Stable Cellularity in Hopfological Algebra
Abstract
Let H be a nontrivial finite-dimensional nonnegatively graded connected Hopf algebra, with ℓ=max{d:Hd≠0}, and let A≥0 be a locally finite H-module algebra with finite-dimensional split semisimple, H-trivial A0. Under the gap Ad=0 for 0<d<ℓ, the shifted standard cells {q−rAex:0≤ r<ℓ} form a simple-minded collection. A socle estimate for positive syzygies proves negative-Hom vanishing, while the gap makes the degree-zero endomorphism algebra semisimple. A bounded t-structure on the compact derived category with these cells is constructed, with standard cells as its simple heart objects. It is also shown that every compact object has a finite-cell representative and that compact K0 is free on [Aex] over 𝕆H=K0(H−underlinegmod). The same connected-Hopf hypotheses give a Keller--Nicolás weight structure on the large derived category, without a claim of boundedness or preservation of compact objects. Neither a silting generator nor a bounded weight structure on compacts is forced by the gap. These arguments require neither cocommutativity nor finite representation type of H. The p-DG case, with deg(∂)=2 and ℓ=2p−2, is treated as a specialization. Separately, under the stronger gap condition A1=⋯=Aℓ=0, literal cellularity of every finitely generated graded-projective hopfological module holds. Explicit p-DG higher-cycle retracts and algebra-valued traces show why additional grading hypotheses are needed; in particular, dropping the gap can produce nonzero p-torsion in the Grothendieck group of compact derived categories.
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BibTeX
@misc{qi2026stable,
title = {{Stable Cellularity in Hopfological Algebra}},
author = {You Qi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28295},
eprint = {2609.28295},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28295},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stable Cellularity in Hopfological Algebra
AU - You Qi
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28295
UR - https://arxiv.org/abs/2609.28295
AB - Let H be a nontrivial finite-dimensional nonnegatively graded connected Hopf algebra, with ℓ=max{d:H_d≠0}, and let A≥0 be a locally finite H-module algebra with finite-dimensional split semisimple, H-trivial A_0. Under the gap A_d=0 for 0<d<ℓ, the shifted standard cells {q^−rAe_x:0≤ r<ℓ} form a simple-minded collection. A socle estimate for positive syzygies proves negative-Hom vanishing, while the gap makes the degree-zero endomorphism algebra semisimple. A bounded t-structure on the compact derived category with these cells is constructed, with standard cells as its simple heart objects. It is also shown that every compact object has a finite-cell representative and that compact K_0 is free on [Ae_x] over 𝕆_H=K_0(H−underlinegmod). The same connected-Hopf hypotheses give a Keller--Nicolás weight structure on the large derived category, without a claim of boundedness or preservation of compact objects. Neither a silting generator nor a bounded weight structure on compacts is forced by the gap. These arguments require neither cocommutativity nor finite representation type of H. The p-DG case, with deg(∂)=2 and ℓ=2p−2, is treated as a specialization. Separately, under the stronger gap condition A_1=⋯=A_ℓ=0, literal cellularity of every finitely generated graded-projective hopfological module holds. Explicit p-DG higher-cycle retracts and algebra-valued traces show why additional grading hypotheses are needed; in particular, dropping the gap can produce nonzero p-torsion in the Grothendieck group of compact derived categories.
ER -