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Steenrod operations on algebraic De Rham cohomology, Hodge cohomology, and spectral sequences
 Author(s): Drury, Ryan Alton
 Advisor(s): Suh, Junecue
 et al.
Abstract
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 GaussManin connection.
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