Motivic Steenrod operations in characteristic $p$

Abstract

Using the recent work of Frankland and Spitzweck, we define Steenrod operations $P^n$ on the mod $p$ motivic cohomology of smooth varieties defined over a base field of characteristic $p$. We show that $P^n$ is the $p$th power on $H^{2n,n}(-,{\mathbb{F}}_p)\cong C{H}^n(-)/p$ and prove an instability result for the operations. Restricted to mod $p$ Chow groups, we show that the operations satisfy the expected Adem relations and Cartan formula. For $p=2$, we use the new Steenrod squares to obtain new results on quadratic forms over a base field of characteristic $2$.

26 pages; v2: added result on base change of operations, added result on compatibility of Steenrod operations with specialization maps on Chow groups, added more applications to quadratic forms in characteristic 2, other minor changes and corrections

References in corpus (2)

• On the first Steenrod square for Chow groups
• Opérations de Steenrod motiviques