A Krause-type recollement for Q-shaped derived categories

Abstract

Let k be a commutative noetherian hereditary ring and let Q be a small, Hom-finite, locally bounded k-linear category admitting a Serre functor and satisfying the strong retraction property, with nilpotent pseudo-radical. For every k-algebra A, we construct a recollement of the Q-shaped homo?topy category KQ(A Inj) of objectwise injective Q-shaped modules by its full subcategory Kac Q (A Inj) of exact objects and the Q-shaped derived category DQ(A). The construction uses complete hereditary cotorsion pairs and does not require A to be noetherian. If A is left noetherian, both KQ(A Inj) and Kac Q (A Inj) are compactly generated by special preenvelopes of finitely generat?ed stalk modules. We identify (Kac Q (A Inj))c with the idempotent completion of (KQ(A Inj))c/LI ((DQ(A))c ), where LI is the fully faithful left adjoint of the localization. This paper builds on and generalizes works of Krause and Liu

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BibTeX RIS

BibTeX

@misc{ma2026krausetype,
  title = {{A Krause-type recollement for Q-shaped derived categories}},
  author = {Yajun Ma and Peiru Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21507},
  eprint = {2610.21507},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21507},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A Krause-type recollement for Q-shaped derived categories
AU  - Yajun Ma
AU  - Peiru Yang
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21507
UR  - https://arxiv.org/abs/2610.21507
AB  - Let k be a commutative noetherian hereditary ring and let Q be a small, Hom-finite, locally bounded k-linear category admitting a Serre functor and satisfying the strong retraction property, with nilpotent pseudo-radical. For every k-algebra A, we construct a recollement of the Q-shaped homo?topy category KQ(A Inj) of objectwise injective Q-shaped modules by its full subcategory Kac Q (A Inj) of exact objects and the Q-shaped derived category DQ(A). The construction uses complete hereditary cotorsion pairs and does not require A to be noetherian. If A is left noetherian, both KQ(A Inj) and Kac Q (A Inj) are compactly generated by special preenvelopes of finitely generat?ed stalk modules. We identify (Kac Q (A Inj))c with the idempotent completion of (KQ(A Inj))c/LI ((DQ(A))c ), where LI is the fully faithful left adjoint of the localization. This paper builds on and generalizes works of Krause and Liu
ER  -

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