Auslander correspondence for higher stable dg categories and cluster Morita theory

@misc{tomonaga2026auslander,
  title = {{Auslander correspondence for higher stable dg categories and cluster Morita theory}},
  author = {Ryu Tomonaga},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30740},
  eprint = {2608.30740},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30740},
}

TY  - UNPB
TI  - Auslander correspondence for higher stable dg categories and cluster Morita theory
AU  - Ryu Tomonaga
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30740
UR  - https://arxiv.org/abs/2608.30740
AB  - The notion of d-stable dg categories axiomatizes d-cluster tilting subcategories of stable dg categories. We establish an Auslander correspondence for d-stable dg categories: we characterize the d-stability of an additive connective dg category in terms of coherence, weak global dimension, and a duality on finitely presented modules. This gives a homological characterization of d-stability and reveals it as a twisted form of (d+1)-Calabi--Yau duality. For locally finite connective dg algebras, this interpretation becomes particularly transparent under Koszul duality, where d-stability corresponds to a shifted self-injectivity condition on the Koszul dual. Following the constructions of Amiot, Guo and Keller, for a d-stable dg category M, we introduce its d-cluster dg category 𝒞_d,dg(M):=per_dgM/^L𝒟^b_fp, dg(M). Using our Auslander correspondence, we show that 𝒞_d,dg(M) contains M as a d-cluster tilting subcategory. In particular, every d-stable dg category can be realized as a d-cluster tilting subcategory of a stable dg category. We then develop cluster Morita theory: a pretriangulated dg category equipped with a d-cluster tilting subcategory M is quasi-equivalent to 𝒞_d,dg(M). Thus the connective dg structure of a cluster tilting subcategory determines its ambient dg category up to quasi-equivalence. As an application of cluster Morita theory, we prove a Morita-theoretic variant of Amiot's conjecture. More precisely, we establish a Calabi--Yau correspondence: for a locally finite d-stable dg category M over a field, right (d+1)-Calabi--Yau structures on 𝒟^b_fp, dg(M) are in bijection with right d-Calabi--Yau structures on 𝒞_d,dg(M).
ER  -