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**On the Frobenius functor for symmetric tensor categories in positive characteristic**

*Journal für die reine und angewandte Mathematik*( IF 1.486 )

**Pub Date : 2020-10-08**

*, DOI:*

*10.1515/crelle-2020-0033*

Pavel Etingof; Victor Ostrik

We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bm{k}$ of characteristic $F:\mathcal{C}\to \mathcal{C}\u22a0{\mathrm{Ver}}_{p}$ , where ${\mathrm{Ver}}_{p}$ is the Verlinde category (the semisimplification of ${Rep}_{\mathbf{k}}\left(\mathbb{Z}/p\right)$ ); a similar construction of the underlying additive functor appeared independently in [K. Coulembier,
Tannakian categories in positive characteristic,
preprint 2019]. This generalizes the usual Frobenius twist functor in modular representation theory and also the one defined in [V. Ostrik,
On symmetric fusion categories in positive characteristic,
Selecta Math. (N.S.) 26 2020, 3, Paper No. 36], where it is used to show that if $\mathcal{C}$ is finite and semisimple, then it admits a fiber functor to ${\mathrm{Ver}}_{p}$ . The main new feature is that when $\mathcal{C}$ is not semisimple, $\mathcal{C}\to {\mathrm{Ver}}_{p}$ . We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of ${\mathcal{C}}_{\mathrm{ex}}$ inside any STC $\mathcal{C}$ with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by ${\mathrm{Ver}}_{p}$ . This is the strongest currently available characteristic ${\mathcal{C}}_{\mathrm{ex}}$ . Also, in characteristic 2 we introduce a slightly weaker notion of an almost Frobenius exact category (namely, one having a fiber functor into the category of representations of the triangular Hopf algebra $\bm{k}\left[d\right]/{d}^{2}$ with $R=1\otimes 1+d\otimes d$ ), and show that a STC with Chevalley property is (almost) Frobenius exact. Finally, as a by-product, we resolve Question 2.15 of [P. Etingof and S. Gelaki,
Exact sequences of tensor categories with respect to a module category,
Adv. Math. 308 2017, 1187–1208].

更新日期：2020-10-20
*p*, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor*F*need not be left or right exact, and in fact this lack of exactness is the main obstruction to the existence of a fiber functor*F*, and use it to show that for categories with finitely many simple objects*F*does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which*F*is exact, and define the canonical maximal Frobenius exact subcategory*F*. One of our main results is that a finite STC is Frobenius exact if and only if it admits a (necessarily unique) fiber functor to*p*version of Deligne’s theorem (stating that a STC of moderate growth in characteristic zero is the representation category of a supergroup). We also show that a sufficiently large power of*F*lands in*d*primitive and*R*-matrix