Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field. The formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the Hochschild cohomology of the original category. We also prove that Serre functors act trivially on Hochschild cohomology, and combine this with our formula to give a useful mechanism for computing the Hochschild cohomology of fractional Calabi–Yau categories.

中文翻译：

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Hochschild 上同调和群作用

给定对域上线性（适当增强的）三角化类别的有限群作用，我们建立不变量类别的 Hochschild 上同调公式，假设群的阶与基域的特征互质。该公式表明，上同调以原始范畴的 Hochschild 上同调的不变子空间给出的一个被加数进行规范分裂。我们还证明了 Serre 函子对 Hochschild 上同调的作用微不足道，并将其与我们的公式相结合，给出了计算分数 Calabi-Yau 类别的 Hochschild 上同调的有用机制。