Dimension Theory in Iterated Local Skew Power Series Rings

Abstract

Many well-known local rings, including soluble Iwasawa algebras and certain completed quantum algebras, arise naturally as iterated skew power series rings. We calculate their Krull and global dimensions, obtaining lower bounds to complement the upper bounds obtained by Wang. In fact, we show that many common such rings obey a stronger property, which we call triangularity, and which allows us also to calculate their classical Krull dimension (prime length). Finally, we correct an error in the literature regarding the associated graded rings of general iterated skew power series rings, but show that triangularity is enough to recover this result.