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Let $X$ be a finite-dimensional, locally compact topological space, and consider the dualizing complex $K_X \in \mathbf{D}^b(X,k)$ (bounded derived category of $k$-sheaves, where $k$ is a noetherian ring). We can define the dualizing functor $$C \mapsto D(C) = \mathbf{R}\mathcal{H}om(C, K_X),$$ (derived internal hom), which leads to a biduality map $C \to D^2(C)$. In SGA 4.5 "Th. de finitude," Deligne shows that, when $k = \mathbb{Z}/n$, the analogous biduality morphism is an isomorphism on the constructible bounded derived category (so, an anti-involution of said category) when one is working with a scheme of finite type over a field or DVR (with $n$ prime to the characteristic). I have heard that the same is true for topological spaces under certain conditions, although I'm not sure what the statement (or proof) should be: first, presumably we are going to want with a nice (Gorenstein?) ring like $\mathbb{Z}/n$, and second, probably there needs to be some analog of the constructible derived category. What is this statement?

I had a look at Kashiwara-Schapira's "Sheaves on Manifolds," but I can't parse the biduality statement given in chapter 3. It's not clear to me how to adapt Deligne's argument to the present case, anyway.

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For an analytic space, you can find this on page 118 of Verdier's article "Classe d'homologie d'un cycle" in Asterisque 36-37. And yes this is on an appropriate constructible derived category with $\mathbb{Z}$-coefficients. I seem to recall that Borel, in his book on intersection cohomology, also discusses this for pseudomanifolds, in case you need something more general.

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Thanks! . – Akhil Mathew Jul 14 '11 at 13:53

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