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In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f_!$ has finite cohomological dimension, then the following holds:

(Verdier duality, local form) There is an additive functor of triangulated categories $f^!:\mathsf{D}^+(Y)\to \mathsf{D}^+(X)$ such that there is a functorial isomorphism $$\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathsf{R}f_! \mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \mathsf{R}f_*\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ in $\mathsf{D}^+(Y)$ for any $\mathscr{F}^\bullet\in\mathsf{D}^b(X)$ and $\mathscr{G}^\bullet\in\mathsf{D}^+(Y)$.

Do we really need all those hypotheses? Perhaps we can use Brown's representability theorem to prove it under more general conditions for the unbounded derived category? (There's a post here on MO about using this theorem but there it is under less general conditions.)

Edit: Let me be clear about my "proposed" proof. Let $f:X\to Y$ be a morphism of (locally compact) ringed spaces and $\mathsf{D}(X),\mathsf{D}(Y)$ be their derived categories of modules. The tag 0F5Y on the Stacks Project implies that any triangulated functor $\mathsf{D}(X)\to\mathsf{D}(Y)$ which preserves direct sums has a right adjoint.

The functor $f_!$ preserves finite direct sums (lemma 6.11(a) in Spaltenstein's paper). As arbitrary direct sums are filtered colimits of finite direct sums, $f_!$ should preserve these as well. But direct sums in $\mathsf{D}(X)$ are obtained termwise (tag 07D9 of the Stacks Project), proving that the functor $\mathsf{R}f_!:\mathsf{D}(X)\to\mathsf{D}(Y)$ also preserves direct sums. We conclude the existence of a functor $f^!:\mathsf{D}(Y)\to\mathsf{D}(X)$ such that $$\hom_{\mathsf{D}(Y)}(\mathsf{R}f_!\mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \hom_{\mathsf{D}(X)}(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ naturally in $\mathscr{F}^\bullet$ and $\mathscr{G}^\bullet$. This yields the local form as usual (for example, prop. 3.1.10 in Sheaves in Manifolds). Please let me know if there's something wrong with it.

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f_!$ has finite cohomological dimension, then the following holds:

(Verdier duality, local form) There is an additive functor of triangulated categories $f^!:\mathsf{D}^+(Y)\to \mathsf{D}^+(X)$ such that there is a functorial isomorphism $$\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathsf{R}f_! \mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \mathsf{R}f_*\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ in $\mathsf{D}^+(Y)$ for any $\mathscr{F}^\bullet\in\mathsf{D}^b(X)$ and $\mathscr{G}^\bullet\in\mathsf{D}^+(Y)$.

Do we really need all those hypotheses? Perhaps we can use Brown's representability theorem to prove it under more general conditions for the unbounded derived category? (There's a post here on MO about using this theorem but there it is under less general conditions.)

Edit: Let me be clear about my "proposed" proof. Let $f:X\to Y$ be a morphism of (locally compact) ringed spaces and $\mathsf{D}(X),\mathsf{D}(Y)$ be their derived categories of modules. The tag 0F5Y on the Stacks Project implies that any triangulated functor $\mathsf{D}(X)\to\mathsf{D}(Y)$ which preserves direct sums has a right adjoint.

If we could prove that $\mathsf{R}f_!$ preserves infinite direct sums, then we would conclude the existence of a functor $f^!:\mathsf{D}(Y)\to\mathsf{D}(X)$ such that $$\hom_{\mathsf{D}(Y)}(\mathsf{R}f_!\mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \hom_{\mathsf{D}(X)}(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ naturally in $\mathscr{F}^\bullet$ and $\mathscr{G}^\bullet$. This yields the local form as usual (for example, prop. 3.1.10 in Sheaves in Manifolds).

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f_!$ has finite cohomological dimension, then the following holds:

(Verdier duality, local form) There is an additive functor of triangulated categories $f^!:\mathsf{D}^+(Y)\to \mathsf{D}^+(X)$ such that there is a functorial isomorphism $$\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathsf{R}f_! \mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \mathsf{R}f_*\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ in $\mathsf{D}^+(Y)$ for any $\mathscr{F}^\bullet\in\mathsf{D}^b(X)$ and $\mathscr{G}^\bullet\in\mathsf{D}^+(Y)$.

Do we really need all those hypotheses? Perhaps we can use Brown's representability theorem to prove it under more general conditions for the unbounded derived category? (There's a post here on MO about using this theorem but there it is under less general conditions.)

Edit: Let me be clear about my "proposed" proof. Let $f:X\to Y$ be a morphism of (locally compact) ringed spaces and $\mathsf{D}(X),\mathsf{D}(Y)$ be their derived categories of modules. The tag 0F5Y on the Stacks Project implies that any triangulated functor $\mathsf{D}(X)\to\mathsf{D}(Y)$ which preserves direct sums has a right adjoint.

The functor $f_!$ preserves direct sums (lemma 6.11(a) in Spaltenstein's paper) and, since direct sums in $\mathsf{D}(X)$ are obtained termwise (tag 07D9 of the Stacks Project), so does the functor $\mathsf{R}f_!:\mathsf{D}(X)\to\mathsf{D}(Y)$. This proves the existence of a functor $f^!:\mathsf{D}(Y)\to\mathsf{D}(X)$ such that $$\hom_{\mathsf{D}(Y)}(\mathsf{R}f_!\mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \hom_{\mathsf{D}(X)}(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ naturally in $\mathscr{F}^\bullet$ and $\mathscr{G}^\bullet$. This yields the local form as usual (for example, prop. 3.1.10 in Sheaves in Manifolds). Please let me know if there's something wrong with it.

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f_!$ has finite cohomological dimension, then the following holds:

(Verdier duality, local form) There is an additive functor of triangulated categories $f^!:\mathsf{D}^+(Y)\to \mathsf{D}^+(X)$ such that there is a functorial isomorphism $$\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathsf{R}f_! \mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \mathsf{R}f_*\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ in $\mathsf{D}^+(Y)$ for any $\mathscr{F}^\bullet\in\mathsf{D}^b(X)$ and $\mathscr{G}^\bullet\in\mathsf{D}^+(Y)$.

Do we really need all those hypotheses? Perhaps we can use Brown's representability theorem to prove it under more general conditions for the unbounded derived category? (There's a post here on MO about using this theorem but there it is under less general conditions.)

Edit: Let me be clear about my "proposed" proof. Let $f:X\to Y$ be a morphism of (locally compact) ringed spaces and $\mathsf{D}(X),\mathsf{D}(Y)$ be their derived categories of modules. The tag 0F5Y on the Stacks Project implies that any triangulated functor $\mathsf{D}(X)\to\mathsf{D}(Y)$ which preserves direct sums has a right adjoint.

The functor $f_!$ preserves finite direct sums (lemma 6.11(a) in Spaltenstein's paper). As arbitrary direct sums are filtered colimits of finite direct sums, $f_!$ should preserve these as well. But direct sums in $\mathsf{D}(X)$ are obtained termwise (tag 07D9 of the Stacks Project), proving that the functor $\mathsf{R}f_!:\mathsf{D}(X)\to\mathsf{D}(Y)$ also preserves direct sums. We conclude the existence of a functor $f^!:\mathsf{D}(Y)\to\mathsf{D}(X)$ such that $$\hom_{\mathsf{D}(Y)}(\mathsf{R}f_!\mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \hom_{\mathsf{D}(X)}(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ naturally in $\mathscr{F}^\bullet$ and $\mathscr{G}^\bullet$. This yields the local form as usual (for example, prop. 3.1.10 in Sheaves in Manifolds). Please let me know if there's something wrong with it.

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f_!$ has finite cohomological dimension, then the following holds:

(Verdier duality, local form) There is an additive functor of triangulated categories $f^!:\mathsf{D}^+(Y)\to \mathsf{D}^+(X)$ such that there is a functorial isomorphism $$\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathsf{R}f_! \mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \mathsf{R}f_*\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ in $\mathsf{D}^+(Y)$ for any $\mathscr{F}^\bullet\in\mathsf{D}^b(X)$ and $\mathscr{G}^\bullet\in\mathsf{D}^+(Y)$.

Do we really need all those hypotheses? Perhaps we can use Brown's representability theorem to prove it under more general conditions for the unbounded derived category? (There's a post here on MO about using this theorem but there it is under less general conditions.)

Edit: Let me be clear about my "proposed" proof. Let $f:X\to Y$ be a morphism of (locally compact) ringed spaces and $\mathsf{D}(X),\mathsf{D}(Y)$ be their derived categories of modules. The tag 0F5Y on the Stacks Project implies that any triangulated functor $\mathsf{D}(X)\to\mathsf{D}(Y)$ which preserves direct sums has a right adjoint.

The functor $f_!$ preserves direct sums (lemma 6.11(a) in Spaltenstein's paper) and, since direct sums in $\mathsf{D}(X)$ are obtained termwise (tag 07D9 of the Stacks Project), so does the functor $\mathsf{R}f_!:\mathsf{D}(X)\to\mathsf{D}(Y)$. This proves the existence of a functor $f^!:\mathsf{D}(Y)\to\mathsf{D}(X)$ such that $$\hom_{\mathsf{D}(Y)}(\mathsf{R}f_!\mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \hom_{\mathsf{D}(X)}(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ naturally in $\mathscr{F}^\bullet$ and $\mathscr{G}^\bullet$. This yields the local form as usual (for example, prop. 3.1.10 in Sheaves in Manifolds). Please let me know if there's something wrong with this proposed proof.

In the section 3.2 of Sheaves in Topology by A. Dimca, the author explains that if $f:X\to Y$ is a continuous map (between locally compact, $\sigma$-compact topological spaces with finite homological dimension) such that $f_!$ has finite cohomological dimension, then the following holds:

(Verdier duality, local form) There is an additive functor of triangulated categories $f^!:\mathsf{D}^+(Y)\to \mathsf{D}^+(X)$ such that there is a functorial isomorphism $$\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathsf{R}f_! \mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \mathsf{R}f_*\mathsf{R}\underline{\operatorname{Hom}}^\bullet(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ in $\mathsf{D}^+(Y)$ for any $\mathscr{F}^\bullet\in\mathsf{D}^b(X)$ and $\mathscr{G}^\bullet\in\mathsf{D}^+(Y)$.

Do we really need all those hypotheses? Perhaps we can use Brown's representability theorem to prove it under more general conditions for the unbounded derived category? (There's a post here on MO about using this theorem but there it is under less general conditions.)

Edit: Let me be clear about my "proposed" proof. Let $f:X\to Y$ be a morphism of (locally compact) ringed spaces and $\mathsf{D}(X),\mathsf{D}(Y)$ be their derived categories of modules. The tag 0F5Y on the Stacks Project implies that any triangulated functor $\mathsf{D}(X)\to\mathsf{D}(Y)$ which preserves direct sums has a right adjoint.

The functor $f_!$ preserves direct sums (lemma 6.11(a) in Spaltenstein's paper) and, since direct sums in $\mathsf{D}(X)$ are obtained termwise (tag 07D9 of the Stacks Project), so does the functor $\mathsf{R}f_!:\mathsf{D}(X)\to\mathsf{D}(Y)$. This proves the existence of a functor $f^!:\mathsf{D}(Y)\to\mathsf{D}(X)$ such that $$\hom_{\mathsf{D}(Y)}(\mathsf{R}f_!\mathscr{F}^\bullet,\mathscr{G}^\bullet)\cong \hom_{\mathsf{D}(X)}(\mathscr{F}^\bullet,f^!\mathscr{G}^\bullet)$$ naturally in $\mathscr{F}^\bullet$ and $\mathscr{G}^\bullet$. This yields the local form as usual (for example, prop. 3.1.10 in Sheaves in Manifolds). Please let me know if there's something wrong with it.