Group Cohomology as a Derived Functor

If $f$ is $\mathbb{Z}[G]$-linear, then $gf(1)=f(g\cdot 1)=f(1)$, so evaluation lands in $M^G$.

For $m\in M^G$, define $f_m(z)=zm$. Then $f_m(g\cdot z)=zm=g{f}_m(z)$, hence $f_m$ is $\mathbb{Z}[G]$-linear; evaluation sends it to $m$, and a homomorphism from $\mathbb{Z}$ is determined by $1$. The two constructions commute with maps $M\to N$.

Applying ${\mathrm{Hom}}_{\mathbb{Z}[G]}(\mathbb{Z},-)$ gives an exact sequence at its first two positions by covariant Hom left exactness.

Transport this sequence through the natural isomorphisms of invariants with Hom from the trivial module. This is exactly the asserted sequence.

The induced map $B_G\to C_G$ is surjective because $B\to C$ is. Let $[b]\in B_G$ map to zero. Then the image of $b$ in $C$ is a finite sum ${\sum }_j(g_j{c}_j-c_j)$. Choose lifts $b_j\in B$ of the finitely many $c_j$ and put $b^{\mathrm{\prime }}=b-{\sum }_j(g_j{b}_j-b_j)$. Then $b^{\mathrm{\prime }}$ maps to zero in $C$, so it lies in the image of $A\to B$, while $[b^{\mathrm{\prime }}]=[b]$ in $B_G$.

Thus the image of $A_G\to B_G$ is exactly the kernel of $B_G\to C_G$, and the latter map is surjective. This proves right exactness without invoking a commutative-ring tensor theorem for the possibly noncommutative group ring.

The zeroth right derived functor of invariants recovers the invariants functor.

The zeroth left derived functor of coinvariants recovers coinvariants. Substitute the two definitions of group (co)homology.

Invariants are left exact. Under the stated Choice and supplied-resolution hypotheses, Right derived functors form a cohomological delta functor makes their right derived functors a cohomological delta functor.

Its connecting maps give precisely the displayed sequence after the definition of $H^n(G;-)$ is substituted; delta-functor naturality gives naturality.

Coinvariants are right exact. Under the stated Choice and supplied-resolution hypotheses, Left derived functors form a homological delta functor makes their left derived functors a homological delta functor.

Its connector has degree $H_n(G;C)\to H_{n-1}(G;A)$, giving the stated order after substituting the definition of $H_n$.

Deleting a coordinate commutes with diagonal left multiplication, so each face and hence $d_n$ is $\mathbb{Z}[G]$-linear.

For $n\ge 2$, each double deletion of positions $i<j$ occurs twice, first as $d_i{d}_j$ and then as $d_{j-1}{d}_i$, with opposite signs. Pairing these terms proves $d_{n-1}{d}_n=0$. For $n=1$, both faces have augmentation one, so $\epsilon (d_0-d_1)=0$.

After forgetting the $G$-action, set $s_n(g_0,\dots ,g_n)=(1,g_0,\dots ,g_n)$ and $s_{-1}(1)=(1)$.

For $n\ge 1$, direct cancellation of the faces gives $d_{n+1}{s}_n+s_{n-1}{d}_n=1_{B_n}$. In degree zero it gives the correctly typed identity $d_1{s}_0+s_{-1}\epsilon =1_{B_0}$, and $\epsilon s_{-1}=1_{\mathbb{Z}}$. Thus every cycle is a boundary and the augmentation is onto. The maps $s_n$ are only $\mathbb{Z}$-linear: left translation changes the inserted $1$.

Every diagonal orbit in $G^{n+1}$ has the unique representative $(1,g_0^{-1}{g}_1,\dots ,g_0^{-1}{g}_n)$; hence $B_n(G)$ is free over $\mathbb{Z}[G]$ on these representatives.

The differential and augmentation are $G$-linear, and the underlying augmented complex is exact. Thus it is a free $\mathbb{Z}[G]$-resolution.

Expand $d(df)$; terms correspond to two operations among acting by the first group element, multiplying adjacent elements, and deleting the last element.

Each pair of operations has two orders with opposite signs; associativity gives the same argument of $f$, and the leading pair agrees because $g_1(g_{2}m)=(g_1{g}_2)m$. All terms cancel.

Send a homogeneous equivariant cochain $F$ to $f(g_1,\dots ,g_n)=F(1,g_1,g_1{g}_2,\dots ,g_1\cdots g_n)$.

Its inverse sends $f$ to $F(g_0,\dots ,g_n)=g_{0}f(g_0^{-1}{g}_1,\dots ,g_{n-1}^{-1}{g}_n)$. The formulas are inverse and $G$-equivariance is immediate; applying alternating face deletion yields exactly the displayed formula for $d$.

Form the finite-diagonal double complex ${\mathrm{Hom}}_{\mathbb{Z}[G]}(P,I)$. Projectivity of each $P_i$ makes the columns exact away from $\mathrm{Hom}(P,M)$.

Injectivity of each $I^j$ makes the rows exact away from $\mathrm{Hom}(\mathbb{Z},I)$. The two acyclic-assembly comparisons identify the cohomology of these edge complexes.

The right edge computes ${\mathrm{Ext}}_{\mathbb{Z}[G]}^n(\mathbb{Z},M)=H^n(G;M)$, while the left edge is homogeneous bar cochains; the homogeneous--inhomogeneous isomorphism identifies it with $C^{\bullet }(G,M)$.

A face of a tuple with $g_i=g_{i+1}$ is again degenerate, except for the two faces deleting $g_i$ or $g_{i+1}$; those give the same remaining tuple with opposite signs. Hence $dD\subseteq D$.

Filter $D_n$ by the least index at which two consecutive vertices agree. On each successive quotient, the signed degeneracy that repeats the vertex at that index contracts the quotient; the simplicial identities give $sd+ds=1$. Combining these homotopies along the finite filtration $0\le i<n$ gives a contraction of $D$.

The standard degeneracy splitting gives the unnormalized complex as normalized representatives plus $D$, and the quotient is projection onto the first summand.

The inclusion of normalized representatives is a chain-map section; the contraction of $D$ supplies a homotopy from its composite with the quotient to the identity. Hence the two maps are homotopy inverses.

A chain-homotopy equivalence remains a cochain-homotopy equivalence after applying ${\mathrm{Hom}}_{\mathbb{Z}[G]}(-,M)$.

Thus normalized and unnormalized bar cochains have isomorphic cohomology; the latter computes $H^n(G;M)$.

Evaluation at $1\otimes m$ sends a $G$-map $F:\mathbb{Z}[G]{\otimes }_{\mathbb{Z}[H]}M\to N$ to the $H$-map $m\mapsto F(1\otimes m)$. Conversely, an $H$-map $f:M\to \mathrm{Res}N$ gives the well-defined $G$-map $g\otimes m\mapsto gf(m)$. These operations are inverse and natural, proving the induction adjunction for arbitrary group rings.

The arbitrary-ring coextension adjunction identifies ${\mathrm{Hom}}_G(N,{\mathrm{Hom}}_H(\mathbb{Z}[G],M))$ with ${\mathrm{Hom}}_H(\mathrm{Res}N,M)$. Both identifications are natural, giving the two adjunctions.

Every $g\in G$ has a unique form $th$ with $t\in T,h\in H$.

Group-ring basis expansion therefore gives $\mathbb{Z}[G]={⨁}_{t\in T}t\mathbb{Z}[H]$ as right modules. The opposite-side assertion follows from a right transversal. For arbitrary cosets, choosing $T$ is the only choice input.

Restriction takes $P$ to a projective $\mathbb{Z}[H]$-resolution because $\mathbb{Z}[G]$ is free over $\mathbb{Z}[H]$.

Coinduction--restriction adjunction identifies ${\mathrm{Hom}}_G(P,\mathrm{Coind}M)$ with ${\mathrm{Hom}}_H(\mathrm{Res}P,M)$ as cochain complexes. Taking cohomology gives the claimed natural isomorphism.

Restriction of $P$ is a projective right $\mathbb{Z}[H]$-resolution of the right trivial module: a left-coset transversal gives $\mathbb{Z}[G]\cong {⨁}_{t\in T}t\mathbb{Z}[H]$ as right $\mathbb{Z}[H]$-modules, so restriction sends free, hence projective, right $\mathbb{Z}[G]$-modules to projective right $\mathbb{Z}[H]$-modules. Exactness and the augmentation are unchanged on restriction.

Tensor associativity gives an isomorphism of chain complexes $P{\otimes }_{\mathbb{Z}[G]}(\mathbb{Z}[G]{\otimes }_{\mathbb{Z}[H]}M)\cong (\mathrm{Res}P){\otimes }_{\mathbb{Z}[H]}M$. The left and right sides compute respectively $H_n(G;{\mathrm{Ind}}_H^{G}M)$ and $H_n(H;M)$, so their homology gives the asserted natural isomorphism.

If ${\mathrm{pd}}_{\mathbb{Z}[G]}\mathbb{Z}\le n$, higher ${\mathrm{Ext}}_{\mathbb{Z}[G]}^q(\mathbb{Z},M)$ vanishes for every $M$.

Conversely, the stated vanishing is exactly the higher-Ext criterion applied to the trivial module. Identify Ext with group cohomology to obtain both implications.

For every $t\in T$ there are unique $t^{\mathrm{\prime }}\in T^{\mathrm{\prime }}$ and $h_t\in H$ with $t=t^{\mathrm{\prime }}{h}_t$. If $m\in M^H$, then $tm=t^{\mathrm{\prime }}{h}_{t}m=t^{\mathrm{\prime }}m$. Thus the two representative sums define the same degree-zero norm $N_H^G:M^H\to M^G$.

Corestriction is defined as the unique morphism of universal cohomological delta functors extending that norm. Since the two transversals give the same degree-zero map, uniqueness forces their corestriction maps to agree in every degree.

In degree zero, if $m\in M^G$, restriction regards $m$ as $H$-fixed and the norm sends it to ${\sum }_{xH\in G\mathrm{/}H}xm=[G:H]m$. Thus ${\mathrm{cor}}_H^G{\mathrm{res}}_H^G$ and multiplication by $[G:H]$ have the same degree-zero component.

Both are morphisms from the universal cohomological delta functor $H^{\ast }(G;-)$ to itself. By A morphism between universal delta functors is determined in degree zero, equality in degree zero forces equality in every degree. Hence the composite is multiplication by $[G:H]$ on $H^n(G;M)$ for all $n\ge 0$.

Every positive normalized bar has an identity entry and hence is zero in the normalized complex.

The normalized cochain complex is therefore zero in positive degrees, so its positive cohomology vanishes.

Restriction to the trivial subgroup maps $x$ to $H^n(1;M)=0$.

Corestriction after restriction is multiplication by $[G:1]=\mathrm{\mid }G\mathrm{\mid }$, so $\mathrm{\mid }G\mathrm{\mid }x=0$.

If $g\in G$ has finite order $m>1$, restrict the resolution to $C=⟨g⟩$; subgroup freeness preserves projectives, so ${\mathrm{cd}}_{\mathbb{Z}}C<\mathrm{\infty }$.

In $\mathbb{Z}[C]$, the alternating maps $g-1$ and $N=1+g+\cdots +g^{m-1}$ form a periodic free resolution: $(g-1)N=N(g-1)=0$, and coefficient comparison gives the two kernels as the two images. With trivial coefficients $\mathbb{Z}\mathrm{/}m$, its Hom complex has zero differentials, so $H^q(C;\mathbb{Z}\mathrm{/}m)\mathrm{\ne }0$ for arbitrarily large $q$. This contradicts finite cohomological dimension.

Coinvariants are right exact, so their left derived functors are indexed homologically.

Those derived functors are $H_n(G;M)$; cohomology is instead the right derived functor of the left exact invariants functor.

The contraction sends $(g_0,\dots ,g_n)$ to $(1,g_0,\dots ,g_n)$.

Thus $s(h{g}_0,\dots ,h{g}_n)=(1,h{g}_0,\dots )$, whereas $hs(g_0,\dots ,g_n)=(h,h{g}_0,\dots )$; they differ when $h\mathrm{\ne }1$.

Here $H^1(G;M)$ is defined as the first right derived invariant functor.

The crossed-homomorphism description is a later low-degree interpretation, not this definition; it is intentionally reserved for the group-theory track.

The definition fixes every degree by $H^n(G;M)=R^n((-{)}^G)(M)$.

The extension classification is a separate low-degree theorem and is not used or defined here.

Cohomology uses the right adjoint ${\mathrm{Coind}}_H^G$ through a Hom-complex comparison.

Homology uses the left adjoint ${\mathrm{Ind}}_H^G$ through a tensor-complex comparison. The functors are not generally interchangeable.