Exceptional inverse image as the adjoint of derived proper-support image

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.