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?)?
> 
> 
> 



Characterisation of minimal projective resolutions via the Euler characteristic
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.


