I am familiar with the Hodge star operator or Hodge duality in the theory of finite-dimensional differentiable manifolds, which gives an isomorphism $\star:\Omega^{i}(M)\longrightarrow\Omega^{n-i}(M)$ if $M$ is an $n$-dimensional manifold. The existence of this star operator arises essentially from the perfect product $\Omega^{i}(M)\wedge\Omega^{n-i}(M)\longrightarrow\Omega^{n}(M)$.
Deligne and Illusie (https://eudml.org/doc/143480, p. 255) make the claim that this can be generalized to the case where $X\longrightarrow S$ is a proper smooth morphism of $\mathbb{F}_{p}$ schemes with $\dim(X/S)=n$ and claim that in this case $\Omega^{i}_{X/S}$ and $\Omega^{n-i}_{X/S}$ are dual over $\Omega^{n}_{X/S}$. This is apparently called Serre duality, but I have difficulties to find an exact proof of this claim. (I am probably too stupid to do a decent literature research). Does anybody have a link or a book reference?