Suppose you are given a bounded chain complex $M$ over a commutative ring $R$.

Is there a clear relation between homological dimensions of $M$ and homological dimensions of its cohomologies?

For example, suppose I know that $injdim(H^i(M))<\infty$ for all $i$, does this imply that $injdim(M)<\infty$? what about the converse? what about projective and flat dimension?

Any reference for this?