Stable Cellularity in Hopfological Algebra

Abstract

Let $H$ be a nontrivial finite-dimensional nonnegatively graded connected Hopf algebra, with $\ell=\max\{d:H_d\ne0\}$, and let $A\ge0$ be a locally finite $H$-module algebra with finite-dimensional split semisimple, $H$-trivial $A_0$. Under the gap $A_d=0$ for $0<d<\ell$, the shifted standard cells $\{q^{-r}Ae_x:0\le r<\ell\}$ 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 $\mathbb{O}_H=K_0(H\mbox{-}\underline{\mathrm{gmod}})$. 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 $\mathrm{deg}(\partial)=2$ and $\ell=2p-2$, is treated as a specialization. Separately, under the stronger gap condition $A_1=\cdots=A_\ell=0$, 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.