Let $k$ be a perfect field of characteristic $p>0$. For any positive integers $n$, let $W\_n(k)$ be the truncated Witt vectors of length $n$ with coefficients in $k$. For any positive integers $a,b$, is $\textrm{Ext}(W\_a(k), W\_b(k))$ as an abelian group well understood? Are there any references? Thank you in advance!


Here $\textrm{Ext}(W\_a(k), W\_b(k))$ is the abelian group that contains all the $W(k)$-modules $M$ (up to isomorphisms) such that $0 \to W\_b(k) \to M \to W\_a(k) \to 0$ is a short exact sequence of $W(k)$-modules.



Extensions of truncated Witt vectors
Answer reposted from comment:


The short exact sequence $0\to W\to W\to W\_a\to 0$, where the map $W\to W$ is multiplication by $p^a$, is a projective resolution of $W\_a$ over $W$. Applying $\mathrm{Hom}\_W(-, W\_b)$ to this resolution, we get that $\mathrm{Ext}\_W(W\_a, W\_b)$ is just $W\_b/p^a W\_b = W\_{min(a, b)}$. 


