Acyclic product fibers stabilize bounded constructible pullback Hom

The local comparison and the fiber model. Let $V\subseteq S$ be a product chart with trivialization $Y{\mathrm{\mid }}_V\cong V\times F$ and projection $\mathrm{pr}:V\times F\to V$. By [F1] there is a natural comparison $R{\mathrm{pr}}_{\ast }{\mathrm{pr}}^{\ast }{B}_V\simeq B_V{\otimes }_k{C}^{\ast }(F;k)$ for bounded $B_V\in D_c^b(V;k)$, compatible with restriction to smaller charts and carrying the adjunction unit to $B_V\otimes (k\to C^{\ast }(F;k))$. By [F2] the chart and the fiber carry finite deleted-boundary models, so $C^{\ast }(F;k)$ is a bounded complex of finite-dimensional vector spaces whose cohomology is $k$ in degree zero, in the sense that the structure map $k\to C^{\ast }(F;k)$ identifies $k$ with $H^0$, and is zero in degrees $1\le j\le r$. The comparison is compatible with the global adjunction $f^{\ast }⊣R{f}_{\ast }$ of [F3] over the chart: restriction to $V$ is inverse image along the open inclusion, inverse image commutes with $f^{\ast }$, $R{f}_{\ast }$ commutes with restriction to the open base for open inclusions by [F3] and [F8], and the unit is natural, so the restriction of the global unit $B\to R{f}_{\ast }{f}^{\ast }B$ to $V$ is the chart unit.

The cone bound. Let $\tilde{C}:=\mathrm{Cone}(k\to C^{\ast }(F;k))$, a bounded complex of finite-dimensional vector spaces with $H^0(\tilde{C})=0$ and $H^j(\tilde{C})=H^j(C^{\ast }(F;k))=0$ for $1\le j\le r$; hence $H^j(\tilde{C})=0$ for all $j\le r$. Now let $B\in D_c^{\ge b}(S;k)$, so $H^p(B)=0$ for $p<b$. By [F7] the cohomology of $B_V{\otimes }_k\tilde{C}$ is the direct sum of the terms $H^p(B_V){\otimes }_k{H}^q(\tilde{C})$ with $p+q=n$, and a nonzero such term needs $p\ge b$ and $q\ge r+1$, hence $n\ge b+r+1$; therefore $H^n(B_V{\otimes }_k\tilde{C})=0$ for $n\le b+r$. Since the unit of the chart comparison is $B_V\otimes (k\to C^{\ast }(F;k))$, its cone is $B_V{\otimes }_k\tilde{C}$ up to canonical isomorphism, so the cone of the chart unit lies in $D^{\ge b+r+1}$ on every chart. The charts cover $S$, the restriction of the global unit to each chart is the chart unit by step 1.1, and vanishing of the cohomology sheaves of a complex is a local condition [F6], so the cone of the global unit $B\to R{f}_{\ast }{f}^{\ast }B$ lies in $D^{\ge b+r+1}$, which proves claim 1.

The Hom comparison. Let $B\to R{f}_{\ast }{f}^{\ast }B\to C\to B[1]$ be the distinguished triangle of the unit, with $C\in D^{\ge b+r+1}$ by step 2.1, and let $A\in D_c^{\le a}(S;k)$ and $m\le b+r-a$. By [F5], $C[m]\in D^{\ge b+r+1-m}$ and $C[m-1]\in D^{\ge b+r+2-m}$; since $m\le b+r-a$ gives $b+r+1-m>a$ and $b+r+2-m>a$, orthogonality $\mathrm{Hom}(D^{\le a},D^{\ge a+1})=0$ gives $\mathrm{Hom}(A,C[m])=0=\mathrm{Hom}(A,C[m-1])$. The long exact Hom sequence of [F4] applied to the triangle, shifted by $[m]$, therefore shows that the canonical map $\mathrm{Hom}(A,B[m])\to \mathrm{Hom}(A,R{f}_{\ast }{f}^{\ast }B[m])$ is a bijection. Under the derived adjunction of [F3] this map corresponds to the composition with the unit, $u\mapsto {\eta }_{B[m]}\circ u$, whose adjoint is exactly pullback $u\mapsto f^{\ast }u$; since the adjunction bijection $\mathrm{Hom}(f^{\ast }A,f^{\ast }B[m])\cong \mathrm{Hom}(A,R{f}_{\ast }{f}^{\ast }B[m])$ carries $f^{\ast }u$ to ${\eta }_{B[m]}\circ u$ by naturality of the unit, the induced pullback map $\mathrm{Hom}(A,B[m])\to \mathrm{Hom}(f^{\ast }A,f^{\ast }B[m])$ is a bijection. This proves claim 2.

The image criterion. Assume $a-b\le r$ and put $\mathrm{\Phi }=f^{\ast }:D_c^{[b,a]}(S;k)\to D_c^{[b,a]}(Y;k)$ and $\mathrm{\Psi }={\tau }^{\le a}\circ R{f}_{\ast }$. For $B\in D_c^{[b,a]}(S;k)$ the cone of the unit $B\to R{f}_{\ast }{f}^{\ast }B$ lies in $D^{\ge b+r+1}\subseteq D^{\ge a+1}$ by step 2.1 and $b+r+1>a$, so the truncated unit $B\to {\tau }^{\le a}R{f}_{\ast }{f}^{\ast }B$ is an isomorphism; its inverse gives $\mathrm{\Psi }\mathrm{\Phi }\cong \mathrm{id}$. Now let $C\in D_c^{[b,a]}(Y;k)$. Since $C\in D^{\ge b}$ and $R{f}_{\ast }$ preserves lower bounds (a bounded-below complex can be resolved in degrees at least $b$ by [F4], and $f_{\ast }$ of a complex vanishing below $b$ vanishes below $b$), we get $R{f}_{\ast }C\in D^{\ge b}(S;k)$, and by the setting $R{f}_{\ast }C$ has bounded constructible cohomology; hence $\mathrm{\Psi }C={\tau }^{\le a}R{f}_{\ast }C\in D_c^{[b,a]}(S;k)$. Let ${\gamma }_C:\mathrm{\Phi }\mathrm{\Psi }C\to C$ be the composite of $\mathrm{\Phi }({\tau }^{\le a}R{f}_{\ast }C\to R{f}_{\ast }C)$ with the counit ${\epsilon }_C:\mathrm{\Phi }R{f}_{\ast }C\to C$ of the adjunction. If ${\gamma }_C$ is an isomorphism, then $C\cong \mathrm{\Phi }(\mathrm{\Psi }C)$ with $\mathrm{\Psi }C\in D_c^{[b,a]}(S;k)$, so $C$ lies in the image of $\mathrm{\Phi }$. Conversely, let $C=\mathrm{\Phi }B=f^{\ast }B$ with $B\in D_c^{[b,a]}(S;k)$. Then $\mathrm{\Psi }C={\tau }^{\le a}R{f}_{\ast }{f}^{\ast }B$, and the truncation ${\tau }^{\le a}R{f}_{\ast }{f}^{\ast }B\to R{f}_{\ast }{f}^{\ast }B$ factors, up to the canonical isomorphism ${\tau }^{\le a}R{f}_{\ast }{f}^{\ast }B\cong B$ supplied by the previous paragraph, as the unit $B\to R{f}_{\ast }{f}^{\ast }B$; the triangle identity ${\epsilon }_{f^{\ast }B}\circ f^{\ast }({\eta }_B)={\mathrm{id}}_{f^{\ast }B}$, together with exactness of $f^{\ast }$, therefore shows that ${\gamma }_{f^{\ast }B}$ is an isomorphism. Hence for $C\in D_c^{[b,a]}(Y;k)$ the counit ${\gamma }_C$ is an isomorphism exactly when $C$ is a pullback, which is the first assertion of claim 4.

Full faithfulness. If $a-b\le r$, then $0\le b+r-a$, so claim 2 with $m=0$ gives that $\mathrm{Hom}(A,B)\to \mathrm{Hom}(f^{\ast }A,f^{\ast }B)$ is a bijection for all objects $A,B$ of $D_c^{[b,a]}(S;k)$. Inverse image is exact and preserves the canonical t-structure and algebraic constructibility by [F3] and [F6], so $f^{\ast }$ maps $D_c^{[b,a]}(S;k)$ into $D_c^{[b,a]}(Y;k)$; a functor between categories that is bijective on all Hom sets is fully faithful, so claim 3 follows.

Retracts. Let $C\in D_c^{[b,a]}(Y;k)$ be a direct summand of $\mathrm{\Phi }B=f^{\ast }B$ for some $B\in D_c^{[b,a]}(S;k)$, with maps $i:C\to \mathrm{\Phi }B$ and $p:\mathrm{\Phi }B\to C$ satisfying $pi={\mathrm{id}}_C$. The counit $\gamma$ is a natural transformation $\mathrm{\Phi }\mathrm{\Psi }\to \mathrm{id}$; naturality in the form $u\circ {\gamma }_X={\gamma }_Y\circ \mathrm{\Phi }\mathrm{\Psi }(u)$ for $u:X\to Y$, applied to $u=i$ and $u=p$, gives $p\circ {\gamma }_{\mathrm{\Phi }B}\circ \mathrm{\Phi }\mathrm{\Psi }(i)=p\circ i\circ {\gamma }_C={\gamma }_C$ and ${\gamma }_C\circ \mathrm{\Phi }\mathrm{\Psi }(p)=p\circ {\gamma }_{\mathrm{\Phi }B}$. By step 3.2 the map ${\gamma }_{\mathrm{\Phi }B}$ is an isomorphism. Define $\delta :=\mathrm{\Phi }\mathrm{\Psi }(p)\circ {\gamma }_{\mathrm{\Phi }B}^{-1}\circ i:C\to \mathrm{\Phi }\mathrm{\Psi }C$. Then $\delta \circ {\gamma }_C=\mathrm{\Phi }\mathrm{\Psi }(p)\circ {\gamma }_{\mathrm{\Phi }B}^{-1}\circ {\gamma }_{\mathrm{\Phi }B}\circ \mathrm{\Phi }\mathrm{\Psi }(i)=\mathrm{\Phi }\mathrm{\Psi }(pi)={\mathrm{id}}_{\mathrm{\Phi }\mathrm{\Psi }C}$ and ${\gamma }_C\circ \delta =p\circ {\gamma }_{\mathrm{\Phi }B}\circ {\gamma }_{\mathrm{\Phi }B}^{-1}\circ i=pi={\mathrm{id}}_C$, using the two displayed identities. Hence ${\gamma }_C$ is an isomorphism and, by step 3.2, $C$ lies in the image of $\mathrm{\Phi }$, which proves closure under direct summands.

Extensions under the strict bound. Assume $r\ge a-b+1$ and let $C^{\mathrm{\prime }}\to C\to C^{\mathrm{\prime }\mathrm{\prime }}\to C^{\mathrm{\prime }}[1]$ be a distinguished triangle in $D_c^{[b,a]}(Y;k)$ whose outer objects lie in the image: $C^{\mathrm{\prime }}=f^{\ast }{B}^{\mathrm{\prime }}$ and $C^{\mathrm{\prime }\mathrm{\prime }}=f^{\ast }{B}^{\mathrm{\prime }\mathrm{\prime }}$ with $B^{\mathrm{\prime }},B^{\mathrm{\prime }\mathrm{\prime }}\in D_c^{[b,a]}(S;k)$. Let $\phi :C^{\mathrm{\prime }\mathrm{\prime }}\to C^{\mathrm{\prime }}[1]$ be its connecting morphism. Because $1\le b+r-a$ by the strict bound, claim 2 applies with $A=B^{\mathrm{\prime }\mathrm{\prime }}\in D_c^{\le a}(S;k)$, $B=B^{\mathrm{\prime }}\in D_c^{\ge b}(S;k)$ and $m=1$; the isomorphism $\mathrm{Hom}(B^{\mathrm{\prime }\mathrm{\prime }},B^{\mathrm{\prime }}[1])\cong \mathrm{Hom}(f^{\ast }{B}^{\mathrm{\prime }\mathrm{\prime }},f^{\ast }{B}^{\mathrm{\prime }}[1])$ being pullback along $f^{\ast }$, there is a unique $\bar{\phi }:B^{\mathrm{\prime }\mathrm{\prime }}\to B^{\mathrm{\prime }}[1]$ with $f^{\ast }\bar{\phi }=\phi$. By TR1 the morphism $\bar{\phi }$ is completed to a distinguished triangle $B^{\mathrm{\prime }}\to \bar{B}\to B^{\mathrm{\prime }\mathrm{\prime }}\to B^{\mathrm{\prime }}[1]$ in $D^b(S;k)$ ([F5]). The long exact cohomology sequence of this triangle and $B^{\mathrm{\prime }},B^{\mathrm{\prime }\mathrm{\prime }}\in D^{[b,a]}$ show $H^j(\bar{B})=0$ for $j<b$ and for $j>a$: the exact segment $H^j(B^{\mathrm{\prime }})\to H^j(\bar{B})\to H^j(B^{\mathrm{\prime }\mathrm{\prime }})$ has zero outer terms in both ranges. By [F6] the cone $\bar{B}$ of a morphism of bounded algebraically constructible complexes is bounded and algebraically constructible, so $\bar{B}\in D_c^{[b,a]}(S;k)$. Exactness of $f^{\ast }$ [F3] turns the triangle on $\bar{\phi }$ into a distinguished triangle with connecting morphism $f^{\ast }\bar{\phi }=\phi$; two distinguished triangles with the same connecting morphism are isomorphic on their third objects, because TR2 and TR3 applied to the identity on the source and target of the connecting morphism give a morphism between the two triangles whose first two components are identities and hence, by the five-lemma applied to the long exact Hom sequences of [F4], an isomorphism on the third objects. Thus $C\cong f^{\ast }\bar{B}$ lies in the image of $\mathrm{\Phi }$. This proves the extension-closure clause of claim 4; at the boundary $r=a-b$ the degree-one comparison $1\le b+r-a$ fails and the argument is deliberately not used.

Conclusion and the local nature of the comparison. Step 2.1 proves the cone claim 1; step 3.1 proves the Hom comparison claim 2, step 4.1 full faithfulness claim 3, and steps 3.2, 4.2 and 4.3 the image criterion and the closure properties of claim 4. The comparison used in step 2.1 is the local product comparison on product charts, not a global splitting: no isomorphism $R{f}_{\ast }{f}^{\ast }B\simeq B\otimes R\mathrm{\Gamma }(F;k)$ is asserted, and the argument uses only the cone of the unit and the degreewise vanishing of the fiber model, so nonzero characteristic-class transgressions of the bundle are preserved. The only choice-theoretic inputs are those recorded in [A1].