I am reading a paper on Hodge cohomology in order to make myself more acquainted with the topic, and the authors take a few facts for granted that are unknown to me.
In particular: If $X\longrightarrow S$ is a proper morphism, why are the sheaves $H^{q}_{*}(X/S)$, where $*\in\left\{\operatorname{de Rham, Hodge}\right\}$, coherent? Why is $\Omega^{q}_{X/S}$ coherent? (In my case $S=\operatorname{Spec}(k)$for a field $k$, but this seems to hold more generally.) Please explain it to me in as much détail as possible, since I am new to the topic. A reference would also be nice.
If $S$ is locally Noetherian and $f:X\to S$ is a morphism of finite type, then the sheaf $\Omega_{X/S}$ of differentials is coherent (and hence so are all of its exterior powers). You should be able to find a proof of this in any introductory text on schemes; for instance, it is Remark II.8.9.1 in Hartshorne's Algebraic Geometry. If you define $\Omega_{X/S}$ locally in terms of modules of differentials in the affine case, the proof is very simple: if $A$ is a Noetherian ring and $B$ is a finitely generated $A$-algebra, then $\Omega_{B/A}$ is a finitely generated $B$-module since it is generated by the elements $db_1\dots,db_n$ if $b_1,\dots,b_n$ generate $B$ as an algebra. (For any $b\in B$, we can write it as a polynomial in $b_1,\dots,b_n$, and thus write $db$ in terms of $db_1,\dots,db_n$ since $d$ is a derivation.)
The other fact you need is that if $S$ is locally Noetherian, $f:X\to S$ is a proper morphism, and $F$ is a coherent sheaf on $X$, then the derived pushforwards $R^if_*F$ are coherent on $S$ for all $i$. This is a harder theorem and is one of the major foundational theorems in the theory of cohomology of coherent sheaves; see for instance the Stacks project, or Hartshorne Theorem III.8.8 for the special case of a projective morphism.