Analytic field sheaves have injective derived models

(Setup.) A sheaf of $k$-vector spaces on $X$ is an ${\mathcal{O}}_X$-module for the ringed space $(X,k_X)$ and conversely, and the stalk $k$ is a local ring, so $(X,k_X)$ is a locally ringed space with ${\mathrm{Sh}}_k(X)=\mathrm{M}\mathrm{o}\mathrm{d}(k_X)$; this category is abelian, and $D^b({\mathrm{Sh}}_k(X))$ is taken with the cochain conventions of Derived category of an abelian category, every Hom-set assertion being under its standing smallness hypothesis. In the second part, $D_{\mathfrak{S}}^b(X;k)$ is the full subcategory of complexes whose cohomology sheaves are bounded and $\mathfrak{S}$-constructible.

(Extensions of local systems.) Let $0\to K\to E\stackrel{\pi }{\to }Q\to 0$ be exact on any space $Y$, with $K,Q$ locally constant of finite rank. Near $y\in Y$ trivialize both, and let $e_1,\dots ,e_q$ be the constant basis sections of $Q$. By [F6], their germs lift to $E_y$; represent these finitely many lifts by sections $t_i$ and shrink to one neighbourhood $N^{\mathrm{\prime }}$ where $\pi (t_i)=e_i$. For any open $W\subseteq N^{\mathrm{\prime }}$ and locally constant coefficients $a_i:W\to k$, define $s({\sum }_i{a}_i{e}_i)={\sum }_i{a}_i{t}_i$ by gluing the ordinary scalar multiples on opens where all $a_i$ are constant. Locality makes this well defined and compatible with restriction, and $\pi s=\mathrm{id}$. Identifying $K$ with the kernel of $\pi$, the maps $(u,v)\mapsto u+s(v)$ and $e\mapsto (e-s\pi (e),\pi (e))$ are inverse sheaf morphisms between $K{\mathrm{\mid }}_{N^{\mathrm{\prime }}}\oplus Q{\mathrm{\mid }}_{N^{\mathrm{\prime }}}$ and $E{\mathrm{\mid }}_{N^{\mathrm{\prime }}}$. Therefore $E$ is locally constant of finite rank. This uses finitely many lifts and no connected-neighbourhood hypothesis.

(Injective models.) By [F1] applied to $(X,k_X)$ every $k$-sheaf receives a specific injective resolution, and the category has enough injectives. AC supplies DC by [F3]. For each bounded-below complex, [F11] now supplies a bounded-below termwise injective replacement; the successive embeddings may be taken from the fixed embedding functor of [F1]. Thus the replacement hypothesis of [F2] is met for complexes as well as sheaves, and $K^{+}(\mathrm{Inj}{\mathrm{Sh}}_k(X))\to D^{+}({\mathrm{Sh}}_k(X))$ is an equivalence. By [F12], the canonical functors from $D^b$ and $D^{+}$ to $D$ are fully faithful, with essential images the bounded-cohomology and bounded-below-cohomology objects, respectively. Their commutation with $D^b\to D^{+}$ therefore makes that functor fully faithful, with essential image precisely the bounded-cohomology objects of $D^{+}$. Since replacement preserves cohomology, restricting the injective-model equivalence to complexes with bounded cohomology models $D^b({\mathrm{Sh}}_k(X))$. Hom groups between bounded complexes are computed by homotopy classes of maps between their bounded-below injective replacements, even when the replacements have unbounded terms.

(Shifts and isomorphisms.) Fix $\mathfrak{S}$ and let $A\in D_{\mathfrak{S}}^b(X;k)$. By [F7], $H^j(A[m]){\mathrm{\mid }}_S=H^{j+m}(A){\mathrm{\mid }}_S$, so the cohomological support is reindexed and every restricted cohomology sheaf remains locally constant of finite rank. Isomorphic derived objects have isomorphic cohomology sheaves: the cohomology functors on complexes send quasi-isomorphisms to isomorphisms and therefore descend through localization. Hence the subcategory is closed under all shifts and isomorphisms.

(Cones on one stratum.) Let $f:A\to B$ be a morphism in $D_{\mathfrak{S}}^b(X;k)$ and fix a stratum $S$. By [F6], inverse image to $S$ is exact, since its stalk maps are the original stalk maps. It preserves finite sums, differentials and shifts, hence cone complexes by [F7], and it preserves quasi-isomorphisms. Applying it to a bounded roof representing $f$ consequently gives a derived morphism and its cone triangle on $S$; no chain-map representative $A\to B$ is assumed. The long exact sequence [F4] yields $0\to \mathrm{coker}(f^j)\to H^j(C(f)){\mathrm{\mid }}_S\to \ker (f^{j+1})\to 0$, where $f^j:H^j(A){\mathrm{\mid }}_S\to H^j(B){\mathrm{\mid }}_S$. To check its end terms near $s\in S$, trivialize the finitely ranked source and target. The images of the finitely many constant basis sections have locally constant coordinate functions. Shrinking around $s$ makes these finitely many functions constant, without requiring a connected neighbourhood. The morphism is then a fixed matrix, whose kernel and cokernel are the constant sheaves on the corresponding finite-dimensional vector spaces, as follows either from locally constant functions or from [F8] stalkwise. Thus both end terms are locally constant of finite rank.

(Cone constructibility.) Steps 2.3 and 1.2 show that each $H^j(C(f)){\mathrm{\mid }}_S$ is locally constant of finite rank. If $A$ and $B$ have cohomology only in finite intervals, their cone has cohomology only in the union of the interval for $B$ and the interval for $A$ shifted down one degree, by [F4]. A bounded roof also gives a bounded cone representative. Thus $C(f)\in D_{\mathfrak{S}}^b(X;k)$.

(Triangulated subcategory.) The zero complex belongs to the subcategory, and finite sums do too because their cohomology sheaves are the finite sums of the restricted local systems. By steps 2.2, 3.1 the full additive subcategory $D_{\mathfrak{S}}^b(X;k)$ contains all isomorphs of its objects, is closed under shifts in both directions, and whenever $f$ is a morphism of the subcategory, a cone of $f$ lies in the subcategory. In a triangulated category the third object of any distinguished triangle with two objects in such a subcategory is isomorphic to a cone of a morphism between objects of the subcategory, so the subcategory is closed under the two-out-of-three operation; by the equivalent characterisation of [F5] it is a triangulated subcategory. If $A$ and $B$ with their cone are only asserted to be constructible with respect to one common finite smooth algebraic frontier stratification, the same argument applies with that stratification in place of $\mathfrak{S}$, giving the final sentence: their cone is bounded and constructible with respect to that common stratification. Only step 2.1 uses AC, exactly as declared in [A1].