Skew power series rings over a prime base ring

2021/12/19 by Jones, Adam, Woods, William
#16W20 #16W25 #16W60 #16W70 #16W80 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2112.10242

Abstract

In this paper, we investigate the structure of skew power series rings of the form S = R[[x;σ,δ]], where R is a complete filtered ring and (σ,δ) is a skew derivation respecting the filtration. Our main focus is on the case in which σδ= δσ, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if P is an invariant prime ideal of R, is PS a prime ideal of S? When R has characteristic p, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case δ= σ- id_R in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is "almost" (σ,δ)-invariant in a certain sense.

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