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PreprintarXiv licencearXiv

Filtrations on Local Cohomology and Injectivity Theorems

  • Qianyu Chen
  • Bradley Dirks
  • Sebastian Olano
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Abstract

We prove that the Hodge filtration on local cohomology is contained in the symbolic Ext filtration, and hence in the usual Ext filtration, answering a question of Mustata and Popa. Our main observation is that the Ext filtration admits a filtered-duality interpretation via Hartshorne's algebraic de Rham complex and its infinitesimal filtration. Comparing this construction with the filtered Du Bois complex yields the containment. The same mechanism proves the higher injectivity conjecture of Popa--Shen--Vo and several variants.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026filtrations,
  title = {{Filtrations on Local Cohomology and Injectivity Theorems}},
  author = {Qianyu Chen and Bradley Dirks and Sebastian Olano},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16196},
  eprint = {2609.16196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16196},
}
RIS
TY  - UNPB
TI  - Filtrations on Local Cohomology and Injectivity Theorems
AU  - Qianyu Chen
AU  - Bradley Dirks
AU  - Sebastian Olano
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.16196
UR  - https://arxiv.org/abs/2609.16196
AB  - We prove that the Hodge filtration on local cohomology is contained in the symbolic Ext filtration, and hence in the usual Ext filtration, answering a question of Mustata and Popa. Our main observation is that the Ext filtration admits a filtered-duality interpretation via Hartshorne's algebraic de Rham complex and its infinitesimal filtration. Comparing this construction with the filtered Du Bois complex yields the containment. The same mechanism proves the higher injectivity conjecture of Popa--Shen--Vo and several variants.
ER  -