On the Frobenius functor for symmetric tensor categories in positive characteristic
Name
10.1515_crelle-2020-0033.pdf
Description
Published version
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450.52 KB
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Adobe PDF
Checksum (MD5)
6654f4e89a7d4ee308471a0623d2d997
Author(s)
Etingof, Pavel
Ostrik, Victor
Date Issued
2021
Journal
Journal fรผr die reine und angewandte Mathematik (Crelles Journal)
Publisher
Walter de Gruyter GmbH
Version
Final published version
Abstract
Abstract
We develop a theory of Frobenius functors for symmetric tensor categories (STC) ๐ {\mathcal{C}} over a field ๐ {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : ๐ โ ๐ โ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep ๐ค ( โค / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{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, 3, Paper No. 36], where it is used to show that if ๐ {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to Ver p {{\rm Ver}_{p}} . The main new feature is that when ๐ {\mathcal{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 ๐ โ Ver p {\mathcal{C}\to{\rm 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 FrobeniusโPerron 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 ๐ ex {\mathcal{C}_{\rm ex}} inside any STC ๐ {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose FrobeniusโPerron 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 {{\rm Ver}_{p}} . This is the strongest currently available characteristic p version of Deligneโs 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 ๐ ex {\mathcal{C}_{\rm 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 ๐ โข [ d ] / d 2 {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix R = 1 โ 1 + d โ d {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, 1187โ1208].
We develop a theory of Frobenius functors for symmetric tensor categories (STC) ๐ {\mathcal{C}} over a field ๐ {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : ๐ โ ๐ โ Ver p {F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}} , where Ver p {{\rm Ver}_{p}} is the Verlinde category (the semisimplification of Rep ๐ค ( โค / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{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, 3, Paper No. 36], where it is used to show that if ๐ {\mathcal{C}} is finite and semisimple, then it admits a fiber functor to Ver p {{\rm Ver}_{p}} . The main new feature is that when ๐ {\mathcal{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 ๐ โ Ver p {\mathcal{C}\to{\rm 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 FrobeniusโPerron 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 ๐ ex {\mathcal{C}_{\rm ex}} inside any STC ๐ {\mathcal{C}} with finitely many simple objects. Namely, this is the subcategory of all objects whose FrobeniusโPerron 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 {{\rm Ver}_{p}} . This is the strongest currently available characteristic p version of Deligneโs 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 ๐ ex {\mathcal{C}_{\rm 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 ๐ โข [ d ] / d 2 {\boldsymbol{k}[d]/d^{2}} with d primitive and R-matrix R = 1 โ 1 + d โ d {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, 1187โ1208].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1515/CRELLE-2020-0033
