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
\begin{equation}
H^{i+2d}(X,K^{\vee})(d) \simeq H^{-i}_c(X,K)^{\vee},
\tag{$*$}
\end{equation}
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?
Best Answer
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.