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

@article{Etingof2019OnTF,
title={On the Frobenius functor for symmetric tensor categories in positive characteristic},
author={Pavel Etingof and Victor Ostrik},
journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
year={2019},
volume={2021},
pages={165 - 198}
}
• Published 30 December 2019
• Mathematics
• Journal für die reine und angewandte Mathematik (Crelles Journal)
Abstract We develop a theory of Frobenius functors for symmetric tensor categories (STC) 𝒞 {\mathcal{C}} over a field 𝒌 {\boldsymbol{k}} of characteristic p, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor F : 𝒞 → 𝒞 ⊠ 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 𝐤 ( ℤ / p ) {\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k…
12 Citations
• Mathematics
Duke Mathematical Journal
• 2023
We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to
• Mathematics
Journal für die reine und angewandte Mathematik (Crelles Journal)
• 2022
Abstract We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an
We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new
• Mathematics
• 2022
. Let G be the group of all order-preserving self-maps of the real line. In previous work, the ﬁrst two authors constructed a pre-Tannakian category
• Mathematics
• 2022
We extend [G1] to the nonsemisimple case. We define and study exact factorizations B = A • C of a finite tensor category B into a product of two tensor subcategories A ,C ⊂ B, and relate exact
• Mathematics
Algebra &amp; Number Theory
• 2022
We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. Benson and Etingof conjectured that a certain limit of finite
• Mathematics
• 2021
. A fundamental theorem of P. Deligne (2002) states that a pre- Tannakian category over an algebraically closed ﬁeld of characteristic zero admits a ﬁber functor to the category of supervector spaces
• Mathematics
• 2021
This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in

## References

SHOWING 1-10 OF 34 REFERENCES

• Mathematics
International Mathematics Research Notices
• 2019
We prove that every finite symmetric integral tensor category $\mathcal{C}$ with the Chevalley property over an algebraically closed field $k$ of characteristic $p>2$ admits a symmetric fiber
• Mathematics
Journal of Algebra and Its Applications
• 2020
We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic
• Mathematics
• 2018
We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite
In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive
• Mathematics
• 2003
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our
• Mathematics
• 2015
We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive
• Mathematics
• 1993
This paper is a part of the series [KL]; however, it can be read independently of the first two parts. In [D3], Drinfeld proved the existence of an equivalence between a tensor category of
Part I. General theory: Schemes Group schemes and representations Induction and injective modules Cohomology Quotients and associated sheaves Factor groups Algebras of distributions Representations