A polyhedral dimension bound for every sheaf

The polyhedron. By [F1] the realization $P=\mathrm{\mid }K\mathrm{\mid }$ is a compact Hausdorff space whose weak topology is the Euclidean subspace topology of a finite-dimensional Euclidean space; in particular $P$ is a compact metric space, with the subspace metric and diameter as in [F2], and it is locally compact [F1]. Every simplex of ${\mathrm{sd}}^{r}K$ is a strict chain of nonempty faces of ${\mathrm{sd}}^{r-1}K$, hence has at most $\dim ({\mathrm{sd}}^{r-1}K)+1$ vertices, because a strictly increasing chain of faces of a complex of dimension $e$ adds at least one vertex at each step and every face has at most $e+1$ vertices; inductively $\dim {\mathrm{sd}}^{r}K\le \dim K=d$, so every simplex of ${\mathrm{sd}}^{r}K$ has at most $d+1$ vertices.

The subcomplex clause. Let $L\subseteq K$ be a subcomplex; $L$ is finite because $K$ is. By [F7] the inclusion $\mathrm{\mid }L\mathrm{\mid }\hookrightarrow P$ is a closed embedding, so $\mathrm{\mid }L\mathrm{\mid }$ is closed in $P$ and $P\setminus \mathrm{\mid }L\mathrm{\mid }$ is open; every open subspace of the subspace $P\setminus \mathrm{\mid }L\mathrm{\mid }$ is a trace of an open subspace of $P$, hence is itself open in $P$ [F7].

Čech cohomology vanishes above $d$ on a cover of order at most $d+1$. Let $\mathcal{V}$ be a finite open cover of a space $X$ every point of which lies in at most $d+1$ members, and fix a linear order on its index set so that the ordered Čech complex of [F4] is defined. If $p>d$ then $p+1\ge d+2$ and any $p+1$ distinct members of $\mathcal{V}$ have empty intersection, because a common point would lie in $p+1\ge d+2$ members; each tuple intersection is therefore empty and each factor of $C^p(\mathcal{V},\mathcal{F})$ is the zero group, so $C^p(\mathcal{V},\mathcal{F})=0$ and hence ${\check{H}}^p(\mathcal{V},\mathcal{F})=0$ for every sheaf $\mathcal{F}$ on $X$.

Star covers refine every cover with multiplicity at most $d+1$. Assume first that $P$ is nonempty, let $\mathcal{U}$ be an open cover of $P$, and choose a Lebesgue number $\delta >0$ for $\mathcal{U}$ [F3]. By [F2] the mesh $m({\mathrm{sd}}^{r}K)$ tends to $0$ as $r\to \mathrm{\infty }$ (it is even $0$ when $d=0$), so choose $r$ with $2m({\mathrm{sd}}^{r}K)<\delta$; then every nonempty open star of $T:={\mathrm{sd}}^{r}K$ has diameter at most $2m(T)<\delta$ [F2], so by the Lebesgue property it lies in some member of $\mathcal{U}$. The open stars of the vertices of $T$ are open, nonempty and cover $\mathrm{\mid }T\mathrm{\mid }=P$ [F2], and there are finitely many of them because $T$ is finite; choosing for each vertex one member of $\mathcal{U}$ containing its star is a finite selection among nonempty sets, and defines a refinement function [F4], so the family of open stars is a finite open cover of $P$ refining $\mathcal{U}$ [F4]. Finally, if a point $x\in P$ lies in the stars of vertices $v_0,\dots ,v_m$ of $T$, then all of $v_0,\dots ,v_m$ belong to the support simplex of $x$, which by [F2] is a simplex of $T$ with at most $d+1$ vertices by [step 1.1]; hence $m+1\le d+1$, that is, every point lies in at most $d+1$ members of the cover. If $P$ is empty the empty cover refines $\mathcal{U}$ and has order $0$.

Clause (3), the cohomological statement. Let $X$ be a compact Hausdorff space in which every open cover admits a finite open refinement of order at most $d+1$. The family of all finite open covers of $X$ of order at most $d+1$ is cofinal among all open covers by hypothesis, and by [step 1.3] every member of the family has vanishing Čech cohomology above $d$ with values in every sheaf; the comparison theorem [F5] applies because $X$ is compact Hausdorff, so $H^q(X,\mathcal{F})=0$ for every $q>d$ and every sheaf of abelian groups $\mathcal{F}$ on $X$.

Clause (1). The star covers produced in [step 2.1] form a family of open covers of $P$: for every open cover $\mathcal{U}$ the construction produces a finite cover by open stars of some ${\mathrm{sd}}^{r}K$ refining $\mathcal{U}$, so the family is cofinal in the directed preorder of covers, and by [step 1.3] each of its members has vanishing Čech cohomology above $d$ with values in every sheaf. Since $P$ is compact Hausdorff by [step 1.1], the cofinal-family corollary of the comparison theorem [F5] gives $H^q(P,\mathcal{F})=0$ for every $q>d$ and every sheaf $\mathcal{F}$ of abelian groups on $P$.

Clause (3), the compact-support statement. Let $X$ be as in [step 2.2], let $U\subseteq X$ be open and let $\mathcal{E}$ be a sheaf on $U$. The compact space $X$ is locally compact because $X$ itself is a compact neighbourhood of every point (Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space), so the transport isomorphisms of [F6] apply with base space $X$: $R^q{\mathrm{\Gamma }}_c(X,j_{!}\mathcal{E})\cong R^q{\mathrm{\Gamma }}_c(U,\mathcal{E})$ with $j:U\hookrightarrow X$ the inclusion, and $H_c^q(X,j_{!}\mathcal{E})=H^q(X,j_{!}\mathcal{E})$; by [step 2.2] the latter vanishes for $q>d$, so $H_c^q(U,\mathcal{E})=0$ for $q>d$.

Clause (2). Let $U\subseteq P$ be open and let $\mathcal{E}$ be a sheaf on $U$; write $j:U\hookrightarrow P$ for the open inclusion. By [F6] there are natural isomorphisms $R^q{\mathrm{\Gamma }}_c(P,j_{!}\mathcal{E})\cong R^q{\mathrm{\Gamma }}_c(U,\mathcal{E})$ for all $q\ge 0$, and since $P$ is compact [step 1.1] the identification $H_c^q(P,j_{!}\mathcal{E})=H^q(P,j_{!}\mathcal{E})$ holds. By [step 3.1] the group on the right vanishes for $q>d$, hence $H_c^q(U,\mathcal{E})=0$ for every $q>d$.

Conclusion. Clause (1) is [step 3.1], clause (2) is [step 4.1], and clause (3) is [step 2.2] together with [step 3.2]; the subcomplex statement is [step 1.2]. The isomorphism of [F5] and the derived transport of [F6] use dependent choice, a consequence of AC [F8]; the cover-theoretic construction of [step 2.1] selects only finitely many members of a cover by finite choice, and no other selection is made.