Cohomology of finite tensor categories: Duality and Drinfeld centers
Cohomology of finite tensor categories: Duality and Drinfeld centers
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有限张量类别的上同调：对偶性和德林菲尔德中心
DOI：
10.1090/tran/8548
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发表时间：
2022
影响因子：
1.3
通讯作者：
Plavnik, Julia
中科院分区：
文献类型：
--
作者：
Negron, Cris;Plavnik, Julia
我们考虑有限张量范畴的上同调的有限生成性质，它要求单位$\OPERATIONAME{Ext}^\Text{\Tiny$∙$}_\Mathscr{C}(\mathbf{1}，\mathbf{1})$的自扩张代数是有限生成的代数，并且对于每个对象，分次扩张群$\OPERATIONAME{Ext}\Text{\Tiny$∙$}_\Mathscr{C}(\Mathbf{1}，V)$是上述代数上的有限生成模。我们证明了这种上同调有限性质在对偶性(关于正合模范畴)和取Drinfeld中心下是保持的，并且在适当的限制下。例如，当是奇数Frobenius-Perron维辫子张量范畴时，所述结果成立。应用我们的一般结果，我们得到了一些具有有限生成上同调的有限张量范畴的新例子。在特征方面，我们证明了单位根处的动力量子群具有有限生成上同调。我们还提供了一类新的有限特征的例子，这些例子是通过无穷小群格式构造的。参考文献
We consider the finite generation property for cohomology of a finite tensor category, which requires that the self-extension algebra of the unit $\operatorname {Ext}^\text {\tiny $∙ $} _\mathscr {C}(\mathbf {1},\mathbf {1}) $ is a finitely generated algebra and that, for each objectin, the graded extension group $\operatorname {Ext}^\text {\tiny $∙ $} _\mathscr {C}(\mathbf {1}, V) $ is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on. For example, the stated result holds whenis a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes. References