  We start with a curve over an algebraically closed ground field of positive
characteristic $p>0$. By using specialization techniques, under suitable
natural coprimality conditions, we prove a cohomological Simpson Correspondence
between the moduli space of Higgs bundles and the one of connections on the
curve. We also prove a new $p$-multiplicative periodicity concerning the
cohomology rings of Dolbeault moduli spaces of degrees differing by a factor of
$p$. By coupling this $p$-periodicity in characteristic $p$ with
lifting/specialization techniques in mixed characteristic, we find, in
arbitrary characteristic, cohomology ring isomorphisms between the cohomology
rings of Dolbeault moduli spaces for different degrees coprime to the rank. It
is interesting that this last result is proved as follows: we prove a weaker
version in positive characteristic; we lift and strengthen the weaker version
to the result in characteristic zero; finally, we specialize the result to
positive characteristic. The moduli spaces we work with admit certain natural
morphisms (Hitchin, de Rham-Hitchin, Hodge-Hitchin), and all the cohomology
ring isomorphisms we find are filtered isomorphisms for the resulting perverse
Leray filtrations.
