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Trace ideals of exterior powers of the module of differentials

  • Ryo Ishizuka
  • Sora Miyashita
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Abstract

For each i ≥ 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.

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Cite

BibTeXRIS
BibTeX
@misc{ishizuka2026trace,
  title = {{Trace ideals of exterior powers of the module of differentials}},
  author = {Ryo Ishizuka and Sora Miyashita},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.00418},
  eprint = {2605.00418},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.00418},
}
RIS
TY  - UNPB
TI  - Trace ideals of exterior powers of the module of differentials
AU  - Ryo Ishizuka
AU  - Sora Miyashita
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2605.00418
UR  - https://arxiv.org/abs/2605.00418
AB  - For each i ≥ 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.
ER  -