Characterisation of minimal projective resolutions via the Euler characteristic

Question

Let $A$ be a finite dimensional $K$-algebra (where $K$ is a field) and $M$ a finitely generated $A$-module.

Let $\psi: 0 \rightarrow P_r \rightarrow ... \rightarrow P_0 \rightarrow M \rightarrow 0$ be a complex of $A$-modules such that $P_0 \rightarrow M$ is the projective cover of $M$ and $P_r \rightarrow P_{r-1}$ is injective and where the $P_l$ are projective for $l=0,1,...,r$ and the maps $d_i :P_i \rightarrow P_{i-1}$ are minimal, that is $d_i(X) \neq 0$ for any direct summand $X$ of $P_i$. (so $\psi$ is nearly a minimal projective resolution of $M$, the only thing missing is exactness)

Set $P_{-1}:=M$, then it is well known that the Euler characteristic $\chi(\psi):=\sum\limits_{i=-1}^{r}{dim(P_i)}$ is equal to zero in case $\psi$ is exact.

I remember that there is a converse to that, namely that $\psi$ is exact in case $\chi(\psi)=0$ under some extra conditions on $\psi$. I forgot where I saw that. Maybe someone knows a reference for what I have in mind.

Question: Are the conditions here enough to prove that $\psi$ is exact in case $\chi(\psi)=0$ or what other conditions are needed (is there a reference in case this is well known?)?

Answer 1

Without more conditions it's not true.

Take the Nakayama algebra with two simples and indecomposable projectives $$P(1)=\matrix{1\\2\\1}\hspace{1cm}\text{and}\hspace{1cm}P(2)=\matrix{2\\1}$$

Then there is a complex $$0\to P(2)\to P(1)\to P(1)\to P(1)\to P(1)\to\matrix{1\\2}\to 0$$ which satisfies your conditions, is not exact, but has zero Euler characteristic.