Proper surjections have cohomological descent for constructible field sheaves

Bounded strict model and finite diagonals. Choose a cochain lower bound $a$ with $F$ represented by a strict complex vanishing in degrees $q<a$, possible because $F$ is cohomologically bounded; by [F1] exact inverse image preserves vanishing there, so the same $a$ bounds every $F_p$, and right-derived pushforward produces a functorial double model in degrees $p\ge 0$, $q\ge a$. In total degree $m$ only $0\le p\le m-a$ occurs, so every diagonal is finite and [F2]'s strict total model may be used with either totalization. Take the cone of the augmented map $F\to {\mathrm{Tot}}_{p}R{f}_{p\ast }{F}_p$, place the augmentation in column $p=-1$, and filter by $p$; each total degree of the cone involves only finitely many columns. At a point $x\in X$ the stalk of the cone is the cone of the stalk map, and because the stalk functor is exact and commutes with the finite column sums, this stalk carries the induced finite filtration, whose first page in internal degree $q$ is the augmented horizontal row of $H^q$ of the stalk complexes.

The stalk row and one chosen point. Fix $x\in X$ and put $K=F_x$, a complex of $k$-vector spaces with $H^q(K)=H^q(F{)}_x$; surjectivity of $f$ makes the fiber $S=Y_x$ nonempty. By the proper base change and fiber-pullback clauses of [F1], the stalk at $x$ of $R{f}_{p\ast }{F}_p$ is $R\mathrm{\Gamma }(S^{p+1},K)$ with $K$ the constant derived pullback, and these identifications are natural for all face and degeneracy maps, so the first page of the stalk row in internal degree $q$ is $H^q(K)\stackrel{{\delta }_{-1}}{\to }{H}^q(S,K)\stackrel{{\delta }_0}{\to }{H}^q(S^2,K)\stackrel{{\delta }_1}{\to }\cdots ,$ where ${\delta }_p$ is the alternating sum of the pullbacks along the $(p+2)$ coordinate-deletion maps $d_i:S^{p+2}\to S^{p+1}$. Choose one point $y\in S$; no assertion that any positive cohomology group of $S$ vanishes, or that $S$ is connected or smooth, is made or used.

Explicit contraction of every augmented row. For $r\ge 0$ let ${\mathrm{i}\mathrm{n}\mathrm{s}}_r:S^r\to S^{r+1}$ insert $y$ as first coordinate, with $S^0=\mathrm{p}\mathrm{t}$; so ${\mathrm{i}\mathrm{n}\mathrm{s}}_0$ is the point $y$ itself. The deletion and insertion maps satisfy $d_0{\mathrm{i}\mathrm{n}\mathrm{s}}_{r+1}=\mathrm{i}\mathrm{d}$ and $d_i{\mathrm{i}\mathrm{n}\mathrm{s}}_{r+1}={\mathrm{i}\mathrm{n}\mathrm{s}}_r{d}_{i-1}$ for $i>0$. Put $h_{p+1}:={\mathrm{i}\mathrm{n}\mathrm{s}}_{p+1}^{\ast }:H^q(S^{p+2},K)\to H^q(S^{p+1},K)$ and $h_0:={\mathrm{i}\mathrm{n}\mathrm{s}}_0^{\ast }:H^q(S,K)\to H^q(K)$, with ${\delta }_{-1}$ the constant pullback of the projection $S\to \mathrm{p}\mathrm{t}$. Applying the two identities termwise gives $h_{p+1}{\delta }_p+{\delta }_{p-1}{h}_p=\mathrm{i}\mathrm{d}$ for every $p\ge 0$: the $i=0$ term of $h_{p+1}{\delta }_p$ is the identity, and the terms $i>0$ are the negatives of ${\delta }_{p-1}{h}_p$; moreover $h_0{\delta }_{-1}=\mathrm{i}\mathrm{d}$, because the projection followed by the chosen point is the identity of $\mathrm{p}\mathrm{t}$. Hence every augmented horizontal row is contractible, in particular exact, in every internal degree $q$, using only functorial pullback on cohomology.

Conclusion and naturality. By step 2.1 the next page of the spectral sequence of the stalk cone vanishes at every stalk, and by step 1.1 its filtration is finite in each total degree, so [F3] identifies the associated graded of the stalk cohomology with zero in every degree; the stalk of the cone at $x$ is therefore acyclic, for every $x\in X$. A complex of sheaves whose stalks are all acyclic has zero cohomology sheaves, hence is acyclic; this proves the asserted quasi-isomorphism $F\stackrel{\sim }{\to }{\mathrm{Tot}}_{p\ge 0}R{f}_{p\ast }{F}_p$. All algebraic base changes of $f$ remain proper surjective, and the identifications of steps 1.2-3.1 are pullbacks of one another by the Cartesian-pasting naturality of [F1], so the pulled-back assertion follows without choosing compatible points; coefficient morphisms act componentwise by the same naturality. For the global statement use [F2]’s simplicial injective model: its levelwise injectives push forward to injectives, since inverse image is exact and left adjoint to direct image. The total is a bounded-below complex of injectives, because each total degree is a finite sum. Thus global sections compute its derived global sections termwise; derived composition in [F1] identifies the columns with $R\mathrm{\Gamma }(Y_p,F_p)$, giving $R\mathrm{\Gamma }(X,F)\simeq {\mathrm{Tot}}_{p}R\mathrm{\Gamma }(Y_p,F_p)$. The Axiom of Choice is inherited exactly through [A1].