Duality maps for constructible inverse and direct images

Dualizing a sheaf twice comes with a map from the original sheaf. Whether that map is invertible determines which duality comparisons can be reversed. We will construct the comparison maps, factor the reversed maps through evaluation, and then identify the geometric hypotheses that make those evaluations invertible. An infinite locally constant coefficient requires a different argument: move it through an internal Hom, rather than evaluate it twice.

Scope. The operation, orientation and evaluation arguments are proved below. The constructible applications additionally require the geometric results identified in the prerequisite section; their full proof chains are not claimed here.

AI-written exposition and exercises: GPT-6 Astra (OpenAI), Ultra. The modern weak-coefficient results are due to Andreas Hohl and Pierre Schapira and are credited at their point of use. Original programme expression is dedicated to the public domain under CC0.

Coefficients, pairings and the operation contracts

Let $k$ be a commutative ring of finite global dimension. Spaces in this lesson are finite-dimensional real analytic manifolds, Hausdorff and countable at infinity, with a uniform dimension bound. Complexes are globally bounded. All tensors and internal Homs are derived. We use

$$\omega_X=a_X^!k,\qquad D_XE=R\mathcal Hom_X(E,\omega_X), \qquad H^j(E[s])=H^{j+s}(E). \qquad\text{(1)}$$

For a $d$-manifold, $\omega_X=\mathrm{or}_X[d]$. Keep the orientation line in this formula unless an orientation has been chosen.

We first work formally with a map $f:Y\to X$. The required operation contracts are ordinary adjunction $f^{-1}\dashv Rf_*$, proper-support adjunction $Rf_!\dashv f^!$, tensor–Hom adjunction, the projection isomorphism

$$Rf_!(B\otimes f^{-1}A)\simeq Rf_!B\otimes A, \qquad\text{(2)}$$

and composition $f^!\omega_X\simeq\omega_Y$, with its compatible trace $Rf_!\omega_Y\to\omega_X$. These contracts include their units, counits, naturality and boundedness. The sections below prove compact-support extension and lifting, the proper-image fibre formulas, base change and composition, the uniform manifold dimension bound, and arbitrary-coefficient projection. The adjunction section then constructs both adjunctions on bounded-below categories and fixes their trace-compatible exceptional composition. The following bounds section proves globally bounded preservation for ordinary direct image, exceptional inverse image and duality. The tensor–Hom section then constructs the remaining algebraic contract, including the actual evaluation map and its signs. The orientation section proves the constant local-support and orientation inputs. Constructible and weakly constructible geometric inputs remain prerequisites. In particular, (2) permits arbitrary bounded $A$; it is not a projection theorem restricted to perfect coefficients. The formal arguments also apply outside the analytic setting whenever these same bounded operation contracts hold.

Evaluation is the pairing $\operatorname{ev}_E:D_XE\otimes E\to\omega_X$. By symmetry and currying it gives

$$\eta_E:E\longrightarrow D_XD_XE. \qquad\text{(3)}$$

We call $E$ reflexive for this duality when this particular map is invertible. An abstract isomorphism between $E$ and $D_XD_XE$ would not be sufficient.

All maps below use the usual symmetry on complexes. On homogeneous tensors that symmetry is $v\otimes w\mapsto(-1)^{|v||w|}w\otimes v$. Currying and evaluation use the same convention, so reversing a pairing introduces no unrecorded sign.

Compact support: extension, lifting and acyclicity

The projection proof needs precise control of support when a section is extended or lifted. We supply that part of its foundation here. Throughout this section, spaces are locally compact Hausdorff and coefficients are arbitrary modules over the fixed commutative ring $k$. Finite global dimension is unnecessary for these support arguments. For a compact subset $K\subset X$, $\Gamma(K;F)$ means sections of the inverse-image sheaf on $K$, not sections on $X$ whose support lies in $K$.

The human source for these results is Pierre Schapira’s freely accessible An Introduction to Sheaves on Grothendieck Topologies, 1 August 2026, §4.3. His word “soft” in that section means what we call c-soft: every section on every compact subset extends to a global section. We organise the proof around two concrete operations—extending with prescribed support and lifting through a quotient—and include the compact-neighbourhood details they require. The following sections supply the proper-image fibre formula and the uniform manifold dimension bound.

Compact subsets remember neighbourhood germs

We first justify the comparison

$$\mathop{\mathrm{colim}}_{K\subset V\text{ open}}\Gamma(V;F) \xrightarrow{\sim}\Gamma(K;F). \qquad\text{(C1)}$$

The elementary topological facts needed below follow from compactness and Hausdorff separation. Disjoint compact sets have disjoint open neighbourhoods: separate each pair of points, take a finite intersection on the neighbourhood of a fixed point and a finite union on the opposite compact set, then repeat over a finite cover of the first compact set. Thus a compact Hausdorff space is normal. Inside a compact neighbourhood, this separation shrinks a neighbourhood of a point so that its closure is compact and lies in any specified open neighbourhood. Applying these pointwise shrinkings and then taking a finite subcover gives compact sets subordinate to any finite open cover of a compact set.

Surjectivity of (C1) is not merely the definition of inverse image, since that definition also sheafifies. A section on $K$ has local representatives $s_i$ on ambient open sets $N_i$. Choose finitely many compact sets $K_i\subset K\cap N_i$ covering $K$, and relatively compact open neighbourhoods $O_i$ of $K_i$ with $\overline{O_i}\subset N_i$. For each pair $i,j$, let $E_{ij}$ be the open subset of $N_i\cap N_j$ where the germs of $s_i,s_j$ agree. It contains $K_i\cap K_j$. The compact set $D_{ij}=(\overline{O_i}\cap\overline{O_j})\setminus E_{ij}$ therefore meets $K_i$ and $K_j$ in disjoint closed subsets. Separate those two subsets by disjoint relatively open subsets of $D_{ij}$. Their complements in $D_{ij}$ are compact, so taking their complements in $X$ gives open neighbourhoods of $K_i,K_j$ whose intersection misses $D_{ij}$. Intersect these neighbourhoods over the finitely many pairs, also intersecting with $O_i$. The resulting opens $W_i$ still contain $K_i$, and the representatives agree on every $W_i\cap W_j$. The sheaf axiom glues them to a section on $\bigcup_iW_i$, a neighbourhood of $K$. This proves surjectivity.

For injectivity, two representatives with equal restriction to $K$ have equal germs at every point of $K$. Their equality locus is open and contains $K$. They are therefore equal after shrinking their common neighbourhood, exactly the equivalence relation in the colimit.

We also use gluing over a finite closed cover of a compact space. If sections on closed sets $K_1,\ldots,K_m$ agree on their intersections, their germs give a section on their union. To check this locally at a point, discard the finitely many closed sets not containing that point. Representatives from the remaining sets have the same germ there, so they agree on a common smaller neighbourhood. This supplies a local representative of the proposed glued section; these representatives agree wherever both are defined. Hence the sheaf axiom proves existence and uniqueness of the glued section. This argument concerns sections of restricted sheaves, so equality on an intersection includes equality of the corresponding ambient germs.

Extending a section while controlling its support

Support extension lemma. Suppose $F$ is c-soft, $K\subset X$ is compact, and $s\in\Gamma(K;F)$. If the support of $s$ in $K$ lies in an open set $V\subset X$, there is $t\in\Gamma_c(X;F)$ such that

$$t|_K=s,\qquad \operatorname{supp}(t)\subset V. \qquad\text{(C2)}$$

Proof. The support of a section is closed in its domain because its zero-germ locus is open. Thus $A=\operatorname{supp}(s)$ is compact. Choose a compact neighbourhood $L$ of $A$ contained in $V$; if $A$ is empty, take $t=0$. On the compact set $(K\cap L)\cup\partial L$, prescribe $s$ on $K\cap L$ and zero on $\partial L$. These prescriptions agree on the intersection because $A\subset\operatorname{int}L$. Finite closed gluing supplies a section there. By c-softness it extends to a global section $u$ of $F$.

The germs of $u$ vanish on $\partial L$, so $u$ vanishes on an open neighbourhood of that boundary. Glue $u$ on $\operatorname{int}L$ to zero on the union of that neighbourhood with $X\setminus L$. These opens cover $X$, and the prescribed sections agree on the overlap. The resulting $t$ has support in $L\subset V$. On $K\cap L$ it agrees with $s$; on $K\setminus L$ both vanish. This proves (C2). $\square$

Several stability properties now have direct proofs.

• Restriction preserves c-softness whenever the subspace in question is locally compact Hausdorff. A compact subset of that subspace is compact in $X$, and iterated inverse image identifies its restricted sections with $\Gamma(K;F)$. Extend in $X$ and then restrict.
• If $j:U\hookrightarrow X$ is open and $G$ is c-soft on $U$, then $j_!G$ is c-soft on $X$. Given a section on a compact $K\subset X$, its support $A$ is a compact subset of $K\cap U$, since the stalks off $U$ are zero. Choose $A\subset\operatorname{int}L\subset L\subset U$ with $L$ compact. The same prescription on $(K\cap L)\cup\partial L$, now made entirely inside $U$, extends using c-softness of $G$. Cut it off at $\partial L$ as in the lemma and extend by zero to $X$. It has the prescribed restriction to $K$.
• An arbitrary coproduct of c-soft sheaves is c-soft. A section of that coproduct on a compact set locally involves finitely many summands, by the sheafification definition of a coproduct. A finite cover of the compact set shows that a single finite set of indices suffices. The section factors through that finite subsum on the compact set, as one can check on stalks. Extend its finitely many components globally and then include the finite subsum in the full coproduct. No assertion about unrestricted sections commuting with infinite coproducts is used.

Finally, injective sheaves are c-soft. For opens $U\subset V$, the map $k_U\to k_V$ is a monomorphism, as its stalks show. The identification $\operatorname{Hom}(k_U,I)=\Gamma(U;I)$ and injectivity of $I$ make every restriction $\Gamma(V;I)\to\Gamma(U;I)$ surjective. Thus $I$ is flabby. A section on a compact set is represented on a neighbourhood by (C1), and flabbiness extends that representative to $X$. The same argument proves directly that every flabby sheaf is c-soft.

Lifting compactly supported sections

Lifting lemma. If $0\to F'\to F\to F\text{”}\to0$ is exact and $F'$ is c-soft, then

$$0\longrightarrow\Gamma_c(X;F')\longrightarrow\Gamma_c(X;F) \longrightarrow\Gamma_c(X;F\text{”})\longrightarrow0 \qquad\text{(C3)}$$

is exact.

Proof. Left exactness follows from kernels and the definition of support. We prove surjectivity first when $X$ is compact. A section $s\text{”}$ has lifts on an open cover because the sheaf map $F\to F\text{”}$ is surjective on stalks. Choose a finite compact closed cover $K_i$ subordinate to that cover, and use the corresponding local lifts on $K_i$.

Suppose lifts have already been glued on $K_1\cup\cdots\cup K_{i-1}$. On its intersection with $K_i$, the difference between the existing lift and the new lift is a section of $F'$: restriction is exact and sections preserve kernels. The intersection is compact, so c-softness extends that difference to a global section of $F'$. Add this extension to the new lift. The adjusted lifts agree on the intersection, and finite closed gluing combines them. Finite induction gives a lift on all of $X$.

For general $X$, let $s\text{”}$ have compact support $A$, and choose a relatively compact open $U$ containing $A$. Write $F_U=j_!(F|_U)$, and similarly for the other two sheaves. Open extension by zero is exact by its stalk formula, so these sheaves give a short exact sequence. Its first term is c-soft by the restriction and extension properties just proved. Each term has zero stalks outside the compact set $\overline U$.

A sheaf supported on a closed set is the direct image of its restriction to that set: the natural map to that direct image is an isomorphism on every stalk. Restrict our sequence to $\overline U$, apply the compact case there, and push the resulting section back to $X$. All its support is compact. The section $s\text{”}$ belongs to $\Gamma_c(X;F\text{”}_U)$, since it vanishes near every point outside its compact support $A\subset U$. The compact-case lift is therefore a compactly supported lift of the original $s\text{”}$. This proves (C3). $\square$

Quotient consequence. In a short exact sequence, if $F'$ and $F$ are c-soft, then so is $F\text{”}$. Given a section of $F\text{”}$ on a compact $K$, restrict the sequence to $K$. Its first term is c-soft there, so the compact case of (C3) lifts that section to $\Gamma(K;F)$. C-softness of $F$ extends the lift globally. Its image is the desired extension of the original section.

The compact-cohomology criterion

We now prove, rather than assume, the equivalence

$$F\text{ is c-soft}\quad\Longleftrightarrow\quad H_c^q(U;F|_U)=0 \quad\text{for every open }U\subset X\text{ and }q>0. \qquad\text{(C4)}$$

Use the usual injective definition of the right derived functor of $\Gamma_c$. If $F$ is c-soft, embed it into an injective and continue to an injective resolution. Every successive cokernel is c-soft by the quotient consequence. The lifting lemma makes each of the resulting short exact sequences exact after $\Gamma_c$. Hence the complex of compactly supported sections of the injective resolution is exact in positive degrees. This proves $\Gamma_c$-acyclicity. Apply the same argument to the c-soft restriction on each open $U$ to obtain the forward implication in (C4).

We will also use the following derived extension identity, including for a sheaf $G$ which is not c-soft:

$$H_c^q(X;j_!G)=H_c^q(U;G),\qquad j:U\hookrightarrow X. \qquad\text{(C5)}$$

Take an injective resolution of $G$ on $U$. Its terms are c-soft, so their open extensions are c-soft on $X$ and therefore $\Gamma_c$-acyclic by the preceding paragraph. Exactness of $j_!$ makes this an acyclic resolution of $j_!G$. Termwise, compactly supported sections on $U$ and on $X$ are identical under extension by zero: a compact support in $U$ is compact in $X$, and a compact support of a section of $j_!G$ lies inside $U$. The acyclic-resolution comparison, applied to these degreewise identical complexes, proves (C5).

Here is the comparison needed in that last step. For an acyclic resolution $0\to M\to A^0\to A^1\to\cdots$, put $Z^1=A^0/M$. The first derived-functor exact sequence identifies $R^1T(M)$ with $\operatorname{coker}(T(A^0)\to T(Z^1))$, which is $H^1(T(A^\bullet))$ because $T$ preserves kernels. For $q\geq2$, the same exact sequence gives $R^qT(M)=R^{q-1}T(Z^1)$. Repeat using the tail resolution of $Z^1$; after finitely many repetitions this is the same degree-one calculation. Degree zero is left exactness. The connecting maps come from the given short exact sequences, so the identifications are natural. Apply this with $T=\Gamma_c$.

For the reverse implication in (C4), fix a compact $K\subset X$, put $U=X\setminus K$, and let $i:K\hookrightarrow X$, $j:U\hookrightarrow X$. Stalks give an exact sequence

$$0\longrightarrow j_!(F|_U)\longrightarrow F \longrightarrow i_*(F|_K)\longrightarrow0. \qquad\text{(C6)}$$

The long exact sequence for compactly supported cohomology and (C5) show that $\Gamma_c(X;F)\to\Gamma_c(X;i_*(F|_K))$ is surjective, because its obstruction lies in $H_c^1(U;F|_U)=0$. The group on the right is $\Gamma(K;F)$, since $K$ is compact. Every section on $K$ thus has a compactly supported global extension, proving c-softness. This completes (C4).

What this proves for proper direct image

The support arguments above also supply proper-image acyclicity from the underived proper-image fibre formula, proved in the next section:

$$(f_!Q)_x\simeq\Gamma_c(f^{-1}(x);Q|_{f^{-1}(x)}). \qquad\text{(C7)}$$

Indeed, restrict an injective resolution of a c-soft $Q$ to the closed fibre. Restriction is exact, all its terms are c-soft there, and all the successive cokernels are c-soft. The lifting lemma makes its compact-section complex exact in positive degrees. Applying (C7) termwise identifies that complex with the stalk of $f_!I^\bullet$. Thus $R^qf_!Q=0$ for $q>0$. This proves the acyclicity implication without assuming the derived fibre formula as an extra theorem.

The next section proves (C7) and the coproduct and open-extension compatibilities of $f_!$, thereby establishing the acyclicity conclusion. The following composition and dimension section proves the uniform compact-cohomology bound. The subsequent bounded c-soft construction shows exactly how it supplies finite cohomological dimension and arbitrary-coefficient projection.

Proper support and the fibre calculation

We now supply the proper-image facts used in (C7) and in the projection argument. Let $f:Y\to X$ be continuous between locally compact Hausdorff spaces. The results in this section hold over an arbitrary fixed commutative coefficient ring $k$; finite generation and finite global dimension are unnecessary. The source for the construction is Pierre Schapira’s freely accessible An Introduction to Sheaves on Grothendieck Topologies, 1 August 2026, §§4.1–4.2. We use compactly supported pieces to make the stalk calculation explicit, including its agreement with the ordinary proper-support definition.

A proper map shrinks neighbourhoods of its fibre

Suppose first that $p:Z\to X$ is proper, meaning that inverse images of compact subsets are compact. Such a map is closed. Indeed, if $A\subset Z$ is closed and $x\notin p(A)$, choose a compact neighbourhood $N$ of $x$. The set $A\cap p^{-1}N$ is compact, so its image is compact and hence closed in $X$. Removing that image from $\operatorname{int}N$ gives a neighbourhood of $x$ disjoint from $p(A)$.

Let $W$ be any open neighbourhood of the compact fibre $Z_x=p^{-1}(x)$. Closedness of $p$ shows that $V=X\setminus p(Z\setminus W)$ is an open neighbourhood of $x$, and $p^{-1}V\subset W$. Consequently the sets $p^{-1}V$, for $V$ ranging over neighbourhoods of $x$, are cofinal among neighbourhoods of $Z_x$. The compact-germ comparison (C1) gives

$$(p_*H)_x =\mathop{\mathrm{colim}}_{x\in V}\Gamma(p^{-1}V;H) \simeq\Gamma(Z_x;H|_{Z_x}). \qquad\text{(F1)}$$

This is the restriction map on germs, and is natural in $H$. If the fibre is empty, the same closedness argument supplies a neighbourhood with empty inverse image, so both sides are zero. No surjectivity or local triviality of $p$ was assumed.

The same formula applies when $p$ is proper only on a closed set supporting $H$: write $H$ as the direct image of its restriction to that closed set and apply the proper case there. In particular it applies to every sheaf supported on a fixed compact set.

Constructing the proper-support subsheaf

For an open $U\subset Y$, let $Q_U$ denote the restriction of $Q$ to $U$, extended by zero. Each $Q_U\to Q$ is monic on stalks. Relatively compact open subsets of $Y$ form a directed family under finite unions. Define the sheaf

$$f_!Q=\mathop{\mathrm{colim}}_{U\Subset Y}f_*(Q_U), \qquad U\Subset Y\text{ means }\overline U\text{ is compact}. \qquad\text{(F2)}$$

The colimit here is a sheaf colimit. The inclusions into $f_*Q$ identify it with a subsheaf: taking stalks gives a directed union of submodules. Thus a section of $f_!Q$ over an open $V\subset X$ is locally represented by sections of $Q_U$ for relatively compact $U$. We prove that this is exactly the familiar condition that its support be proper over $V$.

Take such a section, viewed as $s\in\Gamma(f^{-1}V;Q)$, and write $S=\operatorname{supp}(s)$, a closed subset of $f^{-1}V$. Locally on $V$, the representative through $Q_U$ implies $S\subset\overline U$. Over a compact subset $C$ of that local neighbourhood, the set $S\cap f^{-1}C$ is closed in $Y$ and contained in the compact set $\overline U$; it is therefore compact. For a general compact $C\subset V$, choose finitely many compact subsets covering $C$, each subordinate to one of those local neighbourhoods. This finite shrinking is available by the compact Hausdorff argument preceding (C1). Their inverse images in $S$ are compact and cover the inverse image of $C$. Thus $S\to V$ is proper.

Conversely, suppose $S\to V$ is proper. For $x\in V$, choose a compact neighbourhood $N\subset V$ of $x$. Then $S\cap f^{-1}N$ is compact, so it lies in some relatively compact open $U\subset Y$. On $f^{-1}(\operatorname{int}N)$, the section has zero germs outside $U$. The stalk description of the subsheaf $Q_U\subset Q$ shows that it factors through that subsheaf there. Hence $s$ locally belongs to a term in (F2), so it is a section of $f_!Q$. This proves the claimed identification with proper support.

In particular, when $f$ itself is proper, every section of $f_*Q$ satisfies this condition: a closed support over an open of the target remains proper over that open. The natural inclusion is then an isomorphism $f_!Q=f_*Q$.

The underived fibre formula

Fix $x\in X$ and put $Y_x=f^{-1}(x)$. For every relatively compact open $U\subset Y$, the sheaf $Q_U$ is supported on the compact set $\overline U$. Applying (F1) to that support gives

$$(f_!Q)_x \simeq\mathop{\mathrm{colim}}_{U\Subset Y} \Gamma(Y_x;(Q_U)|_{Y_x}) \simeq\Gamma_c(Y_x;Q|_{Y_x}). \qquad\text{(F3)}$$

Here are the details of the last isomorphism. By the stalk formula for open extension, $(Q_U)|_{Y_x}$ is $Q|_{Y_x}$ restricted to $U\cap Y_x$ and extended by zero within the fibre. A global section of this sheaf has support closed in $Y_x$ and contained in $\overline U\cap Y_x$, a compact set. It therefore gives a compactly supported section of $Q|_{Y_x}$. All transition maps are the inclusions of these sections into the same section module.

Conversely, if $t$ has compact support in $Y_x$, that support is compact in $Y$. Choose a relatively compact open $U$ containing it. The section has zero germs outside $U\cap Y_x$, so it belongs to $\Gamma(Y_x;(Q_U)|_{Y_x})$. The directed union therefore contains exactly all compactly supported sections. This proves (F3), including injectivity, and proves the previously stated formula (C7). No commutation of unrestricted sections with a colimit was used.

Restriction to the fibre is exact, and compactly supported sections preserve kernels: a section in a kernel has the same support whether regarded in the kernel sheaf or in the containing sheaf. Testing on stalks in (F3) consequently proves that $f_!$ is left exact. It also confirms that, for the map to a point, (F2) is exactly $\Gamma_c$.

Coproducts and open extensions

Compactly supported sections commute with arbitrary coproducts. A section with compact support $K$ locally uses only finitely many summands; a finite cover of $K$ gives one finite set of indices which suffices everywhere on $K$. Outside $K$ all its germs are zero. Stalkwise factorization therefore puts the whole section in that finite subsum, and its finitely many components have compact support. Conversely a finite family of compactly supported sections gives such a section of the coproduct. These inverse constructions prove the claim. Inverse image to a fibre preserves coproducts, so (F3) shows that the canonical map

$$\bigoplus_\lambda f_!Q_\lambda \xrightarrow{\sim}f_!\left(\bigoplus_\lambda Q_\lambda\right) \qquad\text{(F4)}$$

is an isomorphism on every stalk. This does not assert that arbitrary sections commute with coproducts.

Let $V\subset X$ be open, $U=f^{-1}V$, and $f_V:U\to V$ the restricted map. The proper-support description of sections immediately gives $(f_!Q)|_V=(f_V)_!(Q|_U)$: over every open subset of $V$, the section modules and proper-support conditions are identical.

There is also the extension comparison

$$(f_!Q)_V\xrightarrow{\sim}f_!(Q_U), \qquad U=f^{-1}V. \qquad\text{(F5)}$$

Both sides are subsheaves of $f_!Q$. The right side is a subsheaf by left exactness. On a stalk inside $V$, formula (F3) identifies each inclusion with the identity of $\Gamma_c(Y_x;Q|_{Y_x})$. On a stalk outside $V$, the fibre misses $U$, so both sides are zero. Their images in $f_!Q$ are thus the same subsheaf, proving (F5) canonically.

The stalk identifications $Q\otimes k_U=Q_U$ and $f_!Q\otimes k_V=(f_!Q)_V$ turn (F5) into the open-generator projection map (P7). It is the map which multiplies a properly supported section by the pulled-back coefficient section. Hence the isomorphism just proved is the specific map used in the projection proof, not merely an unspecified isomorphism of its endpoints.

The derived fibre formula and c-soft acyclicity

Combining (F3) with the compact-support proofs gives, for $B\in D^+(k_Y)$, the natural identity

$$(Rf_!B)_x\simeq R\Gamma_c(Y_x;B|_{Y_x}). \qquad\text{(F6)}$$

To prove it, choose a bounded-below injective representative $I$ of $B$. Stalks are exact, so $(Rf_!B)_x$ is represented by $(f_!I)_x$, which (F3) identifies termwise with $\Gamma_c(Y_x;I|_{Y_x})$. Restriction is exact, hence $I|_{Y_x}$ represents $B|_{Y_x}$. Its terms are c-soft, by injective c-softness and restriction stability, and thus are $\Gamma_c$-acyclic.

For clarity, a bounded-below complex of acyclic terms computes a right derived functor. In any fixed degree $q$, cut the complex off brutally above degree $q+1$. The kernel starts in degree $q+2$; applying the termwise functor or its right derived functor gives no cohomology below that degree. For the derived statement, use a bounded-below injective replacement starting in the same degree, which exists by the usual degreewise resolution construction. The cutoff therefore changes neither computation in degree $q$. The cutoff complex is bounded, and the finite acyclic-complex argument proved after (P4) applies to it. These natural comparisons prove the assertion in each degree. This argument does not use an unbounded totalization or a uniform cohomological-dimension bound.

Apply that comparison to $I|_{Y_x}$ to prove (F6). The maps arise from restriction and resolution comparisons, so the identity is natural in $B$. For an ordinary sheaf $Q$ whose restriction to every fibre is c-soft, (C4) and (F6) imply $R^qf_!Q=0$ for $q>0$. In particular every c-soft sheaf on $Y$ is $f_!$-acyclic. Thus the conditional acyclicity argument after (C7) is now established without an imported fibre theorem.

No bounded-output assertion follows merely from (F6). The next section supplies a uniform compact-cohomology bound and a finite c-soft resolution on the standing manifolds, thereby keeping $Rf_!$ in a globally bounded range. The later adjunction section constructs the bounded-below exceptional adjoint. The subsequent bounds section proves its globally bounded output estimate. The geometric duality inputs remain separate obligations.

Composition and the uniform dimension bound

The fibre calculation alone does not bound cohomology. We now prove the missing bound, using proper-image composition to reduce Euclidean space one coordinate at a time to the interval. We then turn the local bounds into finite c-soft resolutions on manifolds. These arguments work for arbitrary sheaves over the fixed commutative ring $k$; neither finite generation nor finite global dimension is needed in this section.

The human source is Schapira’s free An Introduction to Sheaves on Grothendieck Topologies, 1 August 2026: §4.2 for proper-support composition, §§4.3 and 4.5 for c-soft images and base change, Lemma 3.5.1 for the interval, and Lemma 5.1.1 and Proposition 5.1.2 for the dimension bound. We supply the compact-neighbourhood continuity, closed-interval gluing and dimension-shifting details used in the proof. The dimension argument does not use the projection formula it is about to justify.

Pulling back a proper support

Consider a Cartesian square with $f:Y\to X$, $g:X'\to X$, $Y'=Y\times_X X'$, and projections $f':Y'\to X'$, $g':Y'\to Y$. These spaces are locally compact Hausdorff: the fibre product is closed in $Y\times X'$, since the diagonal of $X$ is closed. Pulling back sections defines

$$g^{-1}f_!Q\xrightarrow{\sim}f'_!g'^{-1}Q. \qquad\text{(D1)}$$

We verify both the map and its invertibility. A section of $f_!Q$ over an open $V\subset X$ is a section of $Q$ on $f^{-1}V$ with a support $S$ proper over $V$. Its pullback has support $g'^{-1}S$. For a compact $C\subset g^{-1}V$, the part of that support over $C$ is a closed subset of $(S\cap f^{-1}g(C))\times C$. Both factors are compact, and the equality $f(y)=g(x')$ is closed. Thus the pulled-back support is proper. Pullback of local sections, followed by sheafification of the inverse-image presheaf, therefore gives the displayed map.

At a point $x'\in X'$, (F3) identifies its two stalks with compactly supported sections on $Y_{g(x')}$ and on $Y'_{x'}$. The projection $g'$ identifies these fibres homeomorphically, with the same restricted sheaf, and the map is the resulting pullback of sections. It is an isomorphism. Stalkwise detection proves (D1). This proof also shows that the comparison respects restriction and consecutive changes of base: in each case it pulls back the same section.

Composing proper images and preserving c-softness

Let $f:Y\to X$ and $g:X\to Z$. Then there is a natural identification

$$g_!f_!Q\simeq(gf)_!Q. \qquad\text{(D2)}$$

To check it on an open $V\subset Z$, compare both sides as subsheaves of $(gf)_*Q$. For a section $s$ of $Q$ on $(gf)^{-1}V$, put $S=\operatorname{supp}(s)$. If $S\to V$ is proper, then $S\to g^{-1}V$ through $f$ is proper: the inverse image of a compact $C\subset g^{-1}V$ is closed in the compact set $S\cap(gf)^{-1}g(C)$. The corresponding section $t$ of $f_!Q$ has support exactly $T=f(S)$. Indeed, (F3) says that its stalk at $x$ is the restricted section on the fibre, which is nonzero exactly when that fibre meets $S$. The set $T$ is closed in $g^{-1}V$, since $f|_S$ is proper and hence closed. Its inverse image under $g$ of a compact $C\subset V$ is the compact image $f(S\cap(gf)^{-1}C)$. Thus $t$ has proper support over $V$.

Conversely, if $s$ represents a section of $g_!f_!Q$, then $S\to T$ and $T\to V$ are proper by the same support description. The inverse image in $S$ of a compact subset of $V$ is compact by two successive properness applications. Hence $S\to V$ is proper. Both sheaves therefore consist of the same sections with the same restriction maps. This proves (D2), and also proves its identity and threefold-associativity compatibilities: every comparison retains the underlying section of $Q$.

In particular, taking the last map to a point gives $\Gamma_c(X;f_!Q)=\Gamma_c(Y;Q)$. We use this to prove that $f_!$ preserves c-soft sheaves.

First, a c-soft sheaf $Q$ has surjective restriction $\Gamma_c(Y;Q)\to\Gamma_c(A;Q|_A)$ for every closed $A\subset Y$. Given a compactly supported section on $A$, choose a compact neighbourhood $L$ of its support. Prescribe that section on $A\cap L$ and zero on $\partial L$. The prescriptions agree, since the support lies in $\operatorname{int}L$. Both sets are compact. The compact extension and cutoff argument proving (C2) gives a global compactly supported section with the required restriction to all of $A$; off $L$, both restrictions vanish.

Now let $K\subset X$ be compact. Base change (D1) for the closed inclusion of $K$, followed by (D2) for the map from $K$ to a point, identifies $\Gamma(K;f_!Q)$ with $\Gamma_c(f^{-1}K;Q|_{f^{-1}K})$. The restriction map from $\Gamma_c(X;f_!Q)=\Gamma_c(Y;Q)$ becomes restriction to the closed set $f^{-1}K$, which is surjective by the preceding paragraph. This proves c-softness of $f_!Q$.

Choose a bounded-below injective representative $I$ of $B\in D^+(k_Y)$. Its terms are c-soft, so $f_!I$ has c-soft terms and computes $Rf_!B$. Those terms are also $g_!$-acyclic by (F6). The bounded-below acyclic-complex comparison proved there therefore gives

$$Rg_!Rf_!B\simeq R(gf)_!B, \qquad R\Gamma_c(X;Rf_!B)\simeq R\Gamma_c(Y;B). \qquad\text{(D3)}$$

The termwise maps are (D2), so they are the natural composition maps and retain identity and threefold-associativity compatibility. This proof does not need a cohomological-dimension bound.

The same method derives (D1). Each term of $g'^{-1}I$ is c-soft on every fibre of $f'$: the fibre is identified with a fibre of $f$, on which the original injective term restricts to a c-soft sheaf. Formula (F6) makes these terms $f'_!$-acyclic. Since inverse image is exact, applying the bounded-below comparison and (D1) termwise proves

$$g^{-1}Rf_!B\simeq Rf'_!g'^{-1}B. \qquad\text{(D4)}$$

The maps still pull back the original local sections before passing to resolutions; their base-change compatibilities follow from that construction. No properness of $f$, beyond the proper supports defining $f_!$, is required.

Cohomology near a compact set and the interval bound

Let $K$ be a compact subset of a locally compact Hausdorff space. There is a natural continuity isomorphism

$$\mathop{\mathrm{colim}}_{K\subset V\text{ open}}H^q(V;Q|_V) \simeq H^q(K;Q|_K). \qquad\text{(D5)}$$

To prove it, take an injective resolution $I$ of $Q$ on the ambient space. Restriction to an open preserves injectives: its left adjoint is the exact extension by zero, so its Hom test for injectivity remains exact. Thus $\Gamma(V;I)$ computes the left-hand groups. The restricted complex $I|_K$ is exact over $Q|_K$, and its terms are c-soft on the compact space $K$. There $\Gamma=\Gamma_c$, so the same terms are acyclic and $\Gamma(K;I)$ computes the right-hand groups. Comparison (C1) identifies these complexes after taking the filtered colimit. Filtered colimits of modules are exact: each equality and each witness for a preimage is witnessed at some common later index. Consequently they commute with kernels modulo images, hence with cohomology. This proves (D5).

The same proof works with compact neighbourhoods instead of open neighbourhoods. Every neighbourhood of a compact set contains a compact neighbourhood, and every compact neighbourhood contains an open one. These interleavings give the same colimit termwise. Restrictions of $I$ to the compact neighbourhoods remain c-soft and acyclic. In particular, closed intervals shrinking down to a closed interval or to an endpoint have this cohomology continuity.

We claim that $H^q([0,1];Q)=0$ for every $q>1$ and every sheaf $Q$. For $0<t<a\leq1$, write $[0,a]=[0,t]\cup[t,a]$. Restrict an injective resolution on $[0,a]$ to these compact sets. Finite closed gluing gives the kernel in the sequence of section complexes, and c-softness gives surjectivity onto the section complex at $\{t\}$: every germ there extends globally. The difference of restrictions therefore gives a termwise short exact sequence. Its long exact cohomology sequence, and exactness of sections on a point, give

$$H^q([0,a];Q)\simeq H^q([0,t];Q)\oplus H^q([t,a];Q) \quad(q>1). \qquad\text{(D6)}$$

Take $s\in H^q([0,1];Q)$, $q>1$, and let $J$ be the set of $a$ for which $s|_{[0,a]}=0$. It is downward closed. The restriction to $\{0\}$ is zero; continuity at that compact set makes the restriction zero on a nontrivial initial interval. Hence $\tau=\sup J$ is positive. Continuity at $\{\tau\}$, considered inside $[0,\tau]$, makes the restriction zero on some terminal interval $[t,\tau]$ with $t<\tau$. The initial interval $[0,t]$ also has zero restriction, by the definition of the supremum and downward closure. Formula (D6) shows that the restriction to $[0,\tau]$ is zero. If $\tau<1$, continuity around the compact set $[0,\tau]$ makes it zero on a strictly larger initial interval, a contradiction. Thus $\tau=1$, and $s=0$. This proves the claim without a constructibility hypothesis.

An open interval is homeomorphic to $\mathbb R$. Extend a sheaf on $(0,1)$ by zero to $[0,1]$. Formula (C5), and compactness of the closed interval, identify its compactly supported cohomology with the ordinary cohomology of that extension. The proved interval bound therefore gives $H_c^q(\mathbb R;Q)=0$ for $q>1$.

Euclidean dimension by one-dimensional fibres

We prove by induction that

$$H_c^q(\mathbb R^n;Q)=0\qquad(q>n) \qquad\text{(D7)}$$

for every sheaf $Q$. For $n=0$, sections on a point are exact. The case $n=1$ was just proved. For $n>1$, use the projection $p:\mathbb R^n\to\mathbb R^{n-1}$. Its fibres are lines. Formula (F6) and the interval result give $R^jp_!Q=0$ for $j\notin\{0,1\}$. The cohomology truncation triangle is therefore $R^0p_!Q\to Rp_!Q\to(R^1p_!Q)[-1]\to(R^0p_!Q)[1]$. Applying $R\Gamma_c(\mathbb R^{n-1};-)$, the induction hypothesis kills its first term in cohomological degrees greater than $n-1$ and its third term in degrees greater than $n$. The long exact sequence kills the middle term in degrees greater than $n$. Composition (D3) identifies that middle term with $R\Gamma_c(\mathbb R^n;Q)$, proving (D7).

If $V\subset\mathbb R^n$ is open and $G$ is any sheaf on $V$, (C5) identifies $H_c^q(V;G)$ with $H_c^q(\mathbb R^n;j_!G)$. The same bound therefore holds on every open subset, with the same integer $n$.

C-softness is local

Suppose a sheaf $Q$ is c-soft on each member $U_i$ of an open cover of a locally compact Hausdorff space. Given a section $s$ on a compact set $K$, choose finitely many compact subsets $K_1,\ldots,K_m$ of $K$ covering $K$, each contained in a member of the cover. Such a closed refinement follows by choosing, at each point of $K$, an open neighbourhood whose closure lies in a cover member, and then taking a finite subcover and intersecting those closures with $K$. We extend $s$ by successive corrections, keeping the already corrected pieces fixed.

Suppose a compactly supported global section already agrees with $s$ on $A=K_1\cup\cdots\cup K_{r-1}$. Its residual section on $K$ has zero germs on $A$, and therefore factors through $Q_{X\setminus A}|_K$, by the stalk formula for extension by zero. On the cover member containing $K_r$, the sheaf $Q_{X\setminus A}$ is c-soft: it is an open extension of a restriction of the locally c-soft sheaf $Q$. Apply (C2) there to extend the residual on $K_r$ to a compactly supported section of that sheaf. Extend the correction by zero to the whole space and add it. It changes no germs on $A$, and corrects $K_r$.

Start with zero and repeat finitely many times. The resulting compactly supported section agrees with $s$ on all of $K$, proving c-softness of $Q$. Only a finite cover of the given compact set was used; no locally finite global partition was assumed.

The uniform manifold bound and finite c-soft resolutions

Let $Y$ be a manifold of the standing kind, with a fixed finite upper bound $N$ on all its local dimensions. Then, for every open $U\subset Y$ and every sheaf $Q$ on $U$,

$$H_c^q(U;Q)=0\qquad(q>N). \qquad\text{(D8)}$$

Here is the finite resolution that proves this globally. For a sheaf $Q$ on $Y$, choose an injective resolution $I$ in nonnegative degrees and put $Z^r=\ker(d:I^r\to I^{r+1})$, with $Z^0=Q$. The sequences $0\to Z^r\to I^r\to Z^{r+1}\to0$ are exact. On an open subset $V$ of a coordinate chart, the terms $I^r|_V$ are c-soft and compact-section acyclic. Their long exact sequences give, for $q>0$,

$$H_c^q(V;Z^N|_V)\simeq H_c^{q+N}(V;Q|_V)=0. \qquad\text{(D9)}$$

The last equality is the Euclidean open-set bound, since the chart dimension is at most $N$. If $N=0$, it is that bound directly, with no dimension-shifting steps. Criterion (C4) makes $Z^N$ c-soft on each chart. Locality, just proved, makes it c-soft on $Y$. We thus have the exact finite c-soft resolution

$$0\longrightarrow Q\longrightarrow I^0\longrightarrow\cdots \longrightarrow I^{N-1}\longrightarrow Z^N\longrightarrow0. \qquad\text{(D10)}$$

For $N=0$, this means that $Q$ itself is c-soft; there are no injective middle terms. Every term is $\Gamma_c$-acyclic, so this resolution gives zero compact cohomology above degree $N$. Applying the same argument to the open manifold $U$, whose local dimensions have the same bound, proves (D8).

Every term of (D10) is also $f_!$-acyclic for any continuous map from $Y$ to a locally compact Hausdorff space, by (F6). Thus the same bound gives $R^qf_!Q=0$ for $q>N$. This proves both dimension inputs used by the bounded c-soft and left-tail constructions below. The bound is uniform because the standing manifold dimensions are uniformly bounded; no claim of a finite global bound is made for an unbounded union of dimensions.

The compact-support, fibre, composition and dimension arguments needed for projection have now been supplied. The later adjunction section supplies the bounded-below exceptional construction and its trace. The later bounds section supplies globally bounded operation estimates. Geometric duality and constructibility remain separate obligations.

Why projection permits arbitrary bounded coefficients

The projection isomorphism (2) is essential to both transport arguments. We prove it here for the standing manifold setting, without assuming finite generation of the coefficient complex. The resolution method is developed from Pierre Schapira’s freely accessible An Introduction to Sheaves on Grothendieck Topologies, 1 August 2026, §4.4. We give the resolution and truncation arguments explicitly; they are needed when a flat resolution has infinitely many terms to the left.

The compact-support section proves injective c-softness, restriction/open-extension/coproduct stability, and the compact-cohomology criterion. The proper-image section proves the fibre formula, c-soft acyclicity for proper direct image and its required underived support compatibilities. The composition and dimension section proves the uniform integer bound $H_c^j(U;Q)=0$ for $j>n$, every open $U\subset Y$ and every sheaf $Q$, where $n$ bounds the standing manifold dimensions. It also proves $R^jf_!Q=0$ for $j>n$. These are the topological inputs used below.

Resolving coefficients by sums of open constant sheaves

For an open subset $U\subset X$, write $k_U$ for its constant sheaf extended by zero to $X$. A sum $\bigoplus_\lambda k_{U_\lambda}$ is flat: its stalk at a point is a direct sum of copies of $k$, and tensor exactness is detected on stalks. Such sums surject onto every sheaf $Q$. Indeed, a section $s\in Q(U)$ gives a map $k_U\to Q$; take the sum over all open sets and sections. Each germ of $Q$ lies in the image of one summand, proving stalkwise surjectivity.

The same generators give a bounded-above flat replacement of every bounded-above complex $G^\bullet$. We include the construction because the generators need not be projective as sheaves. Suppose $P^i$, its differential and a chain map $u^i:P^i\to G^i$ have been constructed for $i>q$. Form the sheaf

$$C^q=\{(x,p)\in G^q\oplus P^{q+1}: d_Gx=u^{q+1}p,\ d_Pp=0\}. \qquad\text{(P1)}$$

Choose a sum of open constant sheaves $P^q$ surjecting onto $C^q$. Compose that epimorphism with the two projections to define $u^q$ and $d_P^q$. Equation (P1) gives $d_P^2=0$ and $d_Gu=ud_P$. Start above the upper bound of $G$ with zero terms and repeat in descending degrees.

To verify that $u:P\to G$ is a quasi-isomorphism, use the convention

$$\operatorname{Cone}(u)^q=G^q\oplus P^{q+1},\qquad d(x,p)=(d_Gx+u(p),-d_Pp). \qquad\text{(P2)}$$

A cycle $(x,p)$ determines $(x,-p)\in C^q$. Locally it lifts to some $z\in P^q$, by the epimorphism just chosen. Then $d(0,z)=(x,p)$. The cone is therefore exact on stalks, which proves the claim. No lifting through a globally projective open-generator sheaf was assumed.

A bounded-above complex of flat sheaves is K-flat for the direct-sum tensor totalization used here. For completeness, its brutal truncations obtained by discarding all terms below a fixed degree are bounded subcomplexes. A bounded flat complex tensored with an exact complex is exact: filter it by its finitely many terms and use flatness and the long exact cohomology sequences of the resulting short exact sequences of complexes. The whole bounded-above flat complex is the directed union of these subcomplexes. Tensor products commute with that union, and directed colimits of sheaves of modules are exact because this can be checked on stalks. Tensoring the union with an exact complex is thus exact. This proves K-flatness and justifies using $P$ to compute a derived tensor product.

Removing an exact left tail before deriving

Finite-dimension lemma. Let $T$ be a left exact additive functor between abelian categories with enough injectives, and assume $R^jT=0$ for $j>r$. Let $C^\bullet$ be bounded above, with $T$-acyclic terms and bounded cohomology. Then the termwise complex $T(C^\bullet)$ computes $RT(C^\bullet)$.

Proof. Choose $a$ such that $H^i(C)=0$ for $i<a$, and write $Z^i=\ker d_C^i$. For $i<a$, the exact left tail gives short exact sequences

$$0\longrightarrow Z^{i-1}\longrightarrow C^{i-1} \longrightarrow Z^i\longrightarrow0. \qquad\text{(P3)}$$

Since the middle terms are $T$-acyclic, the long exact derived-functor sequences give, for $j>0$,

$$R^jT(Z^i)\simeq R^{j+r}T(Z^{i-r})=0. \qquad\text{(P4)}$$

For $r=0$, vanishing is immediate from the hypothesis; no connecting maps are required. Thus every $Z^i$ with $i<a$ is $T$-acyclic. Let $B^a=\operatorname{im}d_C^{a-1}$. The sequence $0\to Z^{a-1}\to C^{a-1}\to B^a\to0$ also makes $B^a$ acyclic for $T$.

Replace the entire left tail by $B^a$ in degree $a-1$, with its inclusion into $C^a$. Call the resulting bounded complex $C'$. The map $C\to C'$, given by the quotient in degree $a-1$ and the identity above it, is a quasi-isomorphism. Applying $T$ preserves that quasi-isomorphism: (P3) and the last short exact sequence remain exact after applying $T$, since their left terms are acyclic. The bounded complex $C'$ has acyclic terms, so the usual bounded acyclic-resolution argument computes $RT(C')$ by $T(C')$. Hence $T(C)$ represents the same derived object. All comparisons are induced by the specified truncation maps, so the result identifies the natural computation map. $\square$

The bounded acyclic-resolution argument used in the last sentence follows by filtering $C'$ by its finitely many brutal truncations. For a single acyclic term it is the definition of acyclicity. At each additional term the short exact sequence of complexes and its derived triangle identify the two computations by the long exact cohomology sequence. Induction over the finite number of terms proves the assertion.

A bounded c-soft model for the source input

Let $B\in D^b(k_Y)$, with cohomology in degrees $[a,b]$. Choose a bounded-below injective representative $I^\bullet$ starting in degree $a$. Its terms are c-soft. Put $Z^i=\ker d_I^i$. Since the cohomology above $b$ is zero, for every $i\geq b$ there is an exact sequence

$$0\longrightarrow Z^i\longrightarrow I^i\longrightarrow Z^{i+1}\longrightarrow0. \qquad\text{(P5)}$$

Restriction to an open $U\subset Y$ preserves these exact sequences and c-softness. Dimension shifting gives, for $j>0$,

$$H_c^j(U;Z^{b+n}|_U) \simeq H_c^{j+n}(U;Z^b|_U)=0. \qquad\text{(P6)}$$

For $n=0$, the vanishing follows directly from the uniform compact-cohomology bound. The c-soft criterion therefore makes $Z^{b+n}$ c-soft. Replace $I^{b+n}$ by this kernel and discard higher degrees. This upper truncation $A^\bullet$ is quasi-isomorphic to $B$, has finitely many nonzero terms, and every term is c-soft. Consequently its terms are $f_!$-acyclic and $f_!A$ computes $Rf_!B$.

For an ordinary sheaf $B$ in degree zero, this construction produces $A$ in degrees $0,\ldots,n$. Its terms are $f_!$-acyclic by the proved fibre and c-soft results. Hence $R^jf_!B=0$ for $j>n$. Thus the same uniform compact-cohomology bound implies finite cohomological dimension of $f_!$; no additional dimension hypothesis on the map is needed for the left-tail lemma.

The projection proof

Choose the bounded c-soft model $A$ of $B$, and an open-generator replacement $P\to E$ for $E\in D^b(k_X)$. Inverse image is exact and sends $k_U$ to $k_{f^{-1}U}$, so $f^{-1}P$ is again a bounded-above K-flat complex. It follows that $A\otimes f^{-1}P$ represents $B\otimes^Lf^{-1}E$.

Each term of that tensor total complex is a finite sum, over the degrees of $A$, of coproducts of open extensions of c-soft sheaves. It is c-soft and hence $f_!$-acyclic. The complex is bounded above. Its cohomology is bounded below as well: stalkwise, the two inputs are bounded complexes of $k$-modules and finite global dimension bounds all Tor degrees. The finite-dimension lemma therefore proves that applying $f_!$ termwise calculates the desired derived image even though $P$ may have an infinite left tail.

For an individual open generator the underived projection map is

$$f_!A^p\otimes k_U\xrightarrow{\sim} f_!(A^p\otimes k_{f^{-1}U}). \qquad\text{(P7)}$$

It is the compatibility of proper-support image with open restriction and extension by zero. Since $f_!$ commutes with coproducts, (P7) holds with any $P^q$ in place of $k_U$. Totalization uses only finitely many terms in each degree, because $A$ is bounded. Thus the termwise maps assemble to an isomorphism of complexes

$$(f_!A)\otimes P\xrightarrow{\sim}f_!(A\otimes f^{-1}P). \qquad\text{(P8)}$$

The left side represents $Rf_!B\otimes^LE$, since $P$ is K-flat. The right side represents $Rf_!(B\otimes^Lf^{-1}E)$, by the preceding paragraph. This proves (2) for arbitrary bounded coefficients.

Finally, these are the canonical projection maps. Before deriving, (P7) multiplies a properly supported section by a pulled-back coefficient section. Multiplying by two coefficients consecutively is the same as multiplying by their tensor product; multiplication by the unit coefficient is the identity. Swapping homogeneous coefficients gives exactly the Koszul sign. These equalities hold termwise in (P8), commute with differentials and pass to the derived category. Hence this proof supplies the unit, associativity and symmetry compatibilities needed by the pairings (5)–(11).

Constructing the adjunction and fixing its trace

We now construct the adjunction used in (1). For this construction let $f:Y\to X$, with $Y$ a manifold whose local dimensions are at most $N$, and $X$ locally compact Hausdorff. The coefficients may be any fixed commutative ring $k$. The construction gives $Rf_!\dashv f^!$ on $D^+$ and a lower cohomological bound. The next section proves the additional upper bound needed to keep its output globally bounded in the standing manifold setting.

A human source for the finite-resolution method is Akhil Mathew’s freely accessible Verdier duality, 29 July 2011, Corollary 3.6, Lemma 3.7 and §§4.1–4.3. That treatment assumes Noetherian coefficients. Here we work first over $\mathbb Z$, prove the needed tensor and generator statements, and then carry the arbitrary $k$-action through the construction. In particular, we do not assume that products of flat modules over an arbitrary ring are flat. The compact-support and dimension inputs are the proofs already given in this lesson.

A finite resolution that works with every coefficient sheaf

The bound (D8), with integral coefficients, has the following consequence. Suppose an abelian sheaf $E$ admits an exact sequence $0\to A\to S_{N-1}\to\cdots\to S_0\to E\to0$, where the $N$ middle terms are c-soft. Restrict to any open $V\subset Y$. Repeated connecting morphisms identify $H_c^q(V;E)$ with $H_c^{q+N}(V;A)$ for $q>0$. The latter vanishes. Criterion (C4) makes $E$ c-soft. If $N=0$, every abelian sheaf is c-soft directly by the same bound and criterion.

Consequently, if $L$ is a flat c-soft abelian sheaf, then $M\otimes_{\mathbb Z}L$ is c-soft for every abelian sheaf $M$. Indeed, resolve $M$ to the left by coproducts of open generators $\mathbb Z_U$, choosing generators for local sections and repeating on the kernel. Flatness preserves exactness after tensoring. The resulting terms are coproducts of $L_U$, hence c-soft by the proved support stability. Use the last $N$ terms and the preceding dimension shift. This proof applies to the underlying abelian sheaf of a $k$-module, without assuming that $k$ is flat over $\mathbb Z$.

We construct a finite resolution of the constant integral sheaf. For an abelian sheaf $E$, put $C(E)(V)=\prod_{y\in V}E_y$, with restrictions deleting coordinates. This is a sheaf and is flabby, because arbitrary coordinate families extend by zero. The germ map $E\to C(E)$ is injective. On the stalk at $y$, evaluating the $y$-coordinate is a retraction: an element of $C(E)_y$ is represented on a neighbourhood containing $y$, so that coordinate is well defined and sends the germ of a genuine section back to itself.

If the stalks of $E$ are torsion-free, so are the products defining $C(E)$, their filtered stalk colimits, and the direct summands $(C(E)/E)_y$. Torsion-free abelian groups are flat: each is the filtered union of its finitely generated subgroups, which are free by integer row reduction, and filtered colimits of tensor functors preserve exactness. Thus $C(E)$ and $C(E)/E$ are flat whenever $E$ is flat over $\mathbb Z$.

Starting from $E^0=\mathbb Z_Y$, form $L^p=C(E^p)$ and $E^{p+1}=L^p/E^p$ for $0\le p<N$, and put $L^N=E^N$. The preceding dimension shift makes the final term c-soft; the other terms are flabby. We obtain

$$0\longrightarrow\mathbb Z_Y\longrightarrow L^0\longrightarrow\cdots \longrightarrow L^N\longrightarrow0, \qquad L^p\text{ flat and c-soft}. \qquad\text{(A1)}$$

For $N=0$, take $L^0=\mathbb Z_Y$. Each constituent short exact sequence of (A1) splits on stalks by the coordinate retraction, so tensoring it with any sheaf preserves exactness. Hence $M\to M\otimes_{\mathbb Z}L^\bullet$ is a quasi-isomorphism for every sheaf $M$. It remains so for a bounded-below complex: the augmented double complex has exact finite rows in the $L$-direction, and in each total degree only finitely many entries occur. Successive elimination along these rows proves its total complex acyclic. Every term of the unaugmented total complex is c-soft by the tensor result. Formula (F6) and its acyclic-complex argument therefore give

$$Rf_!G\simeq f_!\operatorname{Tot}(G\otimes_{\mathbb Z}L^\bullet) \qquad(G\in D^+(k_Y)). \qquad\text{(A2)}$$

Representing the supported pairing on open sets

For a flat c-soft abelian sheaf $L$, write $T_L(M)=f_!(M\otimes_{\mathbb Z}L)$ for $k$-module sheaves $M$. This functor is exact. Tensoring a short exact sequence is exact, and its c-soft terms are $f_!$-acyclic, so the proper-image long exact sequence proves the assertion. It also preserves coproducts, by (F4) and distributivity of tensor.

For an injective $k_X$-module sheaf $I$, define

$$J_L(I)(U)=\operatorname{Hom}_{k_X}\bigl(T_L(k_U),I\bigr) \qquad(U\subset Y\text{ open}). \qquad\text{(A3)}$$

The inclusion $k_V\to k_U$ for $V\subset U$ defines restriction. This presheaf is a sheaf: if $U=\bigcup_iU_i$, the sequence $\bigoplus_{i,j}k_{U_i\cap U_j}\to\bigoplus_i k_{U_i}\to k_U\to0$ is right exact, with the first map the difference of the two inclusions. On a stalk in $U$, this is the presentation of one copy of $k$ by generators indexed by cover members containing the point and relations identifying any two; outside $U$ all terms vanish. Applying the exact coproduct-preserving $T_L$, then $\operatorname{Hom}(-,I)$, gives precisely the sheaf equalizer for (A3). The presentation uses coproducts of sheaves; the resulting equalizer uses products of section modules.

There is a natural bijection

$$\operatorname{Hom}_{k_X}(T_L(M),I) \simeq\operatorname{Hom}_{k_Y}(M,J_L(I)). \qquad\text{(A4)}$$

Given a morphism on the left, precompose with $T_L(k_U)\to T_L(M)$ for each local section of $M$. The resulting compatible linear maps on sections define the morphism on the right. For $M=k_U$ this is the identity in (A3), since $\operatorname{Hom}(k_U,J_L(I))=J_L(I)(U)$. It is therefore a bijection for coproducts of open generators. Every $M$ has a presentation $P_1\to P_0\to M\to0$ by such coproducts: use all local sections to construct the first surjection and repeat for its kernel. Both sides of (A4) send that presentation to the kernel of the corresponding map from the value on $P_0$ to the value on $P_1$. The bijections on the two generators identify the kernels. This proves (A4) and its naturality. No projectivity of $k_U$, or colimit formula omitting additive relations, has been used.

The left side of (A4) is exact as a contravariant functor of $M$. Thus $J_L(I)$ is injective. This construction is covariant in $I$ and contravariant in $L$, as is already visible in (A3).

The complex and the adjunction

Represent $F\in D^+(k_X)$ by a bounded-below injective complex $I$, and use the fixed finite resolution (A1). Define a complex $J(I)$ by

$$J(I)^n=\bigoplus_{p=0}^{N}J_{L^p}(I^{n+p}), \qquad (dh)_p=d_Ih_p-(-1)^n h_{p+1}d_L^p. \qquad\text{(A5)}$$