I have a $p^3$-dimensional (not semisimple) algebra $\mathcal{U}_{q}(sl_2)$ over $\mathbb{C}$ and i know how all its simple modules look like (there are $p$ of them, each $M_i$ has dimension $i$ for $i = \overline{1, p}$, $M_p$ is projective) and now i want to find $\mathrm{Ext}^i(M, N)$ for $M, N$ - simple; or at least $\mathrm{Ext}^1$. The problem is that projective modules in this example are constructed via extensions of simple ones (i've got a hint that each projective is glued out of 2 copies of $M_i$ and 2 copies of $M_{p -i}$, thus we may aswell compute only $\mathrm{Ext}^1(M_i, M_{p - i})$) and i really don't want to write down a huge resolution made out of free modules.
The question can be formulated as follows: are there any ways to compute ext functors if projectives are not known and free modules are too big? I would be happy with either tricks for this particular case or for a more general approaches, even if they don't work with this example.
Definition of the algebra $\mathcal{U}_{q}(sl_2)$ which i use, $q = e^{2\pi i/p}$:
$\mathcal{U}_{q}(sl_2) = \mathbb{C}\langle E, F, H\rangle / \langle E^p = F^p = H^p - 1 = 0,\,\, EF - FE = \frac{H - H^{-1}}{q - q^{-1}},\,\,HF =q^{-2}FH,\,\,\,\,\, HE = q^2EH \rangle$