I'm trying to understand a sketch of a proof that the group ring $k[C_p^2]$ has infinitely many non-isomorphic 2-dimensional (i.e, length 2) indecomposable modules. I need help with some details, and I add my comments, so somebody can correct me if I'm thinking wrong. It goes as follows:
$A=k[G]$ is local. Denote it's maximal ideal by $M$, which is the augmentation ideal. (I don't know if it's important for this proof to know that it's the augmentation ideal.)
$\dim_k(M/M^2)\geq 2$; for otherwise $\dim_k(M/M^2)=1$ implies $\dim_k(A)=\dim_k(A/M^p)=p$ a contradiction. (The implication is because of the sequence $A\supset M\supset M^2 \supset \dots \supset M^p=0$ I guess).
From this point I totally lost track.
When $I$ ranges over the hyperplanes of $M/M^2$, the $A/(M^2+I)$ form pairwise non-isomorphic 2-dimensional indecomposable modules.
As far as I know, hyperplanes of $M/M^2$ are subspaces of $M/M^2$ of codimension 1. Do I need to think them as $I/M^2$ to think $A/(M^2+I)$. Even if that's the case, how can I say there are infinitely many non-isomorphic of these.
If this need geometric concepts, just point me to the right direction with some details of the proof please.
Edit: This proof is from a comment that I took screenshot at mathoverflow from the user Ycor
Let $I\subset M$ be a submodule with $I/M^2$ a hyperplane of $M/M^2$. Then $I$ is in particular an ideal of $A$ and the annihilator of $A/I$ is precisely $I$.
So if $A/I\simeq A/J$, then their annihilators are the same, so $I=J$: they are all pairwise nonisomorphic.
Now the "geometry" part is that since $M/M^2$ is of dimension at least $2$, it has infinitely many hyperplanes, which is a standard fact of linear algebra.
Then you also need to figure out why $A/I$ is indecomposable and of dimension $2$, but you didn't ask specifically for those, so I don't know if you want the proofs. If you do, ask me in the comments.