# Problem

(Weibel's Introduction to Homological Algebra, Exercise 5.7.4,2) Let $\mathbf{Ch}_{\ge0}(\mathcal A)$ be the subcategory of complexes $A$ with $A_p=0$ for $p<0$. Then the hyper-derived functors $\mathbb L_iF$ retricted to $\mathbf{Ch}_{\ge0}$ are the left derived functors of the right exact functor $H_0F$.

# Thoughts

First the $F$ in $H_0F$ should be the functor $\mathcal B=\mathbf{Ch}_{\ge0}(\mathcal A)\to\mathcal B$ induced by $F$, I think. We know that the projective objects in $\mathcal B$ are just almost-acyclic ($H_n=0$ for $n\neq0$) chain complexes of projective objects in $\mathcal A$. However, it's quite hard for me to connect it with Cartan-Eilenberg resolutions. On the other hand, $\mathbb L_iF(A)$, by definition, is $H_i\operatorname{Tot}(F(P))$ where $P\to A$ is a Cartan-Eilenberg resolution. In order to assimilate this with the expression of derived functor of $H_0F$, it seems to me that the projective resolution of $A$ should be chosen from $\operatorname{Tot}(P)$, but I have no idea how to proceed.

Any idea? Thanks!

It's solved now.

Suppose that $$P\to A$$ is a projective resolution in $$\mathbf{Ch}_{\ge0}(\mathcal A)$$, and $$P_{\bullet,\bullet}$$ is the associated double complex (with a sign convention). Take $$Q_i=\operatorname{coker}d_{i,1}^h$$, then $$Q_\bullet$$ is a chain complex of projective objects in $$\mathcal A$$. The first and second spectral sequence associated with $$P_{\bullet,\bullet}$$ converges to $$H_\bullet(\operatorname{Tot}(P_{\bullet,\bullet}))$$, therefore $$Q_\bullet$$ is quasi-isomorphic to $$A_\bullet$$. Now it follows that $$\mathbb L_iF(A)=H_i(FQ_\bullet)=H_i(H_0FP)=L_i(H_0F)(A)$$.

A lemma is used here: Suppose $$Q_\bullet$$ is a chain complex of projective objects quasi-isomorphic to $$A_\bullet$$, then $$\mathbb L_iFA=H_i(FQ_\bullet)$$. A general reference for this is Iversen's Cohomology of Sheaves, Chapter 1, Theorem 6.2.

• This lemma was very helpful! I am wondering if Prof. Weibel had a different proof in his mind avoiding this lemma. It seems that if one starts with Cartan-Eilenberg double complex P.,. for A. and applies F to it, then the II spectral sequence would imply the result, if one can show (H_n F) (P.,k)=0 for n>0. I guess that should follow from the projectivity of P.,. ... Commented Sep 27, 2017 at 14:16
• Why $P_\bullet$ is quasi-isomorphic to $A_\bullet$? We know isomorphisms between homologies, but what is the chain map which induces these isomorphisms? Commented Dec 1, 2017 at 20:24
• @RafaelHolanda I don't have Weibel at hand. I think that what I obtained is that $P_\bullet$ and $A_\bullet$ are both quasi-isomorphic to $\operatorname{Tot}(P_{\bullet,\bullet})$, therefore they are quasi-isomorphic. To be more precise, if there is a chain map $f\colon C_\bullet\to P_\bullet$, where $P_\bullet$ is made up of projective objects, which induces isomorphisms on homologies, then there is also a chain map $P_\bullet\to C_\bullet$ which induces isomorphisms, by considering the isomorphism $[P_\bullet,f]\colon[P_\bullet,C_\bullet]\to[P_\bullet,P_\bullet]$. Commented Dec 2, 2017 at 21:43
• How is the last equality obtained? Commented Jan 28, 2019 at 18:10

Here is an alternative to using the lemma you cite by using the Grothendieck spectral sequence isomorphism. Logically, it's probably worse than your lemma but I find it conceptually helpful.

First show $$L_iH_0=H_i$$ as functors $$C_*(\mathcal{B})\rightarrow{\mathcal{B}}$$, i.e. higher homology is derived zero homology. This can be done by a similar method of comparing the two spectral sequences of a protective resolution in the category of chain complexes.

Then apply the Grothendieck spectral sequence isomorphism to the composition $$C_{\geq 0}(F):C_{\geq 0}(\mathcal{A})\rightarrow C_{\geq 0}(\mathcal{B})$$, $$H_0:C_{\geq 0}(\mathcal{B}) \rightarrow B$$. One obtains $$R_*(H_0\circ F)=H_*\circ(R_*F)$$ where $$R_*F:D_{\geq 0}(\mathcal{A})\rightarrow D_{\geq 0}(\mathcal{B})$$ maps $$A_*$$ to $$Tot(FP_{**})$$ where this time $$P_{**}$$ is a Cartan-Eilenberg projective resolution (another use of the row spectral sequence shows $$Tot(P_{**})$$ is quasi-isomorphic to $$A_*$$, and $$F$$ being additive implies taking total complex commutes with $$F$$). The right hand side gives hyperhomology functors, the left hand side gives derived functors of $$H_0\circ F$$.