Bounded fibrewise-soft resolutions from a uniform all-sheaf dimension bound

Fibres are locally compact Hausdorff. For $y\in Y$ the fibre $F=f^{-1}(y)$ is closed in $X$ because $\{y\}$ is closed in the Hausdorff space $Y$ and $f$ is continuous [F1], so $F$ is a closed subspace of the locally compact space $X$ and is locally compact, and it is Hausdorff as a subspace of a Hausdorff space [F1]; the same holds for every open subspace $V\subseteq F$ [F1]. Consequently, for every sheaf $\mathcal{G}$ on $X$ that is c-soft on $X$, the restriction $\mathcal{G}{\mathrm{\mid }}_F$ is c-soft on $F$ and its further restriction $\mathcal{G}{\mathrm{\mid }}_V$ is c-soft on $V$, by the closed- and open-restriction clauses of [F2]; if $\mathcal{G}$ is injective on $X$ it is c-soft on $X$ by the injectivity clause of [F2].

The resolution and its terms. Fix the supplied functorial injective resolution datum on ${\mathrm{Sh}}_k(X)$ of [F4] (on $\mathrm{A}\mathrm{b}(X)$ for the abelian case) and let $0\to \mathcal{A}\to I^0\to I^1\to \cdots$ be the injective resolution it assigns to $\mathcal{A}$. Each $I^j$ is injective in the chosen sheaf category, hence flasque and c-soft on $X$ by [F2], hence restricts to a c-soft sheaf $I^j{\mathrm{\mid }}_F$ on every fibre $F$ and to a c-soft sheaf $I^j{\mathrm{\mid }}_V$ on every open $V$ of a fibre, by [step 1.1]; in particular each $I^j$ is fibrewise c-soft.

Dimension shifting on an open subspace of a fibre. Let $V\subseteq F$ be open in a fibre $F=f^{-1}(y)$ and define the syzygies $Z^0:=\mathcal{A}$ and, for $j\ge 0$, $Z^{j+1}:=\mathrm{coker}(Z^j\to I^j)$, so that $0\to Z^j\to I^j\to Z^{j+1}\to 0$ is a short exact sequence of sheaves on $X$. The inclusion $V\hookrightarrow X$ need not be open, but inverse-image restriction preserves exactness by the stalk argument of [F5], so $0\to Z^j{\mathrm{\mid }}_V\to I^j{\mathrm{\mid }}_V\to Z^{j+1}{\mathrm{\mid }}_V\to 0$ is a short exact sequence of sheaves on the locally compact Hausdorff space $V$ [F5] whose middle term is c-soft, hence acyclic [F2]; the long exact compactly supported cohomology sequence of [F5] therefore gives isomorphisms $H_c^p(V,Z^{j+1}{\mathrm{\mid }}_V)\stackrel{\sim }{\to }{H}_c^{p+1}(V,Z^j{\mathrm{\mid }}_V)$ for every $p\ge 1$, because the two adjacent groups of the c-soft sheaf $I^j{\mathrm{\mid }}_V$ vanish in positive degrees. Iterating these isomorphisms for $j=0,1,\dots ,d-1$ with $p=1,2,\dots ,d$ gives $H_c^1(V,Z^d{\mathrm{\mid }}_V)\cong H_c^{d+1}(V,\mathcal{A}{\mathrm{\mid }}_V)=0$, the last group vanishing by the dimension hypothesis of the statement because $d+1>d$ and $V$ is an open subspace of a fibre.

The cokernel after $d$ steps is fibrewise c-soft. Let $F=f^{-1}(y)$ be any fibre and let $V\subseteq F$ be open; by [step 3.1] one has $H_c^1(V,Z^d{\mathrm{\mid }}_V)=0$ for every such $V$, so the vanishing criterion [F3] applied to the locally compact Hausdorff space $Z=F$ and the sheaf $\mathcal{E}=Z^d{\mathrm{\mid }}_F$ shows that $Z^d{\mathrm{\mid }}_F$ is c-soft on $F$. Since $y$ was arbitrary, $Z^d$ restricts to a c-soft sheaf on every fibre. Concatenating the short exact sequences $0\to Z^j\to I^j\to Z^{j+1}\to 0$ for $j=0,\dots ,d-1$ gives the exact sequence $0\to \mathcal{A}\to I^0\to I^1\to \cdots \to I^{d-1}\to Z^d\to 0$, a bounded resolution of $\mathcal{A}$ entirely by $k$-linear maps and $k$-sheaves (by abelian maps and sheaves in the abelian case), whose terms $I^0,\dots ,I^{d-1}$ are fibrewise c-soft by [step 2.1] and whose last term $Z^d$ is fibrewise c-soft by the preceding argument; when $d=0$ take ${\mathcal{K}}^0=Z^0=\mathcal{A}$ and the identity sequence $0\to \mathcal{A}\stackrel{\mathrm{i}\mathrm{d}}{\to }\mathcal{A}\to 0$; the isomorphism $H_c^1(V,\mathcal{A}{\mathrm{\mid }}_V)\cong H_c^{d+1}(V,\mathcal{A}{\mathrm{\mid }}_V)=0$ of [step 3.1] with $d=0$ gives the c-softness of $\mathcal{A}$ on every fibre directly by [F3].

Conclusion. The displayed bounded resolution of the statement is [step 4.1], and taking $\mathcal{A}=k_X$ gives the instance of the constant sheaf. The Axiom of Choice is used only through the injective resolution datum of [F4] and the delta-functor and comparison statements of [F5], whose connecting maps use dependent choice, a consequence of AC [F6]; the fibre-by-fibre dimension count of [step 3.1] is finite.