Analytic Derived Six Operations And Duality

Closed inclusions are proper. Let $i:Z\hookrightarrow X$ be the inclusion of a closed subset and let $K\subseteq X$ be compact. Its preimage $K\cap Z$ is closed in the compact space $K$, hence compact by [F3]. The topology on $K\cap Z$ induced from $Z$ is the same as that induced from $K$ or $X$, by transitivity of subspace topology [F6]. Thus $i^{-1}(K)$ is compact in $Z$, and $i$ is proper by [F1].

Composites of proper maps are proper. For $K\subseteq Z$ compact, $(gf{)}^{-1}(K)=f^{-1}(g^{-1}(K))$; the set $g^{-1}(K)$ is compact by [F1] applied to $g$, and then its $f$-preimage is compact by [F1] applied to $f$. Hence $gf$ is proper.

Fibres of a proper map are compact. A one-point space is compact by [F4], so for each $y\in Y$ the preimage $f^{-1}(\{y\})$ is compact by [F1].

Restrictions to closed subspaces are proper. Let $S\subseteq X$ be closed, and give $S$ its subspace topology. For a compact $K\subseteq Y$ one has $(f{\mathrm{\mid }}_S{)}^{-1}(K)=f^{-1}(K)\cap S$. The set $f^{-1}(K)$ is compact by [F1], and $f^{-1}(K)\cap S$ is closed in the compact space $f^{-1}(K)$ because $S$ is closed in $X$; hence $f^{-1}(K)\cap S$ is a compact subset of $f^{-1}(K)$ by [F3]. Its topology as a subset of $S$ agrees with its topology as a subset of $X$ by [F6], so $(f{\mathrm{\mid }}_S{)}^{-1}(K)$ is a compact subset of $S$ and $f{\mathrm{\mid }}_S$ is proper by [F1].

Properness is preserved by base change along a map into a Hausdorff space. Let $K\subseteq Y^{\mathrm{\prime }}$ be compact and put $K_0=u[K]$, a compact subset of $Y$ by [F2]. Then $f^{-1}(K_0)$ is compact by [F1], so the product $f^{-1}(K_0)\times K$ is a compact subspace of $X\times Y^{\mathrm{\prime }}$ by [F4]. The pairing $(f,u):X\times Y^{\mathrm{\prime }}\to Y\times Y$ is continuous by [F7], and ${\mathrm{\Delta }}_Y$ is closed in $Y\times Y$ by [F5]; hence $X{\times }_Y{Y}^{\mathrm{\prime }}=(f,u{)}^{-1}({\mathrm{\Delta }}_Y)$ is closed in $X\times Y^{\mathrm{\prime }}$, and its intersection with the subspace $f^{-1}(K_0)\times K$ is closed in that compact space. Moreover this intersection is exactly $(f^{\mathrm{\prime }}{)}^{-1}(K)$: a pair $(x,y^{\mathrm{\prime }})$ lies in $X{\times }_Y{Y}^{\mathrm{\prime }}$ with $y^{\mathrm{\prime }}\in K$ precisely when $f(x)=u(y^{\mathrm{\prime }})$ and $y^{\mathrm{\prime }}\in K$, and then $u(y^{\mathrm{\prime }})\in K_0$ forces $x\in f^{-1}(K_0)$, so $(f^{\mathrm{\prime }}{)}^{-1}(K)=X{\times }_Y{Y}^{\mathrm{\prime }}\cap (f^{-1}(K_0)\times K)$. By [F3] this is a compact subset of the product $X\times Y^{\mathrm{\prime }}$, and by [F6] its topology as a subset of $X{\times }_Y{Y}^{\mathrm{\prime }}$ agrees with that as a subset of $X\times Y^{\mathrm{\prime }}$; hence $f^{\mathrm{\prime }}$ is proper by [F1].

Proper maps into locally compact Hausdorff spaces are closed. Let $f:X\to Y$ be proper with $Y$ locally compact Hausdorff, let $F\subseteq X$ be closed and let $y\in Y\setminus f[F]$. By [F8] there is an open neighbourhood $V$ of $y$ whose closure $\bar{V}$ is compact. Then $f^{-1}(\bar{V})$ is compact by [F1] and $F\cap f^{-1}(\bar{V})$ is closed in it, hence compact by [F3]; its image under the continuous map $f$ is $f[F]\cap \bar{V}$, which is compact by [F2] and closed in $Y$ by [F8]. Therefore $V\setminus (f[F]\cap \bar{V})$ is open in $Y$, contains $y$ because $y\in V$ and $y\mathrm{\notin }f[F]$, and is disjoint from $f[F]$: a point of it lies in $V$, hence cannot lie in $f[F]\setminus \bar{V}$, and it is disjoint from $f[F]\cap \bar{V}$ by construction. Hence $f[F]$ is closed.

The six stability properties are established: claim 1 by step 1.1, claim 2 by step 1.2, claim 4 by step 1.3, claim 5 by step 1.4, claim 3 by step 1.5 and claim 6 by step 1.6. Every argument uses only the preimage form of properness [F1], the preservation of compactness under continuous maps [F2], closed-in-compact compactness [F3], finite products of compact spaces [F4], the closed-diagonal criterion [F5], the intrinsic subspace conventions [F6], the initial product topology [F7] and the locally compact Hausdorff basis of compact neighbourhoods [F8]; none of these selects points or open sets from an infinite family, so no choice principle is used.

The proper-support functor is a sheaf, additive and left exact; for proper $f$ it equals $f_{\ast }$, for an open inclusion it is extension by zero, and for a closed inclusion it is direct image. For a sheaf $A$ and open $V\subseteq Y$ let $f_{!}A(V)$ be the set of $s\in A(f^{-1}V)$ whose support maps properly to $V$, with restrictions inherited from $A$; these are well defined because for open $W\subseteq V$ the support of $s\mathrm{\mid }$ is $\mathrm{Supp}(s)\cap f^{-1}W$ by [F9], and the preimage of a compact subset of $W$ under $\mathrm{Supp}(s)\cap f^{-1}W\to W$ is the corresponding preimage under the proper map $\mathrm{Supp}(s)\to V$. Additivity is evident from linearity of restriction and gluing. For the sheaf property let $V=\bigcup V_{\alpha }$ and let $s_{\alpha }$ be compatible sections; they glue to $s\in A(f^{-1}V)$ with $\mathrm{Supp}(s)\subseteq {\bigcup }_{\alpha }\mathrm{Supp}(s_{\alpha })$ and $s\mathrm{\mid }=s_{\alpha }$ over $f^{-1}{V}_{\alpha }$. If $L\subseteq V$ is compact, cover $L$ by finitely many open $W_1,\dots ,W_n$ with compact closures $\overline{W_j}$ contained in some $V_{\alpha (j)}$, possible by [F12]; then $\mathrm{Supp}(s)\cap f^{-1}(L)\subseteq {\bigcup }_j(\mathrm{Supp}(s_{\alpha (j)})\cap f^{-1}(\overline{W_j}))$, a finite union of compact sets because $\overline{W_j}\subseteq V_{\alpha (j)}$ is compact and $\mathrm{Supp}(s_{\alpha (j)})\to V_{\alpha (j)}$ is proper; and $\mathrm{Supp}(s)\cap f^{-1}(L)$ is closed in $f^{-1}(L)$ by [F9], so it is a finite union of closed subsets of these compacta, hence compact by [F12]. Thus $f_{!}A$ is a sheaf. Kernels are computed sectionwise, so $(f_{!}\ker \phi )(V)=\ker (f_{!}\phi )(V)$ and $f_{!}$ is left exact. If $f$ is proper, every support map is proper by [F8], so $f_{!}=f_{\ast }$. If $f=j:U\hookrightarrow Y$ is an open inclusion, properness of $\mathrm{Supp}(s)\to V$ is equivalent to closedness of $\mathrm{Supp}(s)$ in $V$ by [F8], which is exactly the extension-by-zero condition of [F9]; if $f=i:Z\hookrightarrow Y$ is a closed inclusion then $\mathrm{Supp}(s)$ is closed in $V\cap Z$, hence in $V$, and $i_{!}=i_{\ast }$.

The fibre comparison is injective. Restriction defines $(f_{!}A{)}_y\to {\mathrm{\Gamma }}_c(X_y,A{\mathrm{\mid }}_{X_y})$: a representative $s\in f_{!}A(V)$ has support $S$ proper over $V$, so $S\cap X_y$ is compact. If its restriction to $X_y$ is zero, every germ $s_x$ for $x\in X_y$ is zero, and therefore $S\cap X_y=\mathrm{\varnothing }$. The proper map $S\to V$ is closed by [F8], so $f(S)$ is closed in $V$ and avoids $y$. On the open neighbourhood $V\setminus f(S)$ the section vanishes identically. Its germ in $(f_{!}A{)}_y$ is consequently zero.

The comparison map is surjective, hence an isomorphism. Let $t\in {\mathrm{\Gamma }}_c(X_y,A{\mathrm{\mid }}_{X_y})$ have compact support $Q$. By [F3], its restriction near $Q$ is represented by an ambient section $s_0$ on a neighbourhood $W$ of $Q$. Shrink $W$ so that $s_0{\mathrm{\mid }}_{W\cap X_y}=t{\mathrm{\mid }}_{W\cap X_y}$: equality holds near every point of $Q$ and its equality locus is open in the fibre, so an ambient neighbourhood of the compact $Q$ can be chosen inside it. By [F12] choose an open $W_0$ containing $Q$ with compact closure contained in $W$, and put $C=\overline{W_0}\setminus W_0$. Since $C\cap X_y$ is disjoint from $Q$, the germs of $s_0$ there are zero. Its zero locus is open, so choose an open $N\subseteq W$ containing $C\cap X_y$ on which $s_0=0$. The compact set $C\setminus N$ is disjoint from $X_y$; its continuous image is compact and hence closed in the Hausdorff $Y$, and does not contain $y$. Let $Y_0=Y\setminus f(C\setminus N)$, an open neighbourhood of $y$. Then $C\cap f^{-1}{Y}_0\subseteq N$. On $f^{-1}{Y}_0$ glue $s_0$ on $W_0\cap f^{-1}{Y}_0$ with zero outside $\overline{W_0}$, using the open vanishing neighbourhood $N\cap f^{-1}{Y}_0$ of the collar to cover and match across it. The resulting section $s$ restricts to $t$ on the fibre and has support contained in $\overline{W_0}\cap f^{-1}{Y}_0$. For every compact $L\subseteq Y_0$, $\mathrm{Supp}(s)\cap f^{-1}L$ is closed in the compact $\overline{W_0}$, so it is compact. Thus $s\in f_{!}A(Y_0)$ represents $t$. No extension of a section from $Q\cup (C\cap X_y)$ to all of $\overline{W_0}$ has been assumed. The comparisons are natural under restriction.

Fibrewise acyclicity and the fibre formula. If $A$ is injective then $A$ is $c$-soft by [F5], and restriction of a $c$-soft sheaf to a closed subspace is $c$-soft by [F4]; a fibre $X_y=f^{-1}(\{y\})$ is closed because $\{y\}$ is closed in $Y$, so $A{\mathrm{\mid }}_{X_y}$ is $c$-soft and hence ${\mathrm{\Gamma }}_c$-acyclic in positive degrees by [F5]. An injective resolution of $A$ on $X$ restricts to a ${\mathrm{\Gamma }}_c$-acyclic resolution of $A{\mathrm{\mid }}_{X_y}$, so $(R{f}_{!}A{)}_y\simeq R{\mathrm{\Gamma }}_c(X_y,A{\mathrm{\mid }}_{X_y})$ by the fibre formula of step 1.3 and exactness of stalks, which evaluates the derived functor on the fibre. The uniform bound $d$ of [F7] makes every fibre $X_y$ have $H_c^q(X_y,-)=0$ for $q>d$, so $R{f}_{!}$ has cohomological dimension at most $d$ fibrewise and maps $D^b(X)$ to $D^b(Y)$: a bounded complex has bounded fibrewise cohomology, and the fibre formula exhibits $R{f}_{!}$ of it as fibrewise compact-support cohomology concentrated in bounded degrees. A fibrewise $c$-soft sheaf is $f_{!}$-acyclic by the same stalk computation. The same formula shows that $f_{!}$ commutes with small direct sums: a compactly supported section of a sheaf coproduct involves finitely many summands locally, and a finite cover of its compact support gives one finite set of summands; its coordinate sections are compactly supported. Thus ${\mathrm{\Gamma }}_c(X_y,⨁A_i{\mathrm{\mid }}_{X_y})=⨁{\mathrm{\Gamma }}_c(X_y,A_i{\mathrm{\mid }}_{X_y})$, and the canonical direct-sum comparison is an isomorphism on stalks.

The complex $H_K(I)$ is a termwise flasque sheaf complex with lower bound $\inf (I)-d$. Let $I$ be a bounded-below injective complex on $Y$ and $K$ the finite fibrewise-soft resolution of [F7] in degrees $0,\dots ,d$, with $K_U=j_{U!}{j}_U^{\ast }K$ for open $U\subseteq X$. For an open cover $U=\bigcup U_i$ of an open $U$, the augmented extension-by-zero Čech complex of $K_U$ is exact on stalks: at a point $x\in U$, the simplex of indices $i$ with $x\in U_i$ has the elementary contracting homotopy that adds a fixed such index, and on $U$ the complex resolves $k_U$ columnwise. Each term is a direct sum of sheaves $j_{W!}(K{\mathrm{\mid }}_W)$, which are fibrewise $c$-soft by [F4] because $K$ is, and hence $f_{!}$-acyclic by step 2.1. The cycles of the augmented complex are again $f_{!}$-acyclic: dimension shifting along its short exact cycle sequences identifies $R^q{f}_{!}$ of a cycle with $R^{q+r}{f}_{!}$ of an earlier cycle for every $r$, and choosing $r>d$ makes the latter vanish by step 2.1; consequently $f_{!}$ carries the resolution to an exact complex. Since each $I^q$ is injective, ${\mathrm{Hom}}_Y(-,I^q)$ preserves exactness, and the first three terms give the sheaf equalizer for the presheaf $U\mapsto {\mathrm{Hom}}_Y^{\bullet }(f_{!}{K}_U,I)$, so $H_K(I)$ is a sheaf complex. For $V\subseteq U$ the map $K_V\to K_U$ is monic and $f_{!}$ is left exact by step 1.1, so $f_{!}{K}_V\to f_{!}{K}_U$ is monic; injectivity of $I^q$ then makes the restriction maps of $H_K(I)$ surjective in every total degree, because ${\mathrm{Hom}}_Y(-,I^q)$ sends a monomorphism to a surjection and finite products over the finite $K$-degree preserve surjectivity. Thus $H_K(I)$ is termwise flasque. Its degree-$n$ term is a finite product of terms ${\mathrm{Hom}}_Y(f_{!}{K}_U,I^{p+n})$ with $p\in [0,d]$, so it vanishes for $n<\inf (I)-d$.

The generator comparison is a derived comparison. Put $E_U=j_{U!}{k}_U$. Exact extension by zero [F9] and the resolution $k_X\to K$ give a quasi-isomorphism $E_U\to K_U$, whose terms are $f_{!}$-acyclic by step 2.1; hence $f_{!}{K}_U$ represents $R{f}_{!}{E}_U$. By construction, ${\mathrm{Hom}}_X^{\bullet }(E_U,H_K(I))=\mathrm{\Gamma }(U,H_K(I))={\mathrm{Hom}}_Y^{\bullet }(f_{!}{K}_U,I)$. The bounded-below flasque complex $H_K(I)$ computes derived sections by [F2], so this is the natural identification $R{\mathrm{Hom}}_X(E_U,H_K(I))\simeq R{\mathrm{Hom}}_Y(R{f}_{!}{E}_U,I)$. Its maps are evaluation and restriction; $H^0$ gives the corresponding derived morphism groups.

A field-linear generator model. For a $k$-sheaf $E$ put $G(E)={⨁}_{(U,s),s\in E(U)}{E}_U$, mapping the generator $1$ to $s$; this is stalk-surjective. Starting above the upper degree of bounded $A$, construct $M$ downwards: once $M^{n+1}$ and its map $\alpha$ to $A^{n+1}$ exist, take $M^n=G(Z^n)$ for $Z^n=\{(a,m)\in A^n\oplus M^{n+1}:d_{A}a=\alpha m,d_{M}m=0\}$, and use the two projections for ${\alpha }^n$ and $d_M^n$. These equations give $d_M^2=0$ and $d_A\alpha =\alpha d_M$. Every stalk cycle $(a,m)$ in $\mathrm{Cone}(\alpha )$ is the boundary of $(0,l)$ for a lift $l$ of $(a,-m)\in Z^n$, so $M\to A$ is a quasi-isomorphism. This supplies the $k$-linear construction rather than interpreting the abelian resolution [F10] as a field-linear resolution. The zigzag $A\leftarrow M\to K{\otimes }_{k}M$ consists of quasi-isomorphisms: the bounded acyclic cone of $k_X\to K$ is split-contractible at each vector-space stalk, and remains contractible after tensoring with $M$. The terms of $K\otimes M$ are direct sums of $K_U$, hence fibrewise c-soft and $f_{!}$-acyclic. Also $f_{!}$ commutes with these direct sums: its fibre-stalk formula reduces this to compactly supported sections, which use only finitely many summands on a finite cover of their compact support. Thus evaluation on summands gives ${\mathrm{Hom}}_X^{\bullet }(M,H_K(I))={\mathrm{Hom}}_Y^{\bullet }(f_{!}(K\otimes M),I)$.

The proper-support counit. For open $V\subseteq Y$ and closed $Q\subseteq f^{-1}V$ proper over $V$, write $K_Q=i_{Q\ast }{i}_Q^{\ast }K$. The sequence $0\to K_{f^{-1}V\setminus Q}\to K_{f^{-1}V}\to K_Q\to 0$ is stalk-exact. Its kernel terms are fibrewise c-soft, so applying $f_{!}$ is exact by step 2.1; applying ${\mathrm{Hom}}_Y(-,I^q)$ is exact by injectivity. Taking the kernel of restriction to the complement of $Q$ therefore identifies the complex of $Q$-supported sections of $H_K(I)$ with ${\mathrm{Hom}}_V^{\bullet }(f_{!}{K}_Q,I{\mathrm{\mid }}_V)$. Since $Q\to V$ is proper, $f_{!}{K}_Q=f_{\ast }{K}_Q$. Precomposition with $k_V\to f_{\ast }{k}_Q\to f_{\ast }{K}_Q$, the constant-section unit followed by coaugmentation, gives a map to $I(V)$. These maps agree when $Q$ is enlarged and when $V$ is restricted, so they assemble to $f_{!}{H}_K(I)\to I$. Its terms are $f_{!}$-acyclic: they are flasque by step 3.1, hence globally c-soft and c-soft on the closed fibres by [F4], so step 2.1 applies. This is the derived counit; on each generator its transposition is the evaluation of step 4.1.

The generator model computes the required derived comparison. If an acyclic complex has $f_{!}$-acyclic terms, its cycle sequences give $R^q{f}_{!}{Z}^{n+1}\cong R^{q+1}{f}_{!}{Z}^n$ for $q\ge 1$. Iterating backwards more than $d$ times kills these groups by step 2.1. All cycles are therefore $f_{!}$-acyclic, the cycle sequences remain exact under $f_{!}$, and the image complex is acyclic. For the bounded-above model $K\otimes M$ of bounded $A$, the same argument in its acyclic low-degree tail shows that a sufficiently low canonical truncation has an $f_{!}$-acyclic first term. It is then a bounded $f_{!}$-acyclic model of $A$, so ordinary acyclic-resolution comparison identifies $f_{!}(K\otimes M)$ with $R{f}_{!}A$. Let $H_K(I)\to J$ be a bounded-below injective replacement. Step 4.1 and flasque acyclicity make ${\mathrm{Hom}}_X^{\bullet }(E_U,H_K(I))\to {\mathrm{Hom}}_X^{\bullet }(E_U,J)$ a quasi-isomorphism. Products of these comparisons are exact over $k$ under AC; the Hom bicomplex of bounded-above $M$ and bounded-below targets has finite diagonals, so its total comparison is a quasi-isomorphism. Finally K-injectivity of $J$ identifies ${\mathrm{Hom}}_X^{\bullet }(A,J)$ with ${\mathrm{Hom}}_X^{\bullet }(M,J)$ up to quasi-isomorphism. Together with step 5.1 this proves $R{\mathrm{Hom}}_X(A,H_K(I))\simeq R{\mathrm{Hom}}_Y(R{f}_{!}A,I)$.

The Hom transposition, the adjoint candidate and its lower bound. Taking $I=I_B$ a bounded-below injective replacement of a bounded-below complex $B$ with $I_B^n=0$ for $n<\inf (B)$ by [F1], step 6.1 gives ${\mathrm{RHom}}_X(A,H_K(I_B))\simeq {\mathrm{RHom}}_Y(R{f}_{!}A,I_B)$, and $H^0$ yields the natural exceptional Hom transposition $\mathrm{Hom}(R{f}_{!}A,B)\simeq \mathrm{Hom}(A,f^{!}B)$ for bounded $A$, with $f^{!}B$ represented by $H_K(I_B)$. By step 3.1 the representative has lower bound $\inf (I_B)-d=\inf (B)-d$, so $f^{!}$ has lower cohomological bound $\inf (B)-d$. The construction uses the chosen finite resolution and the chosen injective model; any two choices differ by the unique identity-transposition comparisons induced by the resolution comparisons of [F2], so the transposition is well defined.

Coherence on the bounded constructible specialization. Step 6.1 gives the Hom transposition and the counit of step 5.2 supplies its evaluation map. When the later geometric theorem proves bounded constructibility of $f^{!}$, this is a full adjunction $R{f}_{!}⊣f^{!}$ on that subcategory. Its unit is the transpose of the identity, and inverse transposition proves both triangle identities. For two choices of $K$ or injective models, the identity transposition gives their comparison; bounded open generators and all their shifts detect its isomorphism through derived sections. Uniqueness of transposition gives the identity and cocycle laws. No unrestricted $D^{+}$-source adjunction is inferred.

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.