下降霍普夫代数的稳定Picard群及其应用
The stable Picard group of Hopf algebras via descent, and an application
摘要：
设$A$是二元域上的有限维余交换霍普夫代数，满足一定的条件.我们建立了一个下降谱序列来计算$A$上模的稳定范畴的Picard群。我们的出发点是观察到$A$ -模的稳定范畴可以被重构为$\infty$ -范畴，作为上单纯$\infty$ -范畴的总和，其层与$A$的准初等子Hopf-代数上的模的稳定范畴有关。这导致计算皮卡德群的谱序列，在某些情况下，皮卡德群是完全理解的。这也导致一个谱序列回答提升问题的类别中的$A$ -模块。然后，我们展示了如何应用这种机器来计算皮卡德群和解决提升问题的情况下，$\mathcal{A}(1)$ -模块，其中$\mathcal{A}(1)$是子代数的Steenrod代数所产生的两个第一Steenrod广场。
英文摘要：
Let $A$ be a cocommutative finite dimensional Hopf algebra over the field with two elements, satisfying some mild hypothesis. We set up a descent spectral sequence which computes the Picard group of the stable category of modules over $A$. The starting point is the observation that the stable category of $A$-modules can be reconstructed, as an $\infty$-category, as the totalization of a cosimplicial $\infty$-category whose layers are related to the stable categories of modules over the quasi-elementary sub-Hopf-algebras of $A$. This leads to a spectral sequence computing the Picard group which, in some cases, is completely understood. This also leads to a spectral sequence answering a lifting problem in the category of $A$-modules. We then show how to apply this machinery to compute Picard groups and solve lifting problems in the case of $\mathcal{A}(1)$-modules, where $\mathcal{A}(1)$ is the subalgebra of the Steenrod algebra generated by the two first Steenrod squares.
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