For C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that the tensor ideal of projective objects in C carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of S-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular S-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

中文翻译：

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可分解有限张量范畴中的射影对象和修正迹

对于 C 是特征为零的代数闭域上的可因式分解且关键的有限张量范畴，我们证明： 1) C 总是包含一个简单的射影对象；2) 如果 C 是额外的色带，则射影模的内部特征跨越射影 SL(2,Z)-动作的子模；3）C的格洛腾迪克环对射影对象内部特征的跨度的作用可以对角化；4) C 的线性化格洛腾迪克环是半单的，当仅当 C 是半单的。在额外的假设下，结果 1-3 在正特性方面仍然是正确的。结果 1 意味着 C 中射影对象的理想张量带有唯一的高达标量修正的跟踪函数。我们用 S 矩阵元素表示由射影着色的开放 Hopf 链接的修改迹。此外，我们给出了一个类似 Verlinde 的公式，用于分解射影对象的张量积，它仅使用仅限于射影对象内部特征的模 S 变换。我们在辛费米子类别的例子中计算修改后的轨迹，并说明了如何在那里应用射影对象的 Verlinde-like 公式。