We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré-Birkhoff-Witt proper...

I will describe a functor A 7→ W (A) from the category of commutative rings to itself. The ring W (A) of ‘Witt vectors’ over A has many applications (to algebraic geometry, local rings, etc.), but I won’t discuss those. Convention: rings have 1’s that are respected by ring homomorphisms. By A I will always denote a commutative ring. The literature on the functor W is in a somewhat unsatisfactor...

Abstract Fix an odd prime $p$. The results in this paper are modeled after work of Hesselholt and Hesselholt–Madsen on the $p$-typical absolute de Rham–Witt complex mixed characteristic. We have two primary results. 1st result is exact sequence that describes kernel restriction map over $A$, where $A$ ring integers algebraic extension $\textbf{Q}_p$ or a $p$-torsion-free perfectoid ring. 2nd de...