Good Reduction of Good Filtrations at Places

Abstract

We consider filtered or graded algebras $A$ over a field $K$. Assume that there is a discrete valuation $O_v$ of $K$ with $m_v$ its maximal ideal and $k_v:=O_v/m_v$ its residue field. Let $Λ$ be $O_v$-order such that $ΛK=A$ and $\barΛ:=k_v\otimes_{O_v}Λ$ the $Λ$-reduction of $A$ at the place $K\leadsto k_v$. Using the filtration of $A$ induced by $Λ$ we shall prove that for certain algebras $A$ their properties are related to $\barΛ$.