Let A be a local ring and M an $A$-module which has a finite free resolution $$0\rightarrow F_{1}\rightarrow F_{0}\rightarrow M\rightarrow 0.$$ In this situation, does a minimal resolution $$0\rightarrow L_{1}\rightarrow L_{0}\rightarrow M\rightarrow 0$$ of M always exist?
In the proof of Theorem 19.6 in Matsumura's "Commutative Ring Theory", I got stucked at this point. Why one can say M has a minimal resolution with a length less than two?