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Quasi-projective dimension for complexes via filtrations

  • Hiroki Matsui
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Abstract

We extend quasi-projective dimension and quasi-projective length from finitely generated modules to homologically finite complexes by using finite filtrations in the derived category. Our definitions recover the original invariants of Gheibi--Jorgensen--Takahashi for modules and behave well under exact functors, which simplifies the proofs of several results. We establish the Auslander--Buchsbaum formula, the derived depth and width formulas, and the dependency formula for complexes of finite quasi-projective dimension. Extending a result of Gheibi--Jorgensen--Takahashi, we show that every homologically finite complex has finite quasi-projective dimension over a suitable complete intersection ring. We also prove a new intersection theorem and a descent theorem for Serre's conditions. Finally, we obtain vanishing results for Tor, Ext, and Tate (co)homology, and also symmetry of eventual Ext vanishing.

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Cite

BibTeXRIS
BibTeX
@misc{matsui2026quasiprojective,
  title = {{Quasi-projective dimension for complexes via filtrations}},
  author = {Hiroki Matsui},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.23289},
  eprint = {2609.23289},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23289},
}
RIS
TY  - UNPB
TI  - Quasi-projective dimension for complexes via filtrations
AU  - Hiroki Matsui
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2609.23289
UR  - https://arxiv.org/abs/2609.23289
AB  - We extend quasi-projective dimension and quasi-projective length from finitely generated modules to homologically finite complexes by using finite filtrations in the derived category. Our definitions recover the original invariants of Gheibi--Jorgensen--Takahashi for modules and behave well under exact functors, which simplifies the proofs of several results. We establish the Auslander--Buchsbaum formula, the derived depth and width formulas, and the dependency formula for complexes of finite quasi-projective dimension. Extending a result of Gheibi--Jorgensen--Takahashi, we show that every homologically finite complex has finite quasi-projective dimension over a suitable complete intersection ring. We also prove a new intersection theorem and a descent theorem for Serre's conditions. Finally, we obtain vanishing results for Tor, Ext, and Tate (co)homology, and also symmetry of eventual Ext vanishing.
ER  -