On the Frobenius functor for symmetric tensor categories in positive characteristic (original) (raw)

We develop a theory of Frobenius functors for symmetric tensor categories (STC) C over a field k of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F W C ! C Ver p , where Ver p is the Verlinde category (the semisimplification of Rep k .Z=p/); 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), no. 3, Paper No. 36], where it is used to show that if C is finite and semisimple, then it admits a fiber functor to Ver p. The main new feature is that when C is not semisimple, 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 C ! Ver p. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of 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 C ex inside any STC C with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius-Perron dimension is preserved by 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 Ver p. This is the strongest currently available characteristic 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 C 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 kOEd =d 2 with d primitive and R-matrix R D 1˝1 C d˝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].