On the Frobenius functor for symmetric tensor categories in positive characteristic
<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> &l...
| 主要な著者: | , |
|---|---|
| その他の著者: | |
| フォーマット: | 論文 |
| 言語: | English |
| 出版事項: |
Walter de Gruyter GmbH
2021
|
| オンライン･アクセス: | https://hdl.handle.net/1721.1/133474 |
| _version_ | 1826217001013477376 |
|---|---|
| author | Etingof, Pavel Ostrik, Victor |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel Ostrik, Victor |
| author_sort | Etingof, Pavel |
| collection | MIT |
| description | <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> |
| first_indexed | 2024-09-23T16:56:46Z |
| format | Article |
| id | mit-1721.1/133474 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:56:46Z |
| publishDate | 2021 |
| publisher | Walter de Gruyter GmbH |
| record_format | dspace |
| spelling | mit-1721.1/1334742023-12-20T16:19:51Z On the Frobenius functor for symmetric tensor categories in positive characteristic Etingof, Pavel Ostrik, Victor Massachusetts Institute of Technology. Department of Mathematics <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> 2021-10-27T19:53:02Z 2021-10-27T19:53:02Z 2021 2021-05-19T18:36:18Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/133474 en 10.1515/CRELLE-2020-0033 Journal für die reine und angewandte Mathematik (Crelles Journal) Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Walter de Gruyter GmbH De Gruyter |
| spellingShingle | Etingof, Pavel Ostrik, Victor On the Frobenius functor for symmetric tensor categories in positive characteristic |
| title | On the Frobenius functor for symmetric tensor categories in positive characteristic |
| title_full | On the Frobenius functor for symmetric tensor categories in positive characteristic |
| title_fullStr | On the Frobenius functor for symmetric tensor categories in positive characteristic |
| title_full_unstemmed | On the Frobenius functor for symmetric tensor categories in positive characteristic |
| title_short | On the Frobenius functor for symmetric tensor categories in positive characteristic |
| title_sort | on the frobenius functor for symmetric tensor categories in positive characteristic |
| url | https://hdl.handle.net/1721.1/133474 |
| work_keys_str_mv | AT etingofpavel onthefrobeniusfunctorforsymmetrictensorcategoriesinpositivecharacteristic AT ostrikvictor onthefrobeniusfunctorforsymmetrictensorcategoriesinpositivecharacteristic |