A Relative Telescope-Type Conjecture: Support and Generation in Half-Bounded Derived Categories

@misc{zhu2026relative,
  title = {{A Relative Telescope-Type Conjecture: Support and Generation in Half-Bounded Derived Categories}},
  author = {Liangyu Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.04690},
  eprint = {2611.04690},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.04690},
}

TY  - UNPB
TI  - A Relative Telescope-Type Conjecture: Support and Generation in Half-Bounded Derived Categories
AU  - Liangyu Zhu
PY  - 2026
DA  - 2026/11/03
PB  - arXiv
DO  - 10.48550/arxiv.2611.04690
UR  - https://arxiv.org/abs/2611.04690
AB  - Let R be a commutative noetherian ring. We ask whether complexes with finitely generated cohomology and a fixed support can be generated by relatively compact objects, using coproducts which exist in the half-bounded derived category. We prove that D^+_fg,W(R)=Loc_ex(D^b_fg,W(R)) whenever W⊆Spec R is specialization closed and has finite dimension. The same equality holds for every specialization-closed subset of Spec B when B is a finite-type algebra over a noetherian domain A of finite character and either A or B is regular. These equalities classify the thick Hom-submodules closed under existing coproducts. We also prove the corresponding bounded-above statement from the support results of Matsui--Takahashi. For an arbitrary noetherian ring, the bounded-below generation conjecture reduces to torsion complexes over quotient domains, with their support unchanged. Finally, over a regular noetherian domain of infinite dimension, we give a bounded-below object which is generated by bounded objects with the same support but has no bounded-below representative by finitely generated modules. The comparison functor from the bounded-below derived category of finite modules is therefore neither full nor essentially surjective.
ER  -