Year: 2026
Type: preprint
Abstract: For each $i \geq 0$, we study the trace ideal of the $i$-th exterior power of the module of differentials. In characteristic zero, we show that these ideals characterize the polynomial rank of graded rings and the formal power series rank of complete local rings with finite residue-field extension, namely the maximal number of variables for a polynomial or formal power series extension over a subring. Moreover, we introduce the top differential trace and prove that it precisely defines the singular locus of reduced equidimensional local or graded rings. Motivated by this, we introduce and investigate nearly regular rings, which are rings whose top differential trace contains the maximal ideal.
Source: arXiv (Cornell University)
Authors: Ishizuka, Ryo, Miyashita, Sora
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7160134844