# Can you detect homological dimensions from homology?

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?

• What do you mean by the homological/injective dimension of a chain complex? – Fernando Muro Aug 24 '15 at 15:28
• If you mean the minimum length of an injective complex that is quasi-isomorphic to the original complex, then I think the answer is yes, by writing the complex up to quasi-isomorphism as an iterated extension of its homology groups and applying the mapping cone to construct an injective resolution of the complex from the injective resolution of the homology groups. – Will Sawin Aug 24 '15 at 15:42
• @WillSawin do you mean complexes of injective modules? By contrast, injective complexes are contractible, I think. – Fernando Muro Aug 24 '15 at 22:24
• @FernandoMuro Yes, I mean precisely that. – Will Sawin Aug 25 '15 at 2:01

Consider the ring $R = \mathbb Z/4$. Then $\mathbb Z/4$ is both injective and projective over itself, whereas $\mathbb Z/2$ has infinite projective and injective dimension.
We could have the chain complex: $0 \to \mathbb Z/4 \to^{\cdot 2} \mathbb Z/4 \to 0$ with the homologies having infinite dimension and the terms of the complex not. Or we could have $0 \to \mathbb Z/2 \to^{\cdot 1} \mathbb Z/2 \to 0$ with the homologies having having $0$ dimension, but the terms of the complex not.