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Understanding conjugation action onthe functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point.

Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of the trivial $G$-module. In Ken Browns book "Cohomology of groups", given a $G$-module $M$, he defines the homology of $M$ to be the homology of the complex $F_\bullet\otimes_G M$, and defines functoriality as follows:

If $\alpha : G\rightarrow G'$ is a homomorphism of groups, let $F_\bullet'$ be a free $\mathbb{Z}G'$-resolution of $\mathbb{Z}$, and let $\tau : F_\bullet\rightarrow \alpha^\# F_\bullet'$ be a morphism of complexes of $G$-modules, extending the identity on $\mathbb{Z}$.

For $M\in Mod_G, M'\in Mod_{G'}$ and $f : M\rightarrow \alpha^\#M'$ a homomorphism of $G$-modules, then the the map $H_*(G,M)\rightarrow H_*(G',M')$ induced by $(\alpha,f)$ is itself by the map $\tau\otimes f : F_\bullet\otimes_G M\rightarrow \alpha^\#F'_\bullet\otimes_G \alpha^\#M' = F'_\bullet\otimes_{G'}M'$ on chains.

My confusion stems from the following. I am used to the definition of group homology as the derived functors of the $G$-coinvariants functor : $Mod_G\rightarrow Ab$ sending $M\mapsto M_G = M\otimes_{\mathbb{Z}G}\mathbb{Z}$, which identifies $H_*(G,M) = Tor_*^{\mathbb{Z}G}(\mathbb{Z},M)$, which is why the complex $F_\bullet\otimes_G M$ computes the homology of $M$. What's unclear is the functorial dependence of the homology on the complex $F_\bullet\otimes_G M$.

From the derived functor definition of group homology, one has the following notion of functoriality in $G$ and $M$:

Given $\alpha : G\rightarrow G'$ as above, $M\in Mod_G, M'\in Mod_{G'}$, and $f : M\rightarrow\alpha^\# M'$, the usual derived functor functoriality gives us a map $$f_* : H_*(G,M)\rightarrow H_*(G,\alpha^\#M')$$ Next, the fact that $H_*(G',-)$ is a universal $\delta$-functor implies that the natural map $(\alpha^\#(-))_G\rightarrow (-)_G$ extends functorially to a morphism of $\delta$-functors: $H_*(G,\alpha^\#(-))\rightarrow H_*(G',-)$, whence a map: $$cor : H_*(G,\alpha^\#M')\rightarrow H_*(G',M')$$

My question can be phrased as follows:

  1. Why is $f_*$ computable using $id\otimes f : F_\bullet\otimes_G M\rightarrow F_\bullet\otimes_G \alpha^\#M'$? (Normally one would take a projective resolution of $M$, then tensor by $\mathbb{Z}$)

  2. Why is the corestriction map $cor$ computable using $\tau\otimes id : F_\bullet\otimes_G \alpha^\# M'\rightarrow \alpha^\#F_\bullet'\otimes_G \alpha^\#M'$?

I've tried to convince myself of the equivalence, to which I've had a little success, though I'd appreciate it if experts could sketch how they would see that these two definitions are equivalent (different perspectives are always valuable), or perhaps explain why it's obvious from some standard facts which I'm missing. References which explain this would also be appreciated.

BEGIN OLD QUESTION

I'm trying to understand Weibel's description of the action of conjugation on group homology (this is p190 in his "Introduction to Homological Algebra"). I'm having trouble understanding exactly what makes his description valid.

Why does this calculate the morphism $\zeta_*\circ(\mu_g)_*$?Why does this calculate the morphism $\zeta_*\circ(\mu_g)_*$?

However, this doesn't seem to directly apply to Weibel's situation, since we have multiple $\delta$-functors at work. I think I can explain why Weibel's construction works by noting that the desired morphism $\zeta_*\circ(\mu_g)_*$ comes from the morphism of left-derived functors: $$L_i(\otimes_H A)\rightarrow L_i(\otimes_Hc_g^\#A)$$ (both functors from $Mod_H\rightarrow Ab$) and sketching an argument that a morphism of left derived functors is also computable using projective resolutions), but I'm not completely convinced of the validity of this argument. Thus, I would very much appreciate it if an expert could describe the situation more clearly for me.

Understanding conjugation action on group homology

I'm trying to understand Weibel's description of the action of conjugation on group homology (this is p190 in his "Introduction to Homological Algebra"). I'm having trouble understanding exactly what makes his description valid.

Why does this calculate the morphism $\zeta_*\circ(\mu_g)_*$?

However, this doesn't seem to directly apply to Weibel's situation, since we have multiple $\delta$-functors at work. I think I can explain why Weibel's construction works by noting that the desired morphism $\zeta_*\circ(\mu_g)_*$ comes from the morphism of left-derived functors: $$L_i(\otimes_H A)\rightarrow L_i(\otimes_Hc_g^\#A)$$ (both functors from $Mod_H\rightarrow Ab$) and sketching an argument that a morphism of left derived functors is also computable using projective resolutions), but I'm not completely convinced of the validity of this argument. Thus, I would very much appreciate it if an expert could describe the situation more clearly for me.

Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point.

Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of the trivial $G$-module. In Ken Browns book "Cohomology of groups", given a $G$-module $M$, he defines the homology of $M$ to be the homology of the complex $F_\bullet\otimes_G M$, and defines functoriality as follows:

If $\alpha : G\rightarrow G'$ is a homomorphism of groups, let $F_\bullet'$ be a free $\mathbb{Z}G'$-resolution of $\mathbb{Z}$, and let $\tau : F_\bullet\rightarrow \alpha^\# F_\bullet'$ be a morphism of complexes of $G$-modules, extending the identity on $\mathbb{Z}$.

For $M\in Mod_G, M'\in Mod_{G'}$ and $f : M\rightarrow \alpha^\#M'$ a homomorphism of $G$-modules, then the the map $H_*(G,M)\rightarrow H_*(G',M')$ induced by $(\alpha,f)$ is itself by the map $\tau\otimes f : F_\bullet\otimes_G M\rightarrow \alpha^\#F'_\bullet\otimes_G \alpha^\#M' = F'_\bullet\otimes_{G'}M'$ on chains.

My confusion stems from the following. I am used to the definition of group homology as the derived functors of the $G$-coinvariants functor : $Mod_G\rightarrow Ab$ sending $M\mapsto M_G = M\otimes_{\mathbb{Z}G}\mathbb{Z}$, which identifies $H_*(G,M) = Tor_*^{\mathbb{Z}G}(\mathbb{Z},M)$, which is why the complex $F_\bullet\otimes_G M$ computes the homology of $M$. What's unclear is the functorial dependence of the homology on the complex $F_\bullet\otimes_G M$.

From the derived functor definition of group homology, one has the following notion of functoriality in $G$ and $M$:

Given $\alpha : G\rightarrow G'$ as above, $M\in Mod_G, M'\in Mod_{G'}$, and $f : M\rightarrow\alpha^\# M'$, the usual derived functor functoriality gives us a map $$f_* : H_*(G,M)\rightarrow H_*(G,\alpha^\#M')$$ Next, the fact that $H_*(G',-)$ is a universal $\delta$-functor implies that the natural map $(\alpha^\#(-))_G\rightarrow (-)_G$ extends functorially to a morphism of $\delta$-functors: $H_*(G,\alpha^\#(-))\rightarrow H_*(G',-)$, whence a map: $$cor : H_*(G,\alpha^\#M')\rightarrow H_*(G',M')$$

My question can be phrased as follows:

  1. Why is $f_*$ computable using $id\otimes f : F_\bullet\otimes_G M\rightarrow F_\bullet\otimes_G \alpha^\#M'$? (Normally one would take a projective resolution of $M$, then tensor by $\mathbb{Z}$)

  2. Why is the corestriction map $cor$ computable using $\tau\otimes id : F_\bullet\otimes_G \alpha^\# M'\rightarrow \alpha^\#F_\bullet'\otimes_G \alpha^\#M'$?

I've tried to convince myself of the equivalence, to which I've had a little success, though I'd appreciate it if experts could sketch how they would see that these two definitions are equivalent (different perspectives are always valuable), or perhaps explain why it's obvious from some standard facts which I'm missing. References which explain this would also be appreciated.

BEGIN OLD QUESTION

I'm trying to understand Weibel's description of the action of conjugation on group homology (this is p190 in his "Introduction to Homological Algebra"). I'm having trouble understanding exactly what makes his description valid.

Why does this calculate the morphism $\zeta_*\circ(\mu_g)_*$?

However, this doesn't seem to directly apply to Weibel's situation, since we have multiple $\delta$-functors at work. I would very much appreciate it if an expert could describe the situation more clearly for me.

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Understanding conjugation action on group homology

I'm trying to understand Weibel's description of the action of conjugation on group homology (this is p190 in his "Introduction to Homological Algebra"). I'm having trouble understanding exactly what makes his description valid.

In general, for a morphism of groups $\rho : H\rightarrow G$ and a $G$-module $A$, we may form the $H$-module $\rho^\#A$ with $H$-module structure given via $\rho$, and this functor $\rho^\#$ is exact.

In our case, let $H\le G$ be a subgroup, and let $c_g : H\rightarrow gHg^{-1}$ be the conjugation isomorphism $h\mapsto ghg^{-1}$.

If $A$ is a $G$-module, the abelian group map $\mu_g : A\rightarrow A$ given by $a\mapsto ga$ is actually an $H$-module map from $A$ to $c_g^\#A$.

Thus, $\mu_g$ determines a natural map $(\mu_g)_* : H_*(H,A)\rightarrow H_*(H,c_g^\#A)$.

Moreover, the functors $Mod_H\rightarrow Ab$ given by $M\mapsto H_*(H,c_g^\#A)$ is a homological $\delta$-functor (since $c_g^\#$ is exact), and hence the natural abelian group homomorphism $$H_0(H,c_g^\#A) = (c_g^\#A)_H\rightarrow A_{gHg^{-1}} = H_0(gHg^{-1},A)$$ extends to a morphism of $\delta$-functors $$\zeta_* : H_*(H,c_g^\#A)\rightarrow H_*(gHg^{-1},A)$$ (This $\zeta_*$ is normally called the corestriction map)

I'd like to understand the composition: $$\zeta_*\circ(\mu_g)_* : H_*(H,A)\rightarrow H_*(gHg^{-1},A)$$

Weibel gives the following way of understanding this composition. He picks a projective $\mathbb{Z}G$-resolution $P_\bullet\rightarrow\mathbb{Z}$ (with $\mathbb{Z}$-the trivial $G$-module), and notes that it is simultaneously projective for $\mathbb{Z}G,\mathbb{Z}H$, and $\mathbb{Z}[gHg^{-1}]$. Then he notes that $\mu_g : P_i\rightarrow P_i$ sending $p\mapsto gp$ is an $H$-module chain map $P_i\rightarrow c_g^\# P_i$ lying over the identity map on $\mathbb{Z}$. He deduces that $\zeta_*\circ(\mu_g)_*$ is induced from $$P\otimes_{\mathbb{Z}H}A\rightarrow P\otimes_{\mathbb{Z}[gHg^{-1}]}A\qquad x\otimes a\mapsto gx\otimes ga$$

Why does this calculate the morphism $\zeta_*\circ(\mu_g)_*$?

My model of what's going on is the following. Suppose you have a right-exact functor $F : \mathcal{A}\rightarrow\mathcal{B}$ where $\mathcal{A}$ has enough projectives. Then for a morphism $f : X\rightarrow Y$ in $\mathcal{A}$, one can calculate $L_iF(f)$ by picking projective resolutions $P_\bullet\rightarrow X$ and $Q_\bullet\rightarrow Y$, extending $f$ to a morphism of complexes $f_\bullet : P_\bullet\rightarrow Q_\bullet$ (unique up to homotopy), applying $F$ to $f_\bullet$, and taking the induced maps on homologies. (Roughly speaking this works because in the derived category one computes everything by replacing what we care about by projective resolutions.)

However, this doesn't seem to directly apply to Weibel's situation, since we have multiple $\delta$-functors at work. I think I can explain why Weibel's construction works by noting that the desired morphism $\zeta_*\circ(\mu_g)_*$ comes from the morphism of left-derived functors: $$L_i(\otimes_H A)\rightarrow L_i(\otimes_Hc_g^\#A)$$ (both functors from $Mod_H\rightarrow Ab$) and sketching an argument that a morphism of left derived functors is also computable using projective resolutions), but I'm not completely convinced of the validity of this argument. Thus, I would very much appreciate it if an expert could describe the situation more clearly for me.