[proofplan] We split the acyclic resolution into short exact sequences involving the kernel sheaves $\mathcal{K}^m$ of the differentials. The cohomology of the global-section complex in positive degree $q$ is first identified with a cokernel of the map $\Gamma(X,\mathcal{A}^{q-1}) \to \Gamma(X,\mathcal{K}^q)$. The long exact sequence in Čech cohomology converts this cokernel into $\check{H}^1(X,\mathcal{K}^{q-1})$, and acyclicity of the sheaves $\mathcal{A}^m$ makes the connecting morphisms into isomorphisms. Iterating these connecting isomorphisms gives the dimension-shifting identification $\check{H}^1(X,\mathcal{K}^{q-1}) \cong \check{H}^q(X,\mathcal{F})$; degree $0$ follows from left exactness of global sections. [/proofplan]

[step:Split the resolution into kernel short exact sequences]Write the acyclic resolution as a cochain complex of sheaves of abelian groups on $X$: \begin{align*} 0 \to \mathcal{F} \xrightarrow{\iota_0} \mathcal{A}^0 \xrightarrow{d_0} \mathcal{A}^1 \xrightarrow{d_1} \mathcal{A}^2 \xrightarrow{d_2} \cdots . \end{align*} By the definition of an acyclic resolution, this complex is exact and, for every integer $m \geq 0$ and every integer $r \geq 1$, \begin{align*} \check{H}^r(X,\mathcal{A}^m)=0. \end{align*} Define $\mathcal{K}^0 := \mathcal{F}$. For each integer $m \geq 1$, define the kernel sheaf \begin{align*} \mathcal{K}^m := \ker\left(d_m:\mathcal{A}^m \to \mathcal{A}^{m+1}\right), \end{align*} and let \begin{align*} \iota_m:\mathcal{K}^m \to \mathcal{A}^m \end{align*} denote the inclusion morphism. For each integer $m \geq 0$, define \begin{align*} \pi_m:\mathcal{A}^m \to \mathcal{K}^{m+1} \end{align*} to be the morphism induced by $d_m$ after restricting the codomain to $\mathcal{K}^{m+1}$. Since the resolution is exact, $\operatorname{im}(d_m)=\mathcal{K}^{m+1}$ and $\ker(\pi_m)=\operatorname{im}(\iota_m)$. Hence for every integer $m \geq 0$ there is a short exact sequence of sheaves \begin{align*} 0 \to \mathcal{K}^m \xrightarrow{\iota_m} \mathcal{A}^m \xrightarrow{\pi_m} \mathcal{K}^{m+1} \to 0. \end{align*}[/step]

[guided]The purpose of the kernel sheaves is to turn the single long resolution into many short exact sequences, because short exact sequences are what produce long exact sequences in cohomology. We write the resolution with named morphisms: \begin{align*} 0 \to \mathcal{F} \xrightarrow{\iota_0} \mathcal{A}^0 \xrightarrow{d_0} \mathcal{A}^1 \xrightarrow{d_1} \mathcal{A}^2 \xrightarrow{d_2} \cdots . \end{align*} The phrase “acyclic resolution” has two parts. First, the displayed complex is exact: $\iota_0$ is injective, $\ker(d_0)=\operatorname{im}(\iota_0)$, and $\ker(d_m)=\operatorname{im}(d_{m-1})$ for every integer $m \geq 1$. Second, each $\mathcal{A}^m$ is Čech-acyclic, meaning that for every integer $r \geq 1$, \begin{align*} \check{H}^r(X,\mathcal{A}^m)=0. \end{align*} Define $\mathcal{K}^0 := \mathcal{F}$. For every integer $m \geq 1$, define \begin{align*} \mathcal{K}^m := \ker\left(d_m:\mathcal{A}^m \to \mathcal{A}^{m+1}\right), \end{align*} and let \begin{align*} \iota_m:\mathcal{K}^m \to \mathcal{A}^m \end{align*} be the inclusion morphism. Because $d_{m+1}\circ d_m=0$, the image of $d_m$ lies inside $\ker(d_{m+1})=\mathcal{K}^{m+1}$. Exactness strengthens this containment to equality: \begin{align*} \operatorname{im}(d_m)=\mathcal{K}^{m+1}. \end{align*} Therefore $d_m$ may be regarded as a morphism with codomain $\mathcal{K}^{m+1}$. We denote this morphism by \begin{align*} \pi_m:\mathcal{A}^m \to \mathcal{K}^{m+1}. \end{align*} The kernel of $\pi_m$ is the same as the kernel of $d_m$, and exactness says this kernel is the image of $\iota_m$. Thus, for every integer $m \geq 0$, we obtain a short exact sequence \begin{align*} 0 \to \mathcal{K}^m \xrightarrow{\iota_m} \mathcal{A}^m \xrightarrow{\pi_m} \mathcal{K}^{m+1} \to 0. \end{align*}[/guided]

[step:Identify positive degree global cohomology with a cokernel of kernel sections]For each integer $m \geq 0$, define the induced maps on global sections \begin{align*} D_m := \Gamma(X,d_m):\Gamma(X,\mathcal{A}^m) \to \Gamma(X,\mathcal{A}^{m+1}), \end{align*} \begin{align*} I_m := \Gamma(X,\iota_m):\Gamma(X,\mathcal{K}^m) \to \Gamma(X,\mathcal{A}^m), \end{align*} and \begin{align*} P_m := \Gamma(X,\pi_m):\Gamma(X,\mathcal{A}^m) \to \Gamma(X,\mathcal{K}^{m+1}). \end{align*} Fix an integer $q \geq 1$. Define the degree-$q$ cocycle and coboundary groups of the global-section complex by \begin{align*} Z^q_\Gamma := \ker(D_q) \end{align*} and \begin{align*} B^q_\Gamma := \operatorname{im}(D_{q-1}). \end{align*} Since $d_q\circ d_{q-1}=0$, we have $B^q_\Gamma \subseteq Z^q_\Gamma$, so the degree-$q$ cohomology group of the global-section complex is \begin{align*} H^q_\Gamma(\mathcal{A}^\bullet) := Z^q_\Gamma/B^q_\Gamma. \end{align*} Because $\mathcal{K}^q=\ker(d_q)$, the left exactness of global sections identifies $I_q$ with an isomorphism \begin{align*} \Gamma(X,\mathcal{K}^q) \xrightarrow{\cong} Z^q_\Gamma. \end{align*} Also $d_{q-1}=\iota_q\circ \pi_{q-1}$, so \begin{align*} D_{q-1}=I_q\circ P_{q-1}. \end{align*} Therefore $I_q$ induces an isomorphism \begin{align*} \Lambda_q: \frac{\Gamma(X,\mathcal{K}^q)}{\operatorname{im}(P_{q-1})} &\xrightarrow{\cong} \frac{Z^q_\Gamma}{B^q_\Gamma}. \end{align*}[/step]

[guided]We now compare two ways of saying that a global section is closed in degree $q$. For each integer $m \geq 0$, the differential $d_m$ induces a map on global sections \begin{align*} D_m := \Gamma(X,d_m):\Gamma(X,\mathcal{A}^m) \to \Gamma(X,\mathcal{A}^{m+1}). \end{align*} Similarly, the inclusion $\iota_m$ and quotient map $\pi_m$ induce \begin{align*} I_m := \Gamma(X,\iota_m):\Gamma(X,\mathcal{K}^m) \to \Gamma(X,\mathcal{A}^m) \end{align*} and \begin{align*} P_m := \Gamma(X,\pi_m):\Gamma(X,\mathcal{A}^m) \to \Gamma(X,\mathcal{K}^{m+1}). \end{align*} Fix an integer $q \geq 1$. The global-section complex has cocycles \begin{align*} Z^q_\Gamma := \ker(D_q) \end{align*} and coboundaries \begin{align*} B^q_\Gamma := \operatorname{im}(D_{q-1}). \end{align*} The containment $B^q_\Gamma \subseteq Z^q_\Gamma$ follows from \begin{align*} D_q\circ D_{q-1} = \Gamma(X,d_q)\circ \Gamma(X,d_{q-1}) = \Gamma(X,d_q\circ d_{q-1}) = 0. \end{align*} Hence the degree-$q$ cohomology of the global-section complex is \begin{align*} H^q_\Gamma(\mathcal{A}^\bullet) := Z^q_\Gamma/B^q_\Gamma. \end{align*} The key point is that $Z^q_\Gamma$ is not mysterious: it is exactly the group of global sections of the kernel sheaf $\mathcal{K}^q$. Indeed, $\mathcal{K}^q=\ker(d_q)$ by definition, so the sequence \begin{align*} 0 \to \mathcal{K}^q \xrightarrow{\iota_q} \mathcal{A}^q \xrightarrow{d_q} \mathcal{A}^{q+1} \end{align*} is exact. Applying the left exactness of global sections gives an exact sequence \begin{align*} 0 \to \Gamma(X,\mathcal{K}^q) \xrightarrow{I_q} \Gamma(X,\mathcal{A}^q) \xrightarrow{D_q} \Gamma(X,\mathcal{A}^{q+1}). \end{align*} Exactness at $\Gamma(X,\mathcal{A}^q)$ says precisely that $I_q$ identifies $\Gamma(X,\mathcal{K}^q)$ with $\ker(D_q)=Z^q_\Gamma$. Next, because $d_{q-1}$ factors through the kernel sheaf $\mathcal{K}^q$, we have \begin{align*} d_{q-1}=\iota_q\circ \pi_{q-1}. \end{align*} Applying $\Gamma(X,-)$ gives \begin{align*} D_{q-1}=I_q\circ P_{q-1}. \end{align*} Thus, under the identification $\Gamma(X,\mathcal{K}^q)\cong Z^q_\Gamma$, the subgroup $\operatorname{im}(P_{q-1})$ corresponds exactly to the subgroup $B^q_\Gamma=\operatorname{im}(D_{q-1})$. Therefore $I_q$ induces an isomorphism of quotients \begin{align*} \Lambda_q: \frac{\Gamma(X,\mathcal{K}^q)}{\operatorname{im}(P_{q-1})} &\xrightarrow{\cong} \frac{Z^q_\Gamma}{B^q_\Gamma}. \end{align*}[/guided]

[step:Use the long exact sequence to turn the cokernel into first cohomology]Fix an integer $q \geq 1$. Apply the long exact sequence in Čech cohomology to the short exact sequence \begin{align*} 0 \to \mathcal{K}^{q-1} \xrightarrow{\iota_{q-1}} \mathcal{A}^{q-1} \xrightarrow{\pi_{q-1}} \mathcal{K}^{q} \to 0. \end{align*} The relevant part is \begin{align*} \Gamma(X,\mathcal{A}^{q-1}) \xrightarrow{P_{q-1}} \Gamma(X,\mathcal{K}^{q}) \xrightarrow{\partial_{q-1}^0} \check{H}^1(X,\mathcal{K}^{q-1}) \to \check{H}^1(X,\mathcal{A}^{q-1}), \end{align*} where $\partial_{q-1}^0$ is the connecting homomorphism. Since $\mathcal{A}^{q-1}$ is acyclic, \begin{align*} \check{H}^1(X,\mathcal{A}^{q-1})=0. \end{align*} Exactness of the displayed sequence gives \begin{align*} \ker(\partial_{q-1}^0)=\operatorname{im}(P_{q-1}) \end{align*} and makes $\partial_{q-1}^0$ surjective. Hence $\partial_{q-1}^0$ descends to an isomorphism \begin{align*} \overline{\partial}_{q-1}^0: \frac{\Gamma(X,\mathcal{K}^{q})}{\operatorname{im}(P_{q-1})} \xrightarrow{\cong} \check{H}^1(X,\mathcal{K}^{q-1}). \end{align*}[/step]

[guided]The previous step reduced the global-section cohomology to the quotient \begin{align*} \frac{\Gamma(X,\mathcal{K}^{q})}{\operatorname{im}(P_{q-1})}. \end{align*} We now identify this quotient using the long exact sequence associated to the short exact sequence ending in $\mathcal{K}^q$: \begin{align*} 0 \to \mathcal{K}^{q-1} \xrightarrow{\iota_{q-1}} \mathcal{A}^{q-1} \xrightarrow{\pi_{q-1}} \mathcal{K}^{q} \to 0. \end{align*} This is a short exact sequence of sheaves of abelian groups on $X$, as proved when we split the resolution. Therefore the hypotheses of the long exact sequence in Čech cohomology are satisfied. The beginning of that long exact sequence is \begin{align*} \Gamma(X,\mathcal{K}^{q-1}) \to \Gamma(X,\mathcal{A}^{q-1}) \xrightarrow{P_{q-1}} \Gamma(X,\mathcal{K}^{q}) \xrightarrow{\partial_{q-1}^0} \check{H}^1(X,\mathcal{K}^{q-1}) \to \check{H}^1(X,\mathcal{A}^{q-1}). \end{align*} The map $\partial_{q-1}^0$ is the connecting homomorphism. Exactness at $\Gamma(X,\mathcal{K}^q)$ gives \begin{align*} \ker(\partial_{q-1}^0)=\operatorname{im}(P_{q-1}). \end{align*} Exactness at $\check{H}^1(X,\mathcal{K}^{q-1})$ says that the image of $\partial_{q-1}^0$ is the kernel of the following map into $\check{H}^1(X,\mathcal{A}^{q-1})$. Since $\mathcal{A}^{q-1}$ is acyclic, we have \begin{align*} \check{H}^1(X,\mathcal{A}^{q-1})=0. \end{align*} Therefore that kernel is all of $\check{H}^1(X,\mathcal{K}^{q-1})$, so $\partial_{q-1}^0$ is surjective. Combining the kernel computation with surjectivity, the quotient by $\operatorname{im}(P_{q-1})$ is exactly the target of the connecting homomorphism: \begin{align*} \overline{\partial}_{q-1}^0: \frac{\Gamma(X,\mathcal{K}^{q})}{\operatorname{im}(P_{q-1})} \xrightarrow{\cong} \check{H}^1(X,\mathcal{K}^{q-1}). \end{align*}[/guided]

[step:Dimension shift first cohomology back to $\check{H}^q(X,\mathcal{F})$]For each integer $m \geq 0$ and each integer $r \geq 1$, apply the long exact sequence in Čech cohomology to \begin{align*} 0 \to \mathcal{K}^{m} \xrightarrow{\iota_m} \mathcal{A}^{m} \xrightarrow{\pi_m} \mathcal{K}^{m+1} \to 0. \end{align*} The relevant segment is \begin{align*} \check{H}^r(X,\mathcal{A}^{m}) \to \check{H}^r(X,\mathcal{K}^{m+1}) \xrightarrow{\partial_m^r} \check{H}^{r+1}(X,\mathcal{K}^{m}) \to \check{H}^{r+1}(X,\mathcal{A}^{m}). \end{align*} Acyclicity gives \begin{align*} \check{H}^r(X,\mathcal{A}^{m})=0 \qquad\text{and}\qquad \check{H}^{r+1}(X,\mathcal{A}^{m})=0, \end{align*} so exactness makes the connecting homomorphism \begin{align*} \partial_m^r: \check{H}^r(X,\mathcal{K}^{m+1}) \xrightarrow{\cong} \check{H}^{r+1}(X,\mathcal{K}^{m}) \end{align*} an isomorphism. For $q=1$, set $\Theta_1$ to be the identity map on $\check{H}^1(X,\mathcal{K}^0)=\check{H}^1(X,\mathcal{F})$. For $q \geq 2$, define \begin{align*} \Theta_q := \partial_0^{q-1} \circ \partial_1^{q-2} \circ \cdots \circ \partial_{q-2}^{1} : \check{H}^1(X,\mathcal{K}^{q-1}) \xrightarrow{\cong} \check{H}^{q}(X,\mathcal{K}^{0}). \end{align*} Since $\mathcal{K}^0=\mathcal{F}$, this gives an isomorphism \begin{align*} \Theta_q: \check{H}^1(X,\mathcal{K}^{q-1}) \xrightarrow{\cong} \check{H}^{q}(X,\mathcal{F}) \end{align*} for every integer $q \geq 1$.[/step]

[guided]This is the dimension-shifting step. The idea is that the acyclic sheaf $\mathcal{A}^m$ sits between $\mathcal{K}^m$ and $\mathcal{K}^{m+1}$, so it contributes no positive cohomology and forces the connecting morphism to identify the cohomology of adjacent kernel sheaves with a degree shift. Fix integers $m \geq 0$ and $r \geq 1$. We apply the long exact sequence in Čech cohomology to the short exact sequence \begin{align*} 0 \to \mathcal{K}^{m} \xrightarrow{\iota_m} \mathcal{A}^{m} \xrightarrow{\pi_m} \mathcal{K}^{m+1} \to 0. \end{align*} The part of the long exact sequence involving the connecting homomorphism is \begin{align*} \check{H}^r(X,\mathcal{A}^{m}) \to \check{H}^r(X,\mathcal{K}^{m+1}) \xrightarrow{\partial_m^r} \check{H}^{r+1}(X,\mathcal{K}^{m}) \to \check{H}^{r+1}(X,\mathcal{A}^{m}). \end{align*} The acyclicity hypothesis on $\mathcal{A}^m$ applies because $r \geq 1$ and $r+1 \geq 1$. Therefore \begin{align*} \check{H}^r(X,\mathcal{A}^{m})=0 \qquad\text{and}\qquad \check{H}^{r+1}(X,\mathcal{A}^{m})=0. \end{align*} Exactness now forces $\partial_m^r$ to be both injective and surjective. Injectivity follows because its kernel is the image of the zero group $\check{H}^r(X,\mathcal{A}^m)$, and surjectivity follows because its image is the kernel of the map into the zero group $\check{H}^{r+1}(X,\mathcal{A}^m)$. Hence \begin{align*} \partial_m^r: \check{H}^r(X,\mathcal{K}^{m+1}) \xrightarrow{\cong} \check{H}^{r+1}(X,\mathcal{K}^{m}) \end{align*} is an isomorphism. Starting with $\check{H}^1(X,\mathcal{K}^{q-1})$, we apply these isomorphisms successively: \begin{align*} \check{H}^1(X,\mathcal{K}^{q-1}) &\xrightarrow{\partial_{q-2}^{1}} \check{H}^{2}(X,\mathcal{K}^{q-2}) \xrightarrow{\partial_{q-3}^{2}} \cdots \xrightarrow{\partial_0^{q-1}} \check{H}^{q}(X,\mathcal{K}^{0}). \end{align*} Thus, for $q \geq 2$, define \begin{align*} \Theta_q := \partial_0^{q-1} \circ \partial_1^{q-2} \circ \cdots \circ \partial_{q-2}^{1}. \end{align*} This is a composition of isomorphisms, hence an isomorphism. For $q=1$, no shifting is needed, so $\Theta_1$ is the identity map on $\check{H}^1(X,\mathcal{K}^0)$. Since $\mathcal{K}^0=\mathcal{F}$, we obtain for every integer $q \geq 1$ an isomorphism \begin{align*} \Theta_q: \check{H}^1(X,\mathcal{K}^{q-1}) \xrightarrow{\cong} \check{H}^{q}(X,\mathcal{F}). \end{align*}[/guided]

[step:Treat degree zero and assemble the displayed isomorphism] For degree $0$, interpret $\Gamma(X,\mathcal{A}^{-1})$ as the zero group and the map $\Gamma(X,\mathcal{A}^{-1}) \to \Gamma(X,\mathcal{A}^{0})$ as the zero map. Since $\ker(d_0)=\operatorname{im}(\iota_0)$ and $\iota_0:\mathcal{F}\to\mathcal{A}^0$ is injective, the left exactness of global sections gives an isomorphism \begin{align*} \Gamma(X,\mathcal{F}) \xrightarrow{\cong} \ker\left(D_0:\Gamma(X,\mathcal{A}^{0})\to \Gamma(X,\mathcal{A}^{1})\right). \end{align*} By zeroth Čech cohomology is global sections, \begin{align*} \check{H}^0(X,\mathcal{F})\cong \Gamma(X,\mathcal{F}), \end{align*} and therefore \begin{align*} \check{H}^0(X,\mathcal{F}) \cong \frac{\ker\left(\Gamma(X,\mathcal{A}^{0})\to \Gamma(X,\mathcal{A}^{1})\right)} {\operatorname{im}\left(\Gamma(X,\mathcal{A}^{-1})\to \Gamma(X,\mathcal{A}^{0})\right)}. \end{align*} Now let $q \geq 1$. Combining the isomorphisms constructed above gives \begin{align*} \frac{\ker\left(\Gamma(X,\mathcal{A}^{q})\to \Gamma(X,\mathcal{A}^{q+1})\right)} {\operatorname{im}\left(\Gamma(X,\mathcal{A}^{q-1})\to \Gamma(X,\mathcal{A}^{q})\right)} &= \frac{Z^q_\Gamma}{B^q_\Gamma} \\ &\cong \frac{\Gamma(X,\mathcal{K}^{q})}{\operatorname{im}(P_{q-1})} \\ &\cong \check{H}^1(X,\mathcal{K}^{q-1}) \\ &\cong \check{H}^{q}(X,\mathcal{F}). \end{align*} Taking the inverse of this composite isomorphism gives the displayed form \begin{align*} \check{H}^q(X,\mathcal{F}) \cong \frac{\ker(\Gamma(X,\mathcal{A}^{q}) \to \Gamma(X,\mathcal{A}^{q+1}))} {\operatorname{im}(\Gamma(X,\mathcal{A}^{q-1}) \to \Gamma(X,\mathcal{A}^{q}))}. \end{align*} This proves that the Čech cohomology of $\mathcal{F}$ is computed by the cohomology of the global-section complex of the acyclic resolution. [/step]