AI 中文总结
本文研究 Hopfological 代数中标准胞腔构成简单-minded 集合的条件，构造有界 t-结构和权结构，并证明在更强间隙下字面细胞性成立，同时展示额外分次假设的必要性。
AI 中文摘要
设 $H$ 为一个非平凡有限维非负分次连通 Hopf 代数，其中 $\ell=\max\{d:H_d\ne0\}$，并设 $A\ge0$ 为一个局部有限 $H$-模代数，其 $A_0$ 为有限维分裂半单且 $H$-平凡。在间隙条件 $A_d=0$（对 $0<d<\ell$）下，平移标准胞腔 $\{q^{-r}Ae_x:0\le r<\ell\}$ 构成一个简单-minded 集合。对正合冲模的 socle 估计证明了负-Hom 消失性，而间隙条件使零次自同态代数半单。利用这些胞腔，在紧致导出范畴上构造了一个有界 $t$-结构，其简单心对象即为标准胞腔。同时证明了每个紧致对象都有一个有限胞腔表示，且紧致 $K_0$ 在 $\mathbb{O}_H=K_0(H\mbox{-}\underline{\mathrm{gmod}})$ 上以 $[Ae_x]$ 为基自由。相同的连通 Hopf 假设在大导出范畴上给出了 Keller--Nicolás 权结构，但不声称有界性或紧致对象的保持。间隙条件既不强制存在 silting 生成子，也不强制紧致对象上的有界权结构。这些论证既不要求 $H$ 余交换，也不要求其有限表示型。$p$-DG 情形，其中 $\mathrm{deg}(\partial)=2$ 且 $\ell=2p-2$，作为特例处理。另外，在更强的间隙条件 $A_1=\cdots=A_\ell=0$ 下，每个有限生成分次投射 hopfological 模的字面细胞性成立。显式的 $p$-DG 高阶循环收缩和代数值迹说明了为何需要额外的分次假设；特别是，去掉间隙条件可能在紧致导出范畴的 Grothendieck 群中产生非零 $p$-挠。
英文摘要
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.
Comments39 pages. Comments welcome
