# Confusion about relative Poincaré duality in the context of $\ell$-adic cohomology

I have recently learned about relative Poincaré duality in the book Weil conjectures, perverse sheaves and $$\ell$$-adic Fourier transform by Kiehl and Weissauer (2001). The reference is section II.7. When trying to apply the statement to some toy examples however, I end up with an absurd conclusion. Thus, there must be something I'm doing wrong, but I can't find it out. Let me explain.

Let $$k$$ be a finite field of characteristic $$p>0$$ (or an algebraic closure thereof) and let $$\ell \not = p$$ be a prime number. For a variety $$X$$ over $$k$$, we denote by $$D_c^b(X,\overline{\mathbb Q_{\ell}})$$ the bounded "derived" category of constructible $$\overline{\mathbb Q_{\ell}}$$-sheaves. For a morphism $$f:X\to S$$ of varieties over $$k$$ and for $$K \in D_c^b(X,\overline{\mathbb Q_{\ell}})$$ and $$L \in D_c^b(S,\overline{\mathbb Q_{\ell}})$$, the relative Poincaré duality is a natural isomorphism $$Rf_{*}R\mathcal{Hom}(K,f^!L) \simeq R\mathcal{Hom}(Rf_!K,L),$$ which is functorial in both $$K$$ and $$L$$. In particular, taking $$S = \mathrm{Spec}(k)$$ and $$L := \overline{\mathbb Q_{\ell}}$$, the identity becomes $$Rf_*(D_X(K)) \simeq D_S(Rf_!(K)),$$ where $$D$$ denote the dualizing functor. Taking cohomology gives $$H^i(X,D_X(K)) \simeq H^{-i}_c(X,K)^{\vee}.$$ If we further assume that $$X$$ is smooth of dimension $$d$$, we have $$D_X(K) = R\mathcal{Hom}(K,\overline{\mathbb Q_{\ell}}[2d](d)) = K^{\vee}[2d](d),$$ where $$K^{\vee} := R\mathcal{Hom}(K,\overline{\mathbb Q_{\ell}})$$. Thus, we obtain $$$$H^{i+2d}(X,K^{\vee})(d) \simeq H^{-i}_c(X,K)^{\vee}, \tag{*}$$$$ which looks like the usual Poincaré duality except that $$K$$ is allowed to be any object of $$D_c^b(X,\overline{\mathbb Q_{\ell}})$$. To me, this sounds doubtful. I always thought that Poincaré duality requires the coefficient sheaf to be smooth (lisse), not just constructible.

Eg. let us assume $$X = \mathbb P^1$$ and take $$K = \mathcal F$$ be just a skyscraper sheaf at a closed point of $$X$$ with value $$\overline{\mathbb Q_{\ell}}$$ in degree $$0$$. Then the RHS of $$(*)$$ is $$1$$-dimensional for $$i=0$$, and vanishes otherwise. On the other hand, $$K^{\vee} \simeq K$$ and the LHS is $$1$$-dimensional for $$i = -2$$, and $$0$$ otherwise. So clearly, something isn't right. Would anybody be able to point out where I wrote something wrong?

In your last paragraph you don't have $$K\simeq K^\vee$$. Let $$i$$ be the inclusion of the closed point, so $$K=i_!\mathbf Q_\ell$$. Then by local Verdier duality $$K^\vee = RHom(i_!\mathbf Q_\ell,\mathbf Q_\ell)=i_\ast RHom(\mathbf Q_\ell,i^!\mathbf Q_\ell)$$which is a skyscraper sheaf concentrated in degree 2.