修正迹与Nakayama函子
Modified traces and the Nakayama functor
摘要：
我们组织的修改后的迹理论与使用的Nakayama函子有限阿贝尔范畴。对于有限阿贝尔范畴$\mathcal{M}$上的线性右正合函子$Σ$，我们在$\mathcal{M}$的投射对象类$\mathrm{Proj}(\mathcal{M})$上引入了$Σ$ -扭曲迹的概念.在我们的框架中，$\mathrm{Proj}(\mathcal{M})$上的$Σ$扭曲迹集与从$Σ$到$\mathcal{M}$的Nakayama函子的自然变换集之间存在一一对应关系.当$\mathcal{M}$是有限张量范畴上的模范畴时，$Σ$ -扭曲迹的非退化性和与模结构的相容性可以用相应的自然变换来表示。作为这一原理的应用，我们给出了修正迹的存在唯一性准则。特别是，么模枢轴有限张量范畴允许非零双侧修改迹当且仅当它是球形的。另外，带状有限张量范畴允许这样的迹当且仅当它是么模的。
英文摘要：
We organize the modified trace theory with the use of the Nakayama functor of finite abelian categories. For a linear right exact functor $Σ$ on a finite abelian category $\mathcal{M}$, we introduce the notion of a $Σ$-twisted trace on the class $\mathrm{Proj}(\mathcal{M})$ of projective objects of $\mathcal{M}$. In our framework, there is a one-to-one correspondence between the set of $Σ$-twisted traces on $\mathrm{Proj}(\mathcal{M})$ and the set of natural transformations from $Σ$ to the Nakayama functor of $\mathcal{M}$. Non-degeneracy and compatibility with the module structure (when $\mathcal{M}$ is a module category over a finite tensor category) of a $Σ$-twisted trace can be written down in terms of the corresponding natural transformation. As an application of this principal, we give existence and uniqueness criteria for modified traces. In particular, a unimodular pivotal finite tensor category admits a non-zero two-sided modified trace if and only if it is spherical. Also, a ribbon finite tensor category admits such a trace if and only if it is unimodular.
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