Finite-dimensional vector spaces with Frobenius action
The early sections of this paper contain a survey of classical results concerning ideals, matrices and modules over noncommutative PIDs. Like their commutative counterparts, these rings are a well-studied subject and there is a similar structure theorem for their modules. The author's interest in this subject stems from the fact that one may naturally identify vector spaces admitting a Frobenius action with modules over a skew-polynomial ring. In case the underlying field is perfect, this skew-polynomial ring is a noncommutative PID.NEWLINENEWLINEThis identification allows one to bring this classical theory into use on current problems concerning vector spaces over rings of positive characteristic. These survey sections provide a brief but thorough and self-contained introduction to the theory noncommutative PIDs for commutative algebraists working with rings of positive characteristic. Also included are ample references for additional consultation and several illustrative examples.NEWLINENEWLINEThe final section of the paper provides applications of the survey material to current research questions. This section nicely shows how the classical theory can be utilized to tackle new problems. In particular, the author focuses on connections with antinilpotent modules and makes a strong case that these tools can be used to help in the study of F-injectivity for generalized Cohen-Macaulay rings. The paper concludes with a proof of the following result along these lines:NEWLINENEWLINE\noindent Assume that \((A, \mathfrak{m})\) is a local, complete domain with a perfect residue field. If \(A\) is F-injective, generalized Cohen-Macaulay, and the test ideal of \(A\) is \(\mathfrak{m}\)-primary, then \(A\) is FH-finite.NEWLINENEWLINEFor the entire collection see [Zbl 1237.13006].
scientific article; zbMATH DE number 6052218