. If your group is small enough that you can produce its basic algebra in Magma (there are commands for this) then you can pass between then to find $PHom$ inside $Hom$.

Finally, $Hom$ can be made into a $kG$-module itself, using $\texttt{HomMod}$. (This requires a recent version of the program.)

If you start off in a basic algebra, you can type $\texttt{SyzygyModule(M,1)}$ for $\Omega(M)$.

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  • $\begingroup$ Thank you very much for the answer. $\endgroup$ May 31, 2020 at 15:19

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