I have recently been looking at algebraic de Rham cohomology of curves in positive characteristic. In particular, I am looking at when the sequence $$0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{dR}(X/k) \rightarrow H^1(X,{\mathcal O}_X) \rightarrow 0 $$ doesn't split (where $k$ is an algebraically closed field of characteristic $p \geq 0$).
I was told that this relates to Hodge theory, and I was considering looking in to this for a postgraduate seminar series we have.
However, I couldn't find a direct link to the algebraic de Rham theory (emphasis on the algebraic), and I am not sure of it's relation to Hodge theory.
I wondered whether anyone might be able to recommend some introductory materials/"light" reading. In particular, material that doesn't/minimally covers the analytic side, and relates to algebraic geometry.