Refinement-limit Čech cohomology computes derived sheaf cohomology on metric spaces

Let $0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0$ be a short exact sequence of abelian sheaves and let $c$ be a Čech $p$-cochain of $\mathcal{C}$ on an open cover $\mathcal{U}$. By [F1], refine $\mathcal{U}$ to a locally finite cover and then use [F2] to choose locally finite covers $\{V_i\}$ and $\{W_i\}$ with $\overline{V_i}\subset W_i\subset \overline{W_i}\subset U_i$. For each point $x$, local surjectivity from [F3] lifts each section $c_{i_0\dots i_p}$ on a neighborhood of $x$ whenever $x\in U_{i_0}\cap \cdots \cap U_{i_p}$. Choose a preliminary neighborhood $O_x$ meeting only finitely many $W_i$, and denote their finite index set by $J_x$. The assigned sets $U_i$ may repeat and need not form a locally finite indexed family. For each tuple of indices in $J_x$ with $x$ in its assigned $U$-intersection, choose a neighborhood on which its cochain component lifts. Shrink a neighborhood $N_x\subseteq O_x$ of $x$ inside all these finitely many lift neighborhoods and, for each $i\in J_x$, inside $U_i$ whenever $x\in U_i$, and inside $W_i$ whenever $x\in W_i$; if $x\mathrm{\notin }{U}_i$, also shrink to avoid $\overline{W_i}$, which is closed and does not contain $x$ because $\overline{W_i}\subset U_i$. Then $N_x\subset U_i$ whenever $N_x\cap W_i\mathrm{\ne }\mathrm{\varnothing }$, and every component indexed by a tuple in $J_x$ whose assigned intersection contains $x$ has a lift on $N_x$. Choose $\rho (x)$ with $x\in W_{\rho (x)}$. Then $N_x\subset W_{\rho (x)}\subset U_{\rho (x)}$, so $\{N_x\}$ refines $\mathcal{U}$. If $N_{x_0}\cap \cdots \cap N_{x_p}\mathrm{\ne }\mathrm{\varnothing }$, then $N_{x_0}$ meets every $W_{\rho (x_j)}$; the construction puts $N_{x_0}$ inside every $U_{\rho (x_j)}$, so $x_0$ belongs to the original intersection indexed by $\rho (x_0),\dots ,\rho (x_p)$. Its cochain component therefore has a lift on $N_{x_0}$ and hence on the full new intersection. Repeated source indices give a zero component of the alternating Čech refinement cochain, which also lifts. Thus every Čech cochain becomes liftable after refinement.

Construct connecting maps on cocycles rather than taking a colimit of cochain complexes. Write $d$ for the Čech differential and $u:\mathcal{A}\to \mathcal{B}$ and $v:\mathcal{B}\to \mathcal{C}$. If $c\in C^p(\mathcal{U},\mathcal{C})$ is closed, step 1.1 supplies on a refinement a lift $b$ of its pullback. Since $v(db)=dc=0$, [F5] gives a unique $a\in C^{p+1}(\mathcal{V},\mathcal{A})$ with $u(a)=db$. Injectivity of $u$ and $d^2=0$ give $da=0$. Define $\mathrm{\partial }[c]$ to be the refinement-limit class of $a$. Two lifts on the same cover differ by an $\mathcal{A}$-cochain, so their resulting $a$ differ by a coboundary. If $c$ is replaced by $c+dt$, lift $t$ after another refinement by step 1.1 and replace $b$ by $b+d\tilde{t}$; the resulting $a$ is unchanged. For $p=0$ there are no negative-degree coboundaries.

Independence of refinement choices requires the cochain homotopy behind [F4]. Extend ordered cochains alternatingly to all tuples, setting repeated-index components to zero. For refinement maps $r,s$ from a cover $\mathcal{V}$ to $\mathcal{U}$, put, for a degree-$p$ cochain with $p\ge 1$, $(Hc{)}_{j_0\dots j_{p-1}}={\sum }_{k=0}^{p-1}(-1{)}^k{c}_{r(j_0),\dots ,r(j_k),s(j_k),\dots ,s(j_{p-1})}{\mathrm{\mid }}_{V_{j_0}\cap \cdots \cap V_{j_{p-1}}},$ and set $H=0$ on degree zero. All restrictions are defined because $V_j\subset U_{r(j)}\cap U_{s(j)}$. Expansion of the alternating differential cancels the terms omitting an index before or after the change from $r$ to $s$ in pairs; the two endpoint terms remain, giving $s^{\ast }-r^{\ast }=dH+Hd$. This identity also holds in degree zero, where $Hdc$ is the difference of the two restrictions of $c$. To compare two constructions in step 2.1, pull their lifts to a common refinement. Their images in $\mathcal{C}$ differ by $(s^{\ast }-r^{\ast })c=dHc$. After lifting $Hc$ on a further refinement, the difference of the two $\mathcal{B}$-lifts minus the differential of that lift lies in $\mathcal{A}$; its differential is the difference of the two connecting cocycles. Thus both constructions define the same limit cohomology class. Common refinements also prove additivity, and lifting the images of chosen cochains proves naturality for morphisms of short exact sequences. No strictly functorial system of cochain maps on the preorder of covers is assumed.

The resulting sequence of limit cohomology groups is exact. At ${\check{H}}^p(\mathcal{B})$, if a closed $b$ maps to a coboundary $dt$ after refinement, lift $t$ by step 1.1 and replace $b$ by $b-d\tilde{t}$; this is a closed $\mathcal{A}$-cochain representing a preimage. In degree zero a zero image already lies in $\mathcal{A}$ by [F5]. At ${\check{H}}^p(\mathcal{C})$, if the connecting cocycle $a$ of step 2.1 becomes $de$, replace its lift $b$ by $b-u(e)$; it is now closed and maps to $c$. Conversely a closed lift has zero connecting class. At ${\check{H}}^{p+1}(\mathcal{A})$, if a closed $a$ maps to a coboundary $db$ after refinement, then $v(b)$ is closed and its connecting class is $[a]$ by construction. Conversely every connecting cocycle maps to $db$. Compositions of consecutive maps vanish by the same formulas; injectivity at degree-zero global sections follows from [F5]. Equality to zero in a limit is witnessed on a common refinement by [F4], so these arguments cover every element of every kernel. Together with [F9] and the naturality in step 3.1, this makes ${\check{H}}^{\bullet }(X,-)$ a cohomological delta functor extending $\mathrm{\Gamma }(X,-)$.

Let $\mathcal{I}$ be injective. By [F7] it and all its restrictions are acyclic, so every open cover is $\mathcal{I}$-acyclic. By [F8], ${\check{H}}^q(\mathcal{U},\mathcal{I})\cong H^q(X,\mathcal{I})=0$ for every cover and every $q>0$. Hence ${\check{H}}^q(X,\mathcal{I})=0$ for $q>0$. Since every sheaf embeds in such an $\mathcal{I}$ by [F6], the Čech delta functor of step 4.1 is effaceable in positive degrees.

For an injective sheaf $\mathcal{I}$, the resolution $0\to \mathcal{I}\stackrel{1}{\to }\mathcal{I}\to 0$ computes $H^q(X,\mathcal{I})=0$ for $q>0$. By [F6], every sheaf embeds in an injective, so [F11] makes $H^{\bullet }(X,-)$ universal, as well as ${\check{H}}^{\bullet }(X,-)$ by step 5.1. Their degree-zero identifications with $\mathrm{\Gamma }(X,-)$ agree by [F9], so [F11] gives a unique natural isomorphism ${\check{H}}^q(X,\mathcal{F})\cong H^q(X,\mathcal{F})$ in every degree. To identify it with the canonical comparison, represent a class by a closed Čech cochain $c$ on a cover $\mathcal{U}$ and refine as in step 1.1 so that $c$ has a lift $b$ to the middle sheaf $\mathcal{B}$ of a short exact sequence $0\to \mathcal{A}\stackrel{i}{\to }\mathcal{B}\stackrel{v}{\to }\mathcal{C}\to 0$. Then $\delta b=i(a)$ for the connecting cocycle $a$ of step 2.1. The product-of-stalks formula in [F13] is exact on sections under AC. For a short exact sequence, the diagram of injective germ maps and the kernel-cokernel exact sequence show that their cokernel sheaves also form a short exact sequence; induction gives exactness on sections in every Godement degree. Products over cover intersections are exact under AC, and each total degree is a finite direct sum, so both the global Godement complexes and the Čech–Godement total complexes form short exact sequences. Their natural quasi-isomorphisms commute with connecting maps by [F12]. By [F13], the comparison uses natural cochain maps to these total complexes; the image of $b$ is a lift of the image of $c$, and its differential is the image of $\delta b=i(a)$. Thus the derived connecting class of the image of $c$ is the comparison image of $[a]$. Compatibility with refinement in [F13] makes this calculation independent of the chosen lift and cover. The canonical comparison is therefore a morphism of delta functors extending the degree-zero identity, so universality identifies it with the unique isomorphism above. Full AC supplies DC for [F10] and the choices in step 1.1; no compact-support cohomology is used.