We advance support variety theory for finite tensor categories. First we show that the dimension of the support variety of an object equals the rate of growth of a minimal projective resolution as measured by the Frobenius-Perron dimension. Then we show that every conical subvariety of the support variety of the unit object may be realized as the support variety of an object. Finally, we show that the support variety of an indecomposable object is connected.

中文翻译：

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支持有限张量类别的变体：复杂性，实现和连通性

我们提出有限张量类别的支持多样性理论。首先，我们证明了对象的支持量的维数等于最小投射分辨率的增长率，如Frobenius-Perron维数所衡量的。然后我们表明，单位对象的支持变体的每个圆锥形子变体都可以实现为对象的支持变体。最后，我们证明了不可分解对象的支持种类是相连的。