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Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients

  • Yizhi Zhang
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Abstract

Let k be a field of characteristic zero, let R be a commutative k-algebra, let I be a proper ideal of R, and assume that A=R/I is Noetherian. We prove that if a=[f] in A is nilpotent and dA/ka=0, then I:f is not contained in (I,f). If R is local with maximal ideal n, then I:f is not contained in (I,f)+n(I:f). As applications, we prove the isolated hypersurface case in arbitrary dimension and extend the conclusion to non-isolated critical loci on smooth varieties. Finally, we exhibit a five-variable isolated singularity satisfying Jf:f⊆Jf̅, thereby disproving the integral-closure strengthening.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026jordan,
  title = {{Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients}},
  author = {Yizhi Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.29516},
  eprint = {2609.29516},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.29516},
}
RIS
TY  - UNPB
TI  - Jordan Blocks, Differential Traces, and Colon-Ideal Noncontainment over Noetherian Quotients
AU  - Yizhi Zhang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.29516
UR  - https://arxiv.org/abs/2609.29516
AB  - Let k be a field of characteristic zero, let R be a commutative k-algebra, let I be a proper ideal of R, and assume that A=R/I is Noetherian. We prove that if a=[f] in A is nilpotent and d_A/ka=0, then I:f is not contained in (I,f). If R is local with maximal ideal n, then I:f is not contained in (I,f)+n(I:f). As applications, we prove the isolated hypersurface case in arbitrary dimension and extend the conclusion to non-isolated critical loci on smooth varieties. Finally, we exhibit a five-variable isolated singularity satisfying J_f:f⊆J_f̅, thereby disproving the integral-closure strengthening.
ER  -