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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!

Yai0Phah
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2 Answers2

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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.

Yai0Phah
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    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.,. ... – Bananeen Sep 27 '17 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? – Rafael Dec 01 '17 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 $[P_\bullet,f]\colon[P_\bullet,C_\bullet]\to[P_\bullet,P_\bullet]$. – Yai0Phah Dec 02 '17 at 21:43
  • How is the last equality obtained? – davik Jan 28 '19 at 18:10
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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 $H_0\circ F$.

Victor Zhang
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