# Relation between Cohomological dimensions of manifolds

Let $$M$$ be a connected manifold of finite type. We denote $$Ch_{\mathbb{Q}}(M),$$ $$Ch_{\mathbb{Z}}(M)$$ and $$Ch_{\mathbb{\pm}\mathbb{Z}}(M)$$ by cohomological dimensions of $$M$$ over $$\mathbb{Q},$$ $$\mathbb{Z}$$ and $$\pm\mathbb{Z}$$ (local coefficients) respectively. Is it always true that $$Ch_{\mathbb{Q}}(M)\leq Ch_{\mathbb{Z}}(M)\leq Ch_{\mathbb{\pm}\mathbb{Z}}(M)?$$ I know that if $$M$$ is orientable then $$Ch_{\mathbb{Z}}(M)= Ch_{\mathbb{\pm}\mathbb{Z}}(M).$$ Can we say that $$Ch_{\mathbb{Z}}(M)< Ch_{\mathbb{\pm}\mathbb{Z}}(M)$$ if $$M$$ is non-orientable?