Truncation of dg categories and connective resolutions

@misc{hanihara2026truncation,
  title = {{Truncation of dg categories and connective resolutions}},
  author = {Norihiro Hanihara},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.08218},
  eprint = {2610.08218},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.08218},
}

TY  - UNPB
TI  - Truncation of dg categories and connective resolutions
AU  - Norihiro Hanihara
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.08218
UR  - https://arxiv.org/abs/2610.08218
AB  - We study the truncation (−)^≤0 of dg categories. We first show that given a dg category 𝒞 with shifts, for example a pretriangulated dg category, the canonical functor 𝒞^≤0→𝒞 is a localization whose kernel is compactly generated by the $0$-th cohomology. Next we demonstrate that the derived category of the truncation serves as a triangulated analogue of the Auslander's category of coherent functors over abelian categories. We give a description of the Auslander-Reiten-Serre duality in terms of the inverse dualizing bimodule of the truncation. Building on truncations, we introduce the notion of connective resolutions of dg categories, defined as a localization functor from a connective dg category. This notion encompasses non-commutative resolutions of algebras in module categories, cluster tilting objects in triangulated categories, and also the truncation as the universal connective resolution. We give a sufficient condition for an object in a triangulated category to give a connective resolution of its enhancement in terms of finiteness of resolution dimension. Furthermore, we show that every proper dg module over a connective dg algebra is a direct summand of a dg module giving a connective resolution. Consequently, for every proper connective dg algebra, its bounded dg derived category has a connective resolution. Finally, we give some explicit description of the truncation for some categories including the derived and cluster categories of a Dynkin quiver, and the Yoneda category of a finite dimensional algebra.
ER  -