Incompressible tensor categories (Q6633105)

Let \(\boldsymbol{k}\) be an algebraically closed field. This paper is concerned with the structure of the category of pretannakian categories over \(\boldsymbol{k}\) [\textit{P. Deligne} and \textit{J. S. Milne}, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004); \textit{P. Etingof} et al., Tensor categories. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1365.18001)]. In characteristic zero, this is governed by Deligne's classical results in [\textit{P. Deligne}, Prog. Math. 87, 111--195 (1990; Zbl 0727.14010); Mosc. Math. J. 2, No. 2, 227--248 (2002; Zbl 1005.18009); in: Algebraic groups and homogeneous spaces. Proceedings of the international colloquium, Mumbai, India, January 6--14, 2004. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research. 209--273 (2007; Zbl 1165.20300)]. For \(\boldsymbol{k}\) of characteristic \(p>0\), the subject has developed rapidly in the last few years [\textit{D. Benson} and \textit{P. Etingof}, Adv. Math. 351, 967--999 (2019; Zbl 1430.18013); \textit{D. Benson} et al., Duke Math. J. 172, No. 1, 105--200 (2023; Zbl 1511.18019); \textit{K. Coulembier}, Duke Math. J. 169, No. 16, 3167--3219 (2020; Zbl 1458.18008); \textit{K. Coulembier} et al., Ann. Math. (2) 197, No. 3, 1235--1279 (2023; Zbl 1537.18019); \textit{P. Etingof} and \textit{S. Gelaki}, Int. Math. Res. Not. 2021, No. 12, 9083--9121 (2021; Zbl 1487.18017); \textit{P. Etingof} and \textit{S. Gelaki}, J. Algebra Appl. 20, No. 1, Article ID 2140010, 18 p. (2021; Zbl 1470.17009); \textit{P. Etingof} and \textit{V. Ostrik}, J. Reine Angew. Math. 773, 165--198 (2021; Zbl 1478.18019); \textit{V. Ostrik}, Sel. Math., New Ser. 26, No. 3, Paper No. 36, 19 p. (2020; Zbl 1440.18032)].\N\NA pretannakian category is said to be \textit{incompressible} if every tensor functor out of it is an embedding of a tensor subcategory. Deligne's results imply that in characteristic zero, the only incompressible categories of moderate growth are \textsf{sVect}\(_{k}\) and its unique proper tensor subcategory \textsf{Vect}\(_{k}\). By sharp contrast, in characteristic \(p>0\), an infinite chain of incompressible tensor categories of moderate growth\N\[\N\mathsf{Vect}_{p}\subset\mathsf{Vect}_{p^{2}}\subset\mathsf{Vect}_{p^{3}}\subset\cdots\subset\mathsf{Vect}_{p^{n}}\subset\cdots\N\]\Nwas constructed in [\textit{D. Benson} et al., Duke Math. J. 172, No. 1, 105--200 (2023; Zbl 1511.18019)], where it was shown that any tensor category of moderate growth admits a tensor functor to the union \(\mathsf{Vect}_{p^{\infty}}=\bigcup_{n}\mathsf{Vect}_{p^{n}}\). The first main result of this paper is\N\NTheorem. For \(\boldsymbol{k}\) of arbitrary characteristic, each pretannakian category of moderate growth admits a tensor functor to some incompressible category of moderate growth.\N\NThe second main result of the paper is\N\NTheorem. If \(\boldsymbol{k}\) is of characteristic zero, then the categories \(\mathsf{PTann}_{k}\) and \(\mathsf{Tens}_{\boldsymbol{k}}\) are not filtered, so in particular have no terminal object, where \(\mathsf{Tens}_{\boldsymbol{k}}\) is the category of all (symmetric) tensor categories over \(\boldsymbol{k}\), while \(\mathsf{PTann}_{k}\) is the full subcategory of pretannakian categories.\N\NIt is easy to show that if a pretannakian category is both subterminal and Bezrukqfnikov [Zbl 1210.20004], then it must be incompressible, and the converse is expected to hold at least in characteristic zero. Whether incompressible categories in positive characteristic must be subterminal and Bezrukqfnikof are interesting open questions, and the authors show that these questions are intimately related to the structure of the category of tensor categories, as is seen in the following theorem (Theorem 5.3.2).\N\NTheorem. For a fixed field \(\boldsymbol{k}\), the following statements are equivalent:\N\N\begin{itemize}\N\item[(1)] The category \(\mathsf{MdGr}_{\boldsymbol{k}}\) of tensor categories of moderate growth is filtered;\N\item[(2)] Every incompressible category in \(\mathsf{MdGr}_{\boldsymbol{k}}\) is subterminal;\N\item[(3)] Every incompressible category in \(\mathsf{MdGr}_{\boldsymbol{k}}\) is subterminal and Bezrukqfnikov.\N\end{itemize}\N\NThe authors are interested in intrinsical characterization of when a tensor category is incompressible. It is shown that\N\NTheorem.\N\begin{itemize}\N\item[(1)] If a finite tensor category is maximally nilpotent, then it is incompressible.\N\item[(2)] If a pretannakian category is maximally nilpotent and geometrically reductive, then it is subterminal.\N\end{itemize}\N\NIt is also shown that\N\NTheorem.\N\begin{itemize}\N\item[(1)] The category \(\mathsf{Vect}_{2^{\infty}}=\bigcup_{n} \mathsf{Vect}_{2^{n}}\) is geometrically reductive and maximally nilpotent, so in particular subterminal.\N\item[(2)] The category \(\mathsf{Vect}_{p}\) is subterminal and Bezrukqfnikov\N\end{itemize}