沿motivic形变生成假设
Generating hypotheses along the motivic deformation
- UCLA Department of Mathematics（加州大学洛杉矶分校数学系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文否证了代数生成假设，构造了具有特定阶和合成性质的幽灵，并通过motivic形变给出反例。
AI中文摘要：
在每个素数 $p$ 处，我们否证了 Hovey 的 $BP_*BP$-余模稳定范畴中紧对象的代数生成假设，并否证了其在每个正高度处的局部类似物，回答了 Barthel 和 Heard 的问题。在这两种设置中，对于每个 $r,n\geq1$，我们构造了一个幽灵，其前 $n$ 个合成幂均非零且具有精确的加法阶 $p^r$。同样的构造适用于偶数 $p$-局部 Landweber 精确理论 $E$（非有理）的 $E_*E$-余模稳定范畴。通过 Gheorghe--Wang--Xu 的 motivic 形变，这些例子给出了 $C\tau$-线性反例，反驳了细胞 $\nathbb{C}$-motivic 生成假设，具有相同的阶和合成性质。我们还构造了 $C\tau$-模之间的非零非 $C\tau$-线性幽灵，以及一个具有非可缩 Betti 实现的 motivic 谱上的非零幽灵。
英文摘要：
At every prime $p$, we disprove the algebraic generating hypothesis for compact objects in Hovey's stable category of $BP_*BP$-comodules and its local analog at every positive height, answering questions of Barthel and Heard. In both settings, for every $r,n\geq1$, we construct a ghost whose first $n$ composition powers are all nonzero and have exact additive order $p^r$. The same construction applies to the stable category of $E_*E$-comodules for even $p$-local Landweber exact theories $E$ that are not rational. Through the motivic deformation of Gheorghe--Wang--Xu, these examples give $Cτ$-linear counterexamples to the cellular $\mathbb{C}$-motivic generating hypothesis, with the same order and composition properties. We also construct nonzero non-$Cτ$-linear ghosts between $Cτ$-modules and a nonzero ghost on a motivic spectrum with noncontractible Betti realization.
