What is the convention on exterior derivatives especially concerning $H^0_{dR}(M)$?
Indeed, $H^0_{dR}(M) = Z^0(M)/B^0(M)$ where $B^0(M) = \Omega^0(M)\cap \text{Im}(d)$. However $d$ is not defined on $\Omega^{-1}(M)$ because $\Omega^{-1}(M)$ is not defined.