Recollements of derived categories from n-term big tilting complexes
Abstract
Let A be a ring, let bf T be a finite-term big tilting complex over A, and choose a bounded projective representative cpxP of bf T. Put B=EndDA(cpxP). We prove that the kernel of the associated derived tensor functor bf R:DB→DA is equivalent, through the derived inclusion, to Dℰ, where ℰ=Ker(bf R)∩ BModcat carries the inherited exact structure. Consequently, DB admits a recollement by Dℰ and DA. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category ℰ is definable and d-symmetric whenever d is at least the projective amplitude of the underlying perfect right B-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when ℰ is an abelian subcategory, or equivalently when the standard t-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.
Related
- A negative answer to a question on tilting objects and two-term complexes
- Recollements of partially wrapped Fukaya categories and surface cuts
- Flat and Ghost dimensions in tensor triangulated categories with a compact tilting generator
- Relative tilting, interlacing categories and rational Dyck posets
- Relative Derived Delooping Levels and τ-Tilting Modules
BibTeX
@misc{pan2026recollements,
title = {{Recollements of derived categories from $n$-term big tilting complexes}},
author = {Shengyong Pan and Huabo Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19921},
eprint = {2608.19921},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19921},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recollements of derived categories from n-term big tilting complexes
AU - Shengyong Pan
AU - Huabo Xu
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2608.19921
UR - https://arxiv.org/abs/2608.19921
AB - Let A be a ring, let bf T be a finite-term big tilting complex over A, and choose a bounded projective representative cpxP of bf T. Put B=End_DA(cpxP). We prove that the kernel of the associated derived tensor functor bf R:DB→DA is equivalent, through the derived inclusion, to Dℰ, where ℰ=Ker(bf R)∩ BModcat carries the inherited exact structure. Consequently, DB admits a recollement by Dℰ and DA. The proof combines a dg double-centralizer argument with a functorial realisation of the kernel on unbounded complexes. The category ℰ is definable and d-symmetric whenever d is at least the projective amplitude of the underlying perfect right B-complex. We also characterise when the kernel is induced by a homological ring epimorphism: this holds precisely when ℰ is an abelian subcategory, or equivalently when the standard t-structure restricts to the kernel. In amplitude at most one, this recovers the universal-localisation description for two-term big tilting complexes. Over semiperfect endomorphism rings, the amplitude is detected by reduction modulo the Jacobson radical. Finally, we construct non-compact examples of arbitrarily large minimal projective amplitude. One family yields algebraic Calkin quotients; another, over connected rings constructed from singular local hypersurfaces, has nonzero, non-abelian exact kernels and cannot be obtained from modules by derived equivalences between ordinary rings.
ER -