Lyndon-Hochschild-Serre spectral sequence

Put $F=(-{)}^N:G\text{-}\mathrm{M}\mathrm{o}\mathrm{d}\to Q\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ and $T=(-{)}^Q:Q\text{-}\mathrm{M}\mathrm{o}\mathrm{d}\to \mathbf{A}\mathbf{b}$. By F1, $TF=(-{)}^G$. F2 proves both functors additive and left exact and verifies the required injective-image acyclicity. These are exactly the Grothendieck hypotheses; no exactness of $N$-invariants is asserted. The supplied resolution systems give the needed injective models in these module categories.

For the supplied $M\to I$, F3 says its restriction resolves $M$ by $N$-injectives. Thus $R^{q}F(M)=H^q(I^N)$ has underlying abelian group $H^q(N,M)$ by F4. Its $Q$-action is the one induced termwise from F1. A $G$-linear resolution comparison and its homotopy restrict to $Q$-linear maps and homotopies on $N$-invariants. They therefore give the same cohomology map, proving this $Q$-module identification is canonical under the declared data convention. Similarly $R^{p}T(V)=H^p(Q,V)$ and $R^n(TF)(M)=H^n(G,M)$.

For a map of extensions as stated, write $U:G^{\mathrm{\prime }}\text{-}\mathrm{M}\mathrm{o}\mathrm{d}\to G\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ and $V:Q^{\mathrm{\prime }}\text{-}\mathrm{M}\mathrm{o}\mathrm{d}\to Q\text{-}\mathrm{M}\mathrm{o}\mathrm{d}$ for restriction of actions. These are exact because underlying groups and maps do not change. There are natural maps $V((-{)}^{N^{\mathrm{\prime }}})\to (U(-){)}^N$ and $(-{)}^{Q^{\mathrm{\prime }}}\to (V(-){)}^Q$, given by inclusion of fixed subgroups: being fixed by all elements of the primed group implies being fixed by their images from the unprimed group. These maps are compatible with the composite inclusion of $G^{\mathrm{\prime }}$-fixed into $G$-fixed elements.

Apply F5 and substitute step 1.2. It gives the displayed page, differential, filtration endpoints and associated graded identification. In particular degree zero is $H^0(G,M)=M^G=(M^N{)}^Q$; all indices are nonnegative, and below total degree zero the target vanishes. Naturality is the comparison naturality of F5 and step 1.2. For $M=0$ the zero models give zero throughout. If $N=1$ or $Q=1$, invariants for the trivial group are the identity exact functor, whose applied resolution is exact in positive degrees; consequently the sequence has only one row or column and reconstructs the remaining group's cohomology. DC is confined to obtaining the countable replacement/comparison choices and the F4 notation. Supplied data give the same construction without that assumption.

Let $M^{\mathrm{\prime }}\to I^{\mathrm{\prime }}$ and $M\to I$ be injective resolutions. Although $U{I}^{\mathrm{\prime }}$ need not be injective, its augmentation is exact. The map $U{M}^{\mathrm{\prime }}\stackrel{u}{\to }M\to I^0$ extends across $U{M}^{\mathrm{\prime }}\hookrightarrow U{I}^{\mathrm{\prime }0}$ by F7. Its differential vanishes on the augmentation image, so it descends to the next image submodule and extends into $I^1$. Repeating gives a map $a:U{I}^{\mathrm{\prime }}\to I$ over $u$. For two lifts, their difference kills the augmentation; factoring through the next image and extending constructs a homotopy recursively, exactly by the same difference-minus-previous-homotopy calculation. These are countably many extensions in fixed Hom sets, supplied or chosen by DC. Hence the resulting cohomology restriction/coefficient map is canonical without assuming $U$ preserves injectives.

For CE resolutions $F^{\mathrm{\prime }}{I}^{\mathrm{\prime }}\to J^{\mathrm{\prime }}$ and $FI\to J$, the composite $V{F}^{\mathrm{\prime }}{I}^{\mathrm{\prime }}\to FU{I}^{\mathrm{\prime }}\stackrel{F(a)}{\to }FI$ is a cochain map. Exactness of $V$ preserves kernels, images and quotients, so $V{J}^{\mathrm{\prime }}$ is an augmented source exact on terms, horizontal boundaries, cycles and cohomology. F6 lifts that map to $V{J}^{\mathrm{\prime }}\to J$. Compose $(J^{\mathrm{\prime }}{)}^{Q^{\mathrm{\prime }}}\to (V{J}^{\mathrm{\prime }}{)}^Q\to J^Q$. It preserves resolution degree and therefore defines a map of the LHS sequences. On $E_2$ it is $H^p(Q^{\mathrm{\prime }},H^q(N^{\mathrm{\prime }},M^{\mathrm{\prime }}))\to H^p(Q,H^q(N,M))$, computed by the same inclusion-of-invariants and resolution comparisons just constructed. On the target the commuting augmentation square identifies it with $H^n(G^{\mathrm{\prime }},M^{\mathrm{\prime }})\to H^n(G,M)$. These descriptions also define the usual derived restriction/coefficient maps for noninjective pullbacks: an exact augmented pullback resolution maps into the chosen injective resolution by step 2.2.

Different choices of $a$ are homotopic and give the same map on $H^q(N,-)$. Different CE lifts give the same $E_2$ and target maps by F6. Equality on later pages follows by taking page homology. Identity and composite extension maps yield the same $E_2$ maps as the identity and composites of these constructions; their target maps agree by the resolution homotopies of step 2.2 and the augmentation square. Thus this is natural for extension and coefficient maps, in addition to fixed-extension module maps. The trivial extension maps and zero coefficient maps are included; no global family of choices is asserted.