Support localization, base change, projection and trace calculus

Closed-support localization. Let $i:Z\hookrightarrow X$ be a closed inclusion with open complement $j$, and put ${\mathrm{\Gamma }}_Z\mathcal{F}=\ker (\mathcal{F}\to j_{\ast }{j}^{\ast }\mathcal{F})$. Its stalks vanish off $Z$ by [F7], so it is $i_{\ast }{i}^{-1}{\mathrm{\Gamma }}_Z\mathcal{F}$ by [F5], and every map $i_{\ast }A\to \mathcal{F}$ lands in this kernel because $j^{\ast }{i}_{\ast }=0$; hence $i_0^{!}\mathcal{F}=i^{-1}{\mathrm{\Gamma }}_Z\mathcal{F}$ is right adjoint to the exact functor $i_{\ast }$. It preserves injectives, so its derived functor is the right adjoint of $i_{!}$ on the derived level where the adjoint exists. The stalk-exact sequence of [F5] gives the first localization triangle $j_{!}{j}^{\ast }A\to A\to i_{\ast }{i}^{\ast }A\to j_{!}{j}^{\ast }A[1]$. For a flasque injective $I$ the map $I\to j_{\ast }{j}^{\ast }I$ is surjective on sections over every open set by [F8] and [F5] with kernel ${\mathrm{\Gamma }}_{Z}I$, and $j^{\ast }$ preserves injectives because $j_{!}$ is exact; hence $i_{\ast }{i}^{!}A\to A\to R{j}_{\ast }{j}^{\ast }A\to i_{\ast }{i}^{!}A[1]$ is the second localization triangle. Finally the tensor identity $H{\otimes }_k{i}_{\ast }{k}_Z=i_{\ast }{i}^{\ast }H$ holds by the stalk formula, and tensor–Hom transposition [F2] together with the closed adjunction gives $\mathrm{Hom}(H,R\mathcal{H}om(i_{\ast }{k}_Z,A))\cong \mathrm{Hom}(i_{\ast }{i}^{\ast }H,A)\cong \mathrm{Hom}(H,i_{\ast }{i}^{!}A)$ locally on each open; hence $R\mathcal{H}om(i_{\ast }{k}_Z,A)\cong R{\mathrm{\Gamma }}_{Z}A$, the actual closed-support object used in the diagonal Thom calculation.

Cartesian base change. In the Cartesian square, inverse image of an injective $I$ on $X$ restricts on every fibre of $f^{\mathrm{\prime }}$ to the restriction of $I$ on the canonically homeomorphic original fibre, so it is fibrewise $c$-soft by [F3] and $f^{\mathrm{\prime }}$-acyclic by [F1]; the section restriction map $u^{\ast }{f}_{!}I\to f_{!}^{\mathrm{\prime }}{v}^{\ast }I$ is an isomorphism on stalks by the fibre-stalk computation of [F1]. Applying this to an injective complex proves $u^{\ast }R{f}_{!}\simeq R{f}_{!}^{\mathrm{\prime }}{v}^{\ast }$. If $f$ is proper then $f^{\mathrm{\prime }}$ is proper by [F4], and using $f_{!}=f_{\ast }$ and $f_{!}^{\mathrm{\prime }}=f_{\ast }^{\mathrm{\prime }}$ this is ordinary proper base change $u^{\ast }R{f}_{\ast }\simeq R{f}_{\ast }^{\mathrm{\prime }}{v}^{\ast }$. These are the canonical restriction/exchange maps, not arbitrarily chosen isomorphisms.

Independent verification for proper maps. For proper $f$ the same identification can be checked through ordinary compactly supported cohomology: restriction of injectives to compact fibres is $c$-soft and $\mathrm{\Gamma }$-acyclic by [F3], compact-neighbourhood sections are delivered by [F6], and preimage neighbourhoods are cofinal because $f$ is closed by [F4], with $V=Y\setminus f(X\setminus U)$ for an open $U$; this recovers the exchange as the canonical derived adjunction map of ordinary proper base change.

The projection formula. For bounded $A$ choose a bounded-below injective replacement $A\to I$. On each fibre $X_y$ the restriction $I{\mathrm{\mid }}_{X_y}$ is $c$-soft by [F3], and $f^{\ast }B$ restricts to the constant complex $B_y$; tensoring with each term $B_y$ is a direct sum of copies of $I{\mathrm{\mid }}_{X_y}$, hence $c$-soft by [F3], and ${\mathrm{\Gamma }}_c(E{\otimes }_{k}V)={\mathrm{\Gamma }}_c(E){\otimes }_{k}V$ because a compactly supported section has a finite local tensor presentation and a finite compact cover uses only finitely many coefficient vectors. This proves the canonical map $R{f}_{!}A{\otimes }_{k}B\to R{f}_{!}(A{\otimes }_k{f}^{\ast }B)$ is an isomorphism on stalks, since both sides compute the fibrewise compact-support cohomology of [F1]; the map is the adjoint of the tensor of the $f^{\ast }⊣R{f}_{\ast }$ counit with the identity, equivalently the sectionwise map $s\otimes b\mapsto s\otimes f^{\ast }b$, and the signed totalization of [F2] supplies the bounded-complex version.

Composition of proper-support direct images. The support definition gives $(gf{)}_{!}=g_{!}{f}_{!}$: a section whose support is proper over $Z$ has support proper over $Y$ because the inverse image of a compact $L\subseteq Y$ is a closed subset of the inverse image of the compact $g(L)$ by [F4], and conversely a support locally proper over $Y$ with proper image over $Z$ has compact preimages of compacta in $Z$. The support of $f_{!}s$ is $f(\mathrm{Supp}s)$ by the fibre-stalk identity [F1] and local properness, so the two descriptions are inverse. Direct images of globally $c$-soft sheaves by $f_{!}$ are $c$-soft: a section over a compact base set is represented on a neighbourhood by [F6], its total support over that compact set is compact by properness of the representing support, and enclosing it in a relatively compact $W$, the support-controlled $c$-soft extension of [F3] supported in $W$ pushes forward and extends the original section over the compact set. Thus $f_{!}$ sends injectives to $g_{!}$-acyclic sheaves, and applying the two ordinary support functors to an injective resolution gives the canonical derived composition $R(gf{)}_{!}=R{g}_{!}R{f}_{!}$; the same comparison proves associativity for triples, the section support bijection being literally associative.

Exceptional composition and traces on the bounded constructible specialization. If the represented right adjoints preserve the bounded constructible subcategory, so that the Hom formula of [F1] is a full adjunction there, then exceptional composition $(gf{)}^{!}\simeq f^{!}{g}^{!}$ is the unique adjoint transpose of the composition isomorphism of step 1.5. The traces ${\mathrm{t}\mathrm{r}}_f:R{f}_{!}{f}^{!}\to \mathrm{i}\mathrm{d}$ are the counits constructed in [F1], and under the composition identification ${\mathrm{t}\mathrm{r}}_{gf}={\mathrm{t}\mathrm{r}}_g\circ R{g}_{!}({\mathrm{t}\mathrm{r}}_f)g^{!}$: both sides have the same transpose, namely the identity of $f^{!}{g}^{!}$, as is checked with the two triangle identities. Thus traces compose without a new axiom, and identity units and counits are identity transpositions.

Pasting coherence, and the boundary of the claim. The two composites of base-change maps in a pasted rectangle restrict a representative section to the same fibre section, hence agree before deriving, and resolution comparisons preserve that equality; projection formulas for composites both send pure tensors $s\otimes b$ to the same pullback tensor; projection/trace compatibility follows by transposition, both composites transposing to evaluation on $s\otimes b$ followed by the same counit. This proves the projection, base-change, unit and trace squares used in pull-push convolution, and the identities hold in the dg resolution model rather than being chosen independently on the triangulated level. The extraordinary proper exchange $u^{!}R{f}_{\ast }\simeq R{f}_{\ast }^{\mathrm{\prime }}{v}^{!}$ is not claimed here: it is not a right-adjoint mate of the ordinary base-change isomorphism and its proof needs the biduality result of the subsequent local duality construction, which therefore is not assumed.