Let $k$ be a field and $V$ a finite-dimensional vector space. Show that for every natural number $n$ there is a a finite projective resolution of $k$ -vector spaces
$0→V_n→V_{n−1} → ··· → V _2 → V _1 → V _0 → V → 0$ with $V _i \neq 0$ for all i = 1 , . . . , n
In general, a resolution is made of short exact sequences. In your case you can start with $V_0=V\oplus k$ and $V_0\to V$ the projection on $V$. The kernel of this map is $k$. Then take $V_1=k\oplus k$ and $V_1\to k$ a projection. The kernel of this map is also $k$, and so on. As a final step you can consider the identity map from $k$ to itself.