Refinement-limit Cech cohomology computes sheaf cohomology on compact Hausdorff spaces

Setup. Write $T^p(\mathcal{F}):={\check{H}}^p(X,\mathcal{F})$ for the refinement-limit Čech groups and $S^p(\mathcal{F}):=H^p(X,\mathcal{F})$ for sheaf cohomology of Sheaf cohomology as right derived global sections, both zero for $p<0$; by [F1] the groups $T^p(\mathcal{F})$ are the filtered colimit of the fixed-cover groups over the directed preorder of open covers, a class being represented by a pair $(\mathcal{U},\alpha )$ with two representatives identified once they agree after refinement to a common cover, and the finite covers are cofinal because $X$ is compact.

The comparison map exists and is natural. For each cover $\mathcal{U}$ and each $p$ the fixed-cover comparison ${\phi }_{\mathcal{U}}^p$ of [F2] is natural in $\mathcal{F}$ and compatible with refinement, and this compatibility is exactly the cocycle condition needed to descend to the filtered colimit of [F1]; hence there is a well-defined map ${\phi }_X^p:T^p(\mathcal{F})\to S^p(\mathcal{F})$, natural in $\mathcal{F}$, induced by the maps ${\phi }_{\mathcal{U}}^p$ on the colimit.

Degree zero. By the degree-zero clause of [F2], each ${\phi }_{\mathcal{U}}^0$ is the identity of $\mathrm{\Gamma }(X,\mathcal{F})$ under the canonical identifications ${\check{H}}^0(\mathcal{U},\mathcal{F})\cong \mathrm{\Gamma }(X,\mathcal{F})\cong H^0(X,\mathcal{F})$; these identifications are compatible with refinement, so they pass to the colimit and exhibit ${\phi }_X^0$ as a natural isomorphism $T^0\stackrel{\sim }{\to }\mathrm{\Gamma }$ and simultaneously $S^0\stackrel{\sim }{\to }\mathrm{\Gamma }$.

Construction of the connecting map ${\mathrm{\partial }}_T$ on the colimit. Let $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence and let $[h]\in T^p({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ be represented by a cocycle $h\in Z^p(\mathcal{U},{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$; replacing $\mathcal{U}$ by a finite subcover represents the same class by [F1], so we may take $\mathcal{U}=(U_i{)}_{i\in I}$ finite. Since the sequence of sheaves is exact, every section $h_{i_0\dots i_p}\in {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}(U_{i_0}\cap \cdots \cap U_{i_p})$ lifts locally: for each point $x$ of that intersection there are an open neighbourhood $W\subseteq U_{i_0}\cap \cdots \cap U_{i_p}$ and a section of $\mathcal{F}$ over $W$ mapping to $h_{i_0\dots i_p}{\mathrm{\mid }}_W$, by stalkwise exactness [F9]; choose such a pair for every tuple and every point, a selection of representatives from nonempty sets licensed by [F11]. Extend the ordered cochain $h$ to all tuples by the alternating convention: repeated indices give zero and permutations give their sign (Ordered and alternating Čech complexes agree, Refinement map of ordered open covers). The compatible-refinement lemma [F10] applied to these tuple-indexed covers produces a finite open cover $(V_j{)}_{j\in J}$ with a map $\alpha :J\to I$ such that $V_j\subseteq U_{\alpha (j)}$ and every nonempty $V_{j_0}\cap \cdots \cap V_{j_p}$ lies in one of the chosen neighbourhoods for the tuple $(\alpha (j_0),\dots ,\alpha (j_p))$. Fix a linear order on the finite $J$. For each increasing tuple with nonempty intersection choose one lift $g_{j_0\dots j_p}\in \mathcal{F}(V_{j_0}\cap \cdots \cap V_{j_p})$ of $h_{\alpha (j_0)\dots \alpha (j_p)}$ restricted to that intersection, use the zero lift on empty intersections, and let $g\in C^p(\mathcal{V},\mathcal{F})$ be the resulting ordered cochain. Then the image of $\delta g\in C^{p+1}(\mathcal{V},\mathcal{F})$ in $C^{p+1}(\mathcal{V},{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ is $\delta ({\alpha }^{\mathrm{♯}}h)=0$, so each component of $\delta g$ lies in the subobject ${\mathcal{F}}^{\mathrm{\prime }}(V_{j_0}\cap \cdots \cap V_{j_{p+1}})\subseteq \mathcal{F}(V_{j_0}\cap \cdots \cap V_{j_{p+1}})$, that is, $\delta g\in C^{p+1}(\mathcal{V},{\mathcal{F}}^{\mathrm{\prime }})$ by the section-level exactness at the middle term in [F9]; being a differential, it is a cocycle, and we define ${\mathrm{\partial }}_T[h]:=[\delta g]\in T^{p+1}({\mathcal{F}}^{\mathrm{\prime }})$.

Exactness of the Godement construction. For an abelian sheaf $\mathcal{E}$, the formula $C^0(\mathcal{E})(W)={\prod }_{x\in W}{\mathcal{E}}_x$ of [F4] shows that $C^0$ sends short exact sequences to sequences exact on every open set: the stalk sequences of the original sheaves are exact [F9], products preserve injectivity and kernels componentwise, and products of surjections are surjective under AC [F11]. Therefore $C^0$ is exact as a sheaf functor. The natural germ maps $\mathcal{E}\to C^0(\mathcal{E})$ are monomorphisms [F4]. Apply the snake lemma [F8] to the diagram with rows $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$ and $0\to C^0({\mathcal{F}}^{\mathrm{\prime }})\to C^0(\mathcal{F})\to C^0({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})\to 0$, and vertical germ maps. Its zero kernels give $0\to Q^0({\mathcal{F}}^{\mathrm{\prime }})\to Q^0(\mathcal{F})\to Q^0({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})\to 0$ exact. Repeat this argument on the exact sequences of $Q^{q-1}$: applying $C^0$ gives exactness of $C^q=C^0{Q}^{q-1}$, and taking cokernels of the germ maps gives exactness of $Q^q$. This proves the assertion in every degree without identifying the stalk of $C^0(\mathcal{E})$ with ${\mathcal{E}}_x$.

Effaceability of $T^{\bullet }$. Let $\mathcal{I}$ be injective and let $\mathcal{U}$ be any open cover; then $\mathcal{I}$ is flasque by [F7], hence every open subspace is $\mathcal{I}$-acyclic by [F5], so $\mathcal{U}$ is $\mathcal{I}$-acyclic; by [F3] both $u$ and $w$ are then quasi-isomorphisms, and therefore each ${\phi }_{\mathcal{U}}^p:{\check{H}}^p(\mathcal{U},\mathcal{I})\to H^p(X,\mathcal{I})$ is an isomorphism; for $p>0$ the target vanishes by the acyclicity clause of [F5] applied with $W=X$, so ${\check{H}}^p(\mathcal{U},\mathcal{I})=0$ for every cover and the colimit gives $T^p(\mathcal{I})=0$ for $p>0$. For every sheaf $\mathcal{F}$ and every $n>0$, [F7] supplies a monomorphism $\mathcal{F}\rightarrowtail \mathcal{I}$ into an injective sheaf, and the induced map $T^n(\mathcal{F})\to T^n(\mathcal{I})=0$ is zero; this is exactly effaceability of $T^{\bullet }$ in positive degrees by injectives.

Identification of the comparison on cohomology. For any of these Godement resolutions write $Z^0=\mathcal{E}$ and $0\to Z^n\to C^n(\mathcal{E})\to Z^{n+1}\to 0$ for its cycle sequences. Its terms are $\mathrm{\Gamma }$-acyclic by [F4, F5], and all its cycles belong to the supplied injective datum by [F7]. The canonical acyclic-resolution isomorphism [F13] is obtained by the successive connecting isomorphisms for these sequences, with the last group $H^1(X,Z^{n-1})$ identified with the cokernel of $\mathrm{\Gamma }{C}^{n-1}(\mathcal{E})\to \mathrm{\Gamma }{Z}^n$. A coaugmentation-preserving comparison $C^{\bullet }(\mathcal{E})\to I^{\bullet }$ gives morphisms from these cycle sequences to the corresponding injective cycle sequences. Naturality of the sheaf long exact sequence [F5] identifies their successive connecting maps, and the induced map of the final cokernels is precisely $H^n(\mathrm{\Gamma }a)$. Thus $H^n(\mathrm{\Gamma }a)$ is the canonical Godement-to-injective identification inverse to [F4], also in degree zero where it is the identity of $\mathrm{\Gamma }\mathcal{E}$. The comparison lemma [F12] makes this identification independent of the comparison map, since two such maps are homotopic and applying the additive section functor preserves the homotopy.

Well-definedness and naturality of ${\mathrm{\partial }}_T$. Two lifts of the same cocycle on the same refinement differ by a cochain $f$ of ${\mathcal{F}}^{\mathrm{\prime }}$, so their differentials differ by $\delta f$. More generally, equal colimit classes have cocycle representatives $h,h^{\mathrm{\prime }}$ which, on a common refinement, differ by $\delta a$ for a $(p-1)$-cochain $a$ of ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ [F1]; for $p=0$ the difference is zero. For $p>0$, the local-lift argument of step 1.4 applied in degree $p-1$ supplies, after another refinement, a lift $b$ of $a$. The chosen lifts $g,g^{\mathrm{\prime }}$ of $h,h^{\mathrm{\prime }}$ then satisfy $g^{\mathrm{\prime }}-g-\delta b\in C^p({\mathcal{F}}^{\mathrm{\prime }})$, so $\delta g^{\mathrm{\prime }}-\delta g$ is a coboundary of ${\mathcal{F}}^{\mathrm{\prime }}$. Thus the boundary class is independent of cocycle representatives and of the lifting covers. Different refinement functions induce homotopic cochain maps by the theorem cited in [F2], hence the same cohomology class, so the same argument handles their nonuniqueness. Additivity is checked on a common refinement by adding lifts. A morphism of short exact sequences carries lifts to lifts and commutes with their differentials, proving naturality.

Two degreewise short exact sequences of complexes. If $0\to \mathcal{A}\to \mathcal{B}\to \mathcal{C}\to 0$ is a short exact sequence of sheaves on a space $X$ and $\mathcal{A}$ is flasque, then for every open $W\subseteq X$ the map $\mathcal{B}(W)\to \mathcal{C}(W)$ is surjective: the long exact sequence of [F5] on the space $W$ gives exactness up to $H^1(W,\mathcal{A}{\mathrm{\mid }}_W)$, and $H^1(W,\mathcal{A}{\mathrm{\mid }}_W)=0$ by the acyclicity clause of [F5]. Consequently, for our short exact sequence with Godement resolutions, the sequence of complexes $0\to \mathrm{\Gamma }(X,C^{\bullet }({\mathcal{F}}^{\mathrm{\prime }}))\to \mathrm{\Gamma }(X,C^{\bullet }(\mathcal{F}))\to \mathrm{\Gamma }(X,C^{\bullet }({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}))\to 0$ is degreewise short exact by step 1.5 and the surjectivity just recorded; and for every open cover $\mathcal{U}$ the same argument applied to each open tuple intersection $W$ and each degree $q$ shows that $0\to C^p(\mathcal{U},C^q({\mathcal{F}}^{\mathrm{\prime }}))\to C^p(\mathcal{U},C^q(\mathcal{F}))\to C^p(\mathcal{U},C^q({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}))\to 0$ is exact for all $p,q$, because $C^p(\mathcal{U},-)$ is a product of section functors over the tuple intersections and each of its component maps is onto and AC [F11] supplies simultaneous preimages; summing over the finite diagonals gives the degreewise short exact sequence of total complexes $0\to \mathrm{Tot}D({\mathcal{F}}^{\mathrm{\prime }})\to \mathrm{Tot}D(\mathcal{F})\to \mathrm{Tot}D({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})\to 0$.

A comparison diagram with injective resolutions. Dualizing the degreewise split projective horseshoe in the opposite abelian category [F12], choose a degreewise split short exact sequence $0\to I^{\mathrm{\prime }\bullet }\to I^{\bullet }\to I^{\mathrm{\prime }\mathrm{\prime }\bullet }\to 0$ of injective resolutions of $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}\to 0$. There is a coaugmentation-preserving morphism from the short exact sequence of Godement resolutions of step 1.5 to this sequence, as follows. A split triple $(J^{\mathrm{\prime }},J^{\mathrm{\prime }}\oplus J^{\mathrm{\prime }\mathrm{\prime }},J^{\mathrm{\prime }\mathrm{\prime }})$ with $J^{\mathrm{\prime }},J^{\mathrm{\prime }\mathrm{\prime }}$ injective admits extensions across any componentwise monomorphism of short exact triples: extend the first coordinate of the middle map to the larger middle object using injectivity of $J^{\mathrm{\prime }}$, extend the right-hand map to the larger right-hand object using injectivity of $J^{\mathrm{\prime }\mathrm{\prime }}$, and define the new middle map by these two coordinates, the second composed with the quotient map; restriction of the first coordinate to the larger left-hand object supplies the new left-hand map. These maps extend the old triple map and commute strictly with both arrows. Start with the coaugmentation triple and extend to degree zero. Inductively, the composites with the target differentials kill the source cycles and factor through the next source cycle triple, so extend them into the next split injective degree triple by the same construction. The source cycle triples are short exact because they are the successive exact Godement cokernel triples proved in step 1.5; the factorization is the kernel-cokernel property in the abelian category. DC from [F11] supplies the successive extensions. This gives cochain maps $a^{\mathrm{\prime }},a,a^{\mathrm{\prime }\mathrm{\prime }}$ commuting strictly with the short exact sequence arrows.

Exactness of the long sequences. Work on cocycle representatives in finite common refinements [F1]. At $T^p({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$, the boundary of an image from $T^p(\mathcal{F})$ is zero, since its cocycle itself is a lift. Conversely, if ${\mathrm{\partial }}_T[h]=0$, choose a lift $g$ as in step 1.4; after refinement $\delta g=\delta f$ for a cochain $f$ of ${\mathcal{F}}^{\mathrm{\prime }}$, and $g-f$ is a cocycle lifting $h$. At $T^{p+1}({\mathcal{F}}^{\mathrm{\prime }})$, every boundary is a differential in the $\mathcal{F}$-complex and dies there; conversely, if an ${\mathcal{F}}^{\mathrm{\prime }}$-cocycle becomes $\delta g$ in $\mathcal{F}$ after refinement, the image of $g$ in ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ is a cocycle whose boundary is the original class. At $T^p(\mathcal{F})$, suppose a cocycle $a$ has image $\delta c$ in ${\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ after refinement. If $p>0$, lift the $(p-1)$-cochain $c$ after further refinement by step 1.4, obtaining $b$; then $a-\delta b$ is a cocycle in ${\mathcal{F}}^{\mathrm{\prime }}$ and represents $[a]$. For $p=0$ the image is zero already, so $a$ lies in ${\mathcal{F}}^{\mathrm{\prime }}$ by [F9]. Reverse containments follow from the zero composite of the sheaf maps. Finally $T^0({\mathcal{F}}^{\mathrm{\prime }})\to T^0(\mathcal{F})$ is injective, since $T^0=\mathrm{\Gamma }$ by step 1.3 and section maps of a monomorphism are injective [F9]. These arguments give the entire long exact sequence.

Naturality of $u$ and $w$. The maps $u$ and $w$ in [F3] are built from restriction maps and the functorial Godement coaugmentations [F4], so they commute with sheaf morphisms. In particular, $u$ gives a morphism from the short exact sequence of global Godement complexes to that of total complexes in step 2.2. The fixed-cover Čech complexes of the original sheaves need not be short exact on the right; only the natural squares for $w$ and the individual cochain lifts of step 1.4 will be used.

Identification of the model boundary. Apply $\mathrm{\Gamma }(X,-)$ to the strict comparison diagram of step 2.3. Its Godement row is degreewise short exact by step 2.2, and its injective row remains degreewise split short exact. By [F8], the induced cohomology comparisons commute with the two connecting maps. Step 1.7 identifies the vertical maps with the canonical Godement-to-sheaf identifications, while the injective-row connecting maps are those of [F5], which allows comparison between injective resolutions. Hence the boundary $\mathrm{\partial }$ on the global Godement complexes corresponds to the sheaf-cohomology connecting map ${\mathrm{\partial }}_S$. This establishes boundary compatibility without extending injective-resolution independence to arbitrary resolutions.

Consequence: $u$ commutes with the boundary maps. By step 3.2 and the naturality of long exact sequences in [F8], the cochain map $u:\mathrm{\Gamma }(X,C^{\bullet }(\mathcal{F}))\to \mathrm{Tot}D(\mathcal{F})$ induces a morphism of the two long exact sequences of cohomology attached to the short exact sequences of complexes of step 2.2; that is, $H^q(u)\circ \mathrm{\partial }={\mathrm{\partial }}^{\mathrm{\prime }}\circ H^q(u)$ for the two boundary maps in every degree $q$.

Consequence: $w$ commutes with the constructed boundary maps of the colimit Čech functors. Let $[h]\in T^p({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ be represented by a Čech $p$-cocycle $h$ on a cover $\mathcal{U}$, pass to the finite lifting refinement of step 1.4 and rename it $\mathcal{U}$, with $h$ replaced by its pullback and $g\in C^p(\mathcal{U},\mathcal{F})$ a lift, and consider $w(h)$ and $w(g)$ in the total complex of [F3], where $w(h)$ is a cocycle and $w(g)$ is a lift of it through the epimorphism $\mathrm{Tot}D(\mathcal{F})\to \mathrm{Tot}D({\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$, by the naturality of $w$ in the sheaf from step 3.2. The boundary of the class of $w(h)$ computed by the snake lemma from this lift is the class of $\delta (w(g))=w(\delta g)$, using that $w$ is a cochain map; and the image under $H(w)$ of the Čech boundary class ${\mathrm{\partial }}_T[h]=[\delta g]\in T^{p+1}({\mathcal{F}}^{\mathrm{\prime }})$ is the class of $w(\delta g)$ as well. Hence $H(w)\circ {\mathrm{\partial }}_T={\mathrm{\partial }}_D\circ H(w)$, where ${\mathrm{\partial }}_D$ is the boundary on the total complexes.

The comparison is a morphism of delta functors extending the identity. By steps 1.4 to 1.6 the functors $T^p$ together with the maps ${\mathrm{\partial }}_T$ form a cohomological delta functor on the abelian sheaves on $X$, and by step 1.3 the degree-zero part of ${\phi }_X$ is the canonical identification with $\mathrm{\Gamma }$. The compatibility $H(w)\circ {\mathrm{\partial }}_T={\mathrm{\partial }}_D\circ H(w)$ of step 4.2, together with $H(u)\circ \mathrm{\partial }={\mathrm{\partial }}_D\circ H(u)$ of step 4.1 and the invertibility of $H(u)$ in every degree, gives ${\phi }_X\circ {\mathrm{\partial }}_T=\mathrm{\partial }\circ {\phi }_X$; by step 3.3 the model boundary is the connecting homomorphism ${\mathrm{\partial }}_S$ of sheaf cohomology, so ${\phi }_X$ commutes with the connecting homomorphisms. Hence ${\phi }_X:T^{\bullet }\to S^{\bullet }$ is a morphism of cohomological delta functors extending the canonical degree-zero identification.

Universality and the isomorphism. By steps 3.1, 5.1 and 1.6 and the effaceability theorem of [F6], $T^{\bullet }$ is a universal cohomological delta functor. The functors $S^p=H^p(X,-)$ are a cohomological delta functor by [F5]. Every sheaf embeds into an injective by [F7], and positive $S^p$ vanish on injectives because they are flasque [F7] and acyclic [F5]. Thus $S^{\bullet }$ is effaceable by injectives and universal by [F6]. Both delta functors extend $\mathrm{\Gamma }$ in degree zero with the identifications of step 1.3; by the uniqueness clause of [F6] there is an isomorphism $T^{\bullet }\to S^{\bullet }$ of cohomological delta functors extending that identification, and by the degree-zero determination clause of [F6] every morphism of universal delta functors extending it equals that isomorphism. Since ${\phi }_X$ is such a morphism by step 5.1, every ${\phi }_X^p$ is an isomorphism.

The cofinal-family consequence. Let a cofinal family of covers satisfy ${\check{H}}^p(\mathcal{U},\mathcal{F})=0$ for all its members and all $p>d$. Given a class in $T^p(\mathcal{F})$ with $p>d$, represent it by a pair $(\mathcal{U},\alpha )$ and choose a cover $\mathcal{V}$ of the family refining $\mathcal{U}$, which exists by cofinality; the class maps to ${\check{H}}^p(\mathcal{V},\mathcal{F})=0$ in the colimit, hence is zero by the description of the colimit in [F1]. Thus $T^p(\mathcal{F})=0$ for $p>d$, and step 6.1 gives $H^p(X,\mathcal{F})\cong T^p(\mathcal{F})=0$ for $p>d$.

Conclusion. The comparison map ${\phi }_X^p$ is an isomorphism for every $p\ge 0$, natural in $\mathcal{F}$; the collection is an isomorphism of cohomological delta functors extending the identity in degree $0$ and commuting with the connecting homomorphisms of short exact sequences, by steps 1.2, 1.3 and 6.1; and the cofinal-family consequence holds by step 7.1. The choice principles used are the Axiom of Choice for the injective-resolution data, for the well-orderings behind the colimit and for the local lifting selections of step 1.4, and dependent choice for the derived-functor comparison theorems, as recorded in [F11].