Steenrod operations on algebraic De Rham cohomology, Hodge cohomology, and spectral sequences

Let $X$ be a topological space and $k$ a field of characteristic $p$. Let $A^\cdot$ be a bounded below complex of sheaves of differential graded commutative $\F_p$-algebras. We show that there exist Steenrod operations canonically defined on the sheaf hypercohomology groups, $\mathbf{H}^\cdot(X, A^\cdot)$. These Steenrod operations satisfy most of their usual properties, including the Cartan formula and the Adem relations. Suppose further that $A^\cdot$ is equipped with a filtration $F^\cdot$, which is finite in each degree, and compatible with the product on $A^\cdot$. The filtration on $F^\cdot A^\cdot$ induces a spectral sequence that converges to $\mathbf{H}^\cdot(X, A^\cdot)$, and we prove that the constructed Steenrod operations also have a compatible action on the $E_1$ and $E_\infty$ pages of this spectral sequence. When $X$ is a smooth projective variety over $k$, we obtain Steenrod operations on the algebraic De Rham cohomology groups, $H^\cdot_\text{DR}(X/k)$, as well as the Hodge cohomology groups. The Steenrod operations on $H^\cdot_\text{DR}(X/k)$ have a compatible action on the first and infinite pages of the Hodge to De Rham spectral sequence, as well as the spectral sequence from Katz and Oda related to the Gauss-Manin connection