Long exact sequence of sheaf cohomology

Regard the three sheaves as cochain complexes concentrated in degree zero. The mapping cone of $i$ has ${\mathcal{F}}_1$ in degree $-1$, ${\mathcal{F}}_2$ in degree $0$, and differential $i$. Its only possibly nonzero cohomology is $H^0(\mathrm{Cone}(i))=\mathrm{coker}(i)\cong {\mathcal{F}}_3$, since $i$ is a monomorphism and the original sequence is short exact. The map $\mathrm{Cone}(i)\to {\mathcal{F}}_3$ induced by $p$ in degree zero and zero in degree $-1$ is therefore a quasi-isomorphism [F5]. By the cone triangulation and localization at quasi-isomorphisms, the short exact sequence determines the distinguished triangle ${\mathcal{F}}_1\to {\mathcal{F}}_2\to {\mathcal{F}}_3\to {\mathcal{F}}_1[1]$ in the bounded-below derived category [F3, F5].

The bounded-below right derived functor $R_I\mathrm{\Gamma }$ exists for the supplied resolution datum [F2], and $\mathrm{\Gamma }(X,-)$ is additive [F1]. It is exact as a functor of triangulated categories [F4], so applying it to the triangle of [step 1.1] gives a distinguished triangle $R_I\mathrm{\Gamma }({\mathcal{F}}_1)\to R_I\mathrm{\Gamma }({\mathcal{F}}_2)\to R_I\mathrm{\Gamma }({\mathcal{F}}_3)\to R_I\mathrm{\Gamma }({\mathcal{F}}_1)[1]$. The induced maps on cohomology in the first two positions are the maps $H^q(X,i)$ and $H^q(X,p)$ because the derived functor is formed from the same supplied datum $I$ that defines sheaf cohomology [F2].

The long exact cohomology sequence of the distinguished triangle of [step 2.1] exists and is natural [F3]. By [F2], its terms are exactly $H^q(X,{\mathcal{F}}_1)$, $H^q(X,{\mathcal{F}}_2)$ and $H^q(X,{\mathcal{F}}_3)$, and its connecting map gives ${\mathrm{\partial }}^q:H^q(X,{\mathcal{F}}_3)\to H^{q+1}(X,{\mathcal{F}}_1)$. This is the asserted long exact sequence.

A morphism of short exact sequences gives a commutative morphism between their mapping-cone triangles in [step 1.1]. The functor $R_I\mathrm{\Gamma }$ and the long exact sequence construction of [F3] preserve this morphism, so all maps, including ${\mathrm{\partial }}^q$, are natural in the short exact sequence. If a different supplied injective resolution datum is used, [F6] gives the canonical natural isomorphism of right total derived functors compatible with coaugmentations. Write $j_K:K\to I_K$ and $j{\mathrm{\prime }}_K:K\to J_K$ for the two replacements. The comparison class $c_K:I_K\to J_K$ is characterized by $Q(c_K)=Q(j{\mathrm{\prime }}_K)Q(j_K{)}^{-1}$, uniquely in the homotopy category by [F7]. Let $t_I:I_{K[1]}\to I_K[1]$ and $t_J:J_{K[1]}\to J_K[1]$ be the shift comparison classes characterized in the same way by $j_K[1]$ and $j{\mathrm{\prime }}_K[1]$. The two classes $c_K[1]t_I$ and $t_J{c}_{K[1]}$ from $I_{K[1]}$ to $J_K[1]$ have the same image under $Q$, namely $Q(j{\mathrm{\prime }}_K[1])Q(j_{K[1]}{)}^{-1}$. The target is again K-injective, so [F7] identifies them in the homotopy category. Applying the additive functor $\mathrm{\Gamma }$ preserves that homotopy equality. These are exactly the transported shift comparisons of [F4], so the natural comparison commutes with shifts and hence with the connecting arrow. Therefore it identifies the resulting long exact sequences, proving independence of the resolutions used. ∎