I would like to understand the category of modules over the group algebra $\mathbb{F}_p[C_p]\cong\mathbb{F}_p[X]/(x^p-1)$.
I am interested in computing the group cohomology of $C_p$ with coefficient in some $\mathbb{F}_p$ on which $C_p$ acts. I thought a good understanding of the category might help me (e.g. deep understanding of the Ext groups).
Ultimately, I am aiming for a decomposition theorem for such spaces. For example, for $p=2$, any $\mathbb{F}_2[C_2]$-module $V$ decomposes as a direct sum
$$ \mathbb{F}_2^a\oplus \mathbb{F}_2[C_2]^b, \qquad a+2b=\dim_{\mathbb{F}_2}V. $$
Any help is very welcome.
First, note that there is an isomorphism $$ \mathbb{F}_p[y]/(y^p) \cong \mathbb{F}_p[x]/(x^p-1) $$ sending $y$ to $x-1$. Next, as long as you're willing to stick with finitely generated modules, then you have a classification: finitely generated modules over $\mathbb{F}_p[y]/(y^p) $ are just finitely generated modules over $\mathbb{F}_p[y]$ on which $y^p$ acts trivially. Since $\mathbb{F}_p[y]$ is a PID, there is a classification of modules over it. (We could skip most of this and just skip to classifying modules over $\mathbb{F}_p[x]$ on which $x^p-1$ acts trivially, but it may be easier to use this alternate form for the ring.)
The remaining step is computing cohomology for each indecomposable module, and that's easy to do by hand.