| Summary: | <jats:title>Abstract</jats:title>
<jats:p>We develop a theory of Frobenius functors for symmetric tensor categories (STC) <jats:inline-formula id="j_crelle-2020-0033_ineq_9999">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi mathvariant="script">𝒞</m:mi>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1473.png" />
<jats:tex-math>{\mathcal{C}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> over a field <jats:inline-formula id="j_crelle-2020-0033_ineq_9998">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi>𝒌</m:mi>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1395.png" />
<jats:tex-math>{\boldsymbol{k}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> of characteristic <jats:italic>p</jats:italic>, and give its applications to classification of such categories. Namely, we define a twisted-linear symmetric monoidal functor <jats:inline-formula id="j_crelle-2020-0033_ineq_9997">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>F</m:mi>
<m:mo>:</m:mo>
<m:mrow>
<m:mi mathvariant="script">𝒞</m:mi>
<m:mo>→</m:mo>
<m:mrow>
<m:mi mathvariant="script">𝒞</m:mi>
<m:mo>⊠</m:mo>
<m:msub>
<m:mi>Ver</m:mi>
<m:mi>p</m:mi>
</m:msub>
</m:mrow>
</m:mrow>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1142.png" />
<jats:tex-math>{F:\mathcal{C}\to\mathcal{C}\boxtimes{\rm Ver}_{p}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>, where <jats:inline-formula id="j_crelle-2020-0033_ineq_9996">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msub>
<m:mi>Ver</m:mi>
<m:mi>p</m:mi>
</m:msub>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1693.png" />
<jats:tex-math>{{\rm Ver}_{p}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is the Verlinde category (the semisimplification of <jats:inline-formula id="j_crelle-2020-0033_ineq_9995">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:msub>
<m:mo>Rep</m:mo>
<m:mi>𝐤</m:mi>
</m:msub>
<m:mrow>
<m:mo stretchy="false">(</m:mo>
<m:mrow>
<m:mi>ℤ</m:mi>
<m:mo>/</m:mo>
<m:mi>p</m:mi>
</m:mrow>
<m:mo stretchy="false">)</m:mo>
</m:mrow>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1507.png" />
<jats:tex-math>{\mathop{\mathrm{Rep}}\nolimits_{\mathbf{k}}(\mathbb{Z}/p)}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>); 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 <jats:inline-formula id="j_crelle-2020-0033_ineq_9994">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi mathvariant="script">𝒞</m:mi>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1473.png" />
<jats:tex-math>{\mathcal{C}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is finite and semisimple, then it admits a fiber functor to <jats:inline-formula id="j_crelle-2020-0033_ineq_9993">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msub>
<m:mi>Ver</m:mi>
<m:mi>p</m:mi>
</m:msub>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1693.png" />
<jats:tex-math>{{\rm Ver}_{p}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. The main new feature is that when <jats:inline-formula id="j_crelle-2020-0033_ineq_9992">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi mathvariant="script">𝒞</m:mi>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1473.png" />
<jats:tex-math>{\mathcal{C}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> is not semisimple, <jats:italic>F</jats:italic> 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 <jats:inline-formula id="j_crelle-2020-0033_ineq_9991">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi mathvariant="script">𝒞</m:mi>
<m:mo>→</m:mo>
<m:msub>
<m:mi>Ver</m:mi>
<m:mi>p</m:mi>
</m:msub>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1453.png" />
<jats:tex-math>{\mathcal{C}\to{\rm Ver}_{p}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. We show, however, that there is a 6-periodic long exact sequence which is a replacement for the exactness of <jats:italic>F</jats:italic>, and use it to show that for categories with finitely many simple objects <jats:italic>F</jats:italic> does not increase the Frobenius–Perron dimension. We also define the notion of a Frobenius exact category, which is a STC on which <jats:italic>F</jats:italic> is exact, and define the canonical maximal Frobenius exact subcategory <jats:inline-formula id="j_crelle-2020-0033_ineq_9990">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msub>
<m:mi mathvariant="script">𝒞</m:mi>
<m:mi>ex</m:mi>
</m:msub>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1467.png" />
<jats:tex-math>{\mathcal{C}_{\rm ex}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> inside any STC <jats:inline-formula id="j_crelle-2020-0033_ineq_9989">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mi mathvariant="script">𝒞</m:mi>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1473.png" />
<jats:tex-math>{\mathcal{C}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with finitely many simple objects. Namely, this is the subcategory of all objects whose Frobenius–Perron dimension is preserved by <jats:italic>F</jats:italic>. 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 <jats:inline-formula id="j_crelle-2020-0033_ineq_9988">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msub>
<m:mi>Ver</m:mi>
<m:mi>p</m:mi>
</m:msub>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1693.png" />
<jats:tex-math>{{\rm Ver}_{p}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. This is the strongest currently available characteristic <jats:italic>p</jats:italic> 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 <jats:italic>F</jats:italic> lands in <jats:inline-formula id="j_crelle-2020-0033_ineq_9987">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:msub>
<m:mi mathvariant="script">𝒞</m:mi>
<m:mi>ex</m:mi>
</m:msub>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1467.png" />
<jats:tex-math>{\mathcal{C}_{\rm ex}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. 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 <jats:inline-formula id="j_crelle-2020-0033_ineq_9986">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mrow>
<m:mi>𝒌</m:mi>
<m:mo></m:mo>
<m:mrow>
<m:mo stretchy="false">[</m:mo>
<m:mi>d</m:mi>
<m:mo stretchy="false">]</m:mo>
</m:mrow>
</m:mrow>
<m:mo>/</m:mo>
<m:msup>
<m:mi>d</m:mi>
<m:mn>2</m:mn>
</m:msup>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1394.png" />
<jats:tex-math>{\boldsymbol{k}[d]/d^{2}}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> with <jats:italic>d</jats:italic> primitive and <jats:italic>R</jats:italic>-matrix <jats:inline-formula id="j_crelle-2020-0033_ineq_9985">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>R</m:mi>
<m:mo>=</m:mo>
<m:mrow>
<m:mrow>
<m:mn>1</m:mn>
<m:mo>⊗</m:mo>
<m:mn>1</m:mn>
</m:mrow>
<m:mo>+</m:mo>
<m:mrow>
<m:mi>d</m:mi>
<m:mo>⊗</m:mo>
<m:mi>d</m:mi>
</m:mrow>
</m:mrow>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2020-0033_eq_1264.png" />
<jats:tex-math>{R=1\otimes 1+d\otimes d}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>), 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].
</jats:p>
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