Suppose $\Delta(n,k)$ is the algebra of upper triangular $n$ by $n$ matrices over a field $k$. Furthermore, let $M$ is an artinian module over $\Delta(n,k)$, and let $ \cdots\to C_n\stackrel{d_n}{\to}C_{n-1}\to\cdots\to C_1\stackrel{d_1}{\to}C_0\stackrel{\epsilon}{\to} M\to 0 $ be a projective resolution of $M$, so $\epsilon$ is a homomorphism and $\epsilon d_1=0$.
Edited: Based on the comments I received, I'm trying to revise the question to something less abstract. Although I see now that the projective resolution need not be finite, is it always possible to truncate it to yield a projective resolution of length two? Is there some justification for this, or have I misunderstood what I've been told?
Thanks.