--- date: 2021-05-12 tags: [ web/quick-notes ] --- # 2021-05-12 ## 10:28 See [Serre's uniformity conjecture](Serre's%20uniformity%20conjecture). ## Pavel Etingof, Frobenius exact symmetric tensor categories > Source: Frobenius exact symmetric tensor categories - Pavel Etingof. IAS Geometric/modular representation theory seminar. Tags: #projects/notes/seminars #lie-theory #higher-algebra/category-theory #higher-algebra/monoidal Refs: [tensor category](tensor%20category.md) - Looking at [modular representations](modular%20representations) of finite groups. - [irreducible](irreducible.md) representations: hard, but a lot is known. - [indecomposable objects of a category](indecomposable%20objects%20of%20a%20category) representations: very hard, very little is known. Hard - See [tensor ideal](tensor%20ideal), [Krull-Schmidt theorem](Krull-Schmidt%20theorem) : decomposition into indecomposables is essentially unique. - Can take [split Grothendieck ring](split%20Grothendieck%20ring). - [symmetric tensor category](symmetric%20tensor%20category) $\cat{C}$: - $k\dash$linear, so enriched in $\Vect_{/k}$ for $k= \bar{k}$ (no assumption on characteristic) So morphisms are vector spaces and composition is bilinear. - [abelian category](abelian%20category.md) - [Artinian category](Artinian%20category.md) : objects have finite length and $\dim_K \cat{C}(X, Y) < \infty$ - [monoidal category](monoidal%20category.md) : $(\tensor, \one)$ satisfying associativity and the pentagon axiom - [symmetric monoidal category](symmetric%20monoidal%20category) : a symmetric braiding $X\tensor Y \mapsvia{\tau_{XY}} Y\tensor X$ such that $\tau_{YX} \circ \tau_{XY} = \id$. - [Unsorted/rigid (objects in a category)](Unsorted/rigid%20(objects%20in%20a%20category).md) : existence of duals and morphisms $X\to X\dual$ plus rigidity axioms - Compatibility of additive/multiplicative structures: implied when $\tensor$ is bilinear on morphisms. - $\Endo_{\cat C}(\one) = k$. - Example: $\Rep_k(G)$ the category of finite-dimensional representations of $G$. Take $G=1$ to recover $\Vect_{/k}$. - Can replace a group here by an affine [group scheme](group%20scheme.md) - Such a category is [tannakian](tannakian.md) if there exists a [fiber functor](fiber%20functor) : a symmetric tensor functor $F: \cat{C} \to \Vect_{/k}$. - Preserves tensor structure (hexagon axiom), preserves braiding, is exact. - Implies automatically faithful. - Can take forgetful functor from representations to underlying vector space. - Called "fiber" because for spaces and local systems, one can take a fiber at a point - Deligne-Milne show this is unique. - Can define scheme of tensor automorphisms, $G = \ul{\Aut}^\tensor(F) \in \Grp\Sch_{/k}$. - For an [additive](additive%20category) rigid symmetric monoidal category $\cat{C}_{/k}$ with $\Endo_{\cat C}(\one) = k$, for any $f\in \cat{C}(X, X)$ we can define its [trace](trace%20(monoidal%20categories).md) $\Tr(f) \in \Endo_{\cat{C}}(\one)$: \begin{tikzcd} \one && {X\tensor X\dual} && {X\tensor X\dual} && {X\dual \tensor X} && \one \arrow["{\coev_X}", from=1-1, to=1-3] \arrow["{f\tensor \id}", from=1-3, to=1-5] \arrow["{\tau_{X, X\dual}}", from=1-5, to=1-7] \arrow["{\ev_X}", from=1-7, to=1-9] \arrow["{\Tr(f)}"', curve={height=30pt}, dashed, from=1-1, to=1-9] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCJcXG9uZSJdLFsyLDAsIlhcXHRlbnNvciBYXFxkdWFsIl0sWzQsMCwiWFxcdGVuc29yIFhcXGR1YWwiXSxbNiwwLCJYXFxkdWFsIFxcdGVuc29yIFgiXSxbOCwwLCJcXG9uZSJdLFswLDEsIlxcY29ldl9YIl0sWzEsMiwiZlxcdGVuc29yIFxcaWQiXSxbMiwzLCJcXHRhdV97WCwgWFxcZHVhbH0iXSxbMyw0LCJcXGV2X1giXSxbMCw0LCJcXFRyKGYpIiwyLHsiY3VydmUiOjUsInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) - Can define a categorical [dimension of a category](dimension%20of%20a%20category) $\dim X \in K$ as $\dim X\da \Tr(\id_X)$. - See theory of [semisimplification](semisimplification). - A morphism $f\in \cat{C}(X, Y)$ is **negligible** if for all $g\in \cat{C}(Y, X)$ we have $\Tr(f\circ g) = 0$. - Negligible morphisms form a [tensor ideal](tensor%20ideal) of morphisms: stable under composition and tensor product with other morphisms. - Can form quotient $\bar{\cat{C}} = \cat{C} / \cat{N}$: full subcategory where $\bar{\cat{C}}(X, Y) \da \cat{C}(X, Y) / \cat{N}(X, y)$, i.e. form the vector space quotient of the hom sets. - Still monoidal. - Generally nasty, but if trace of any nilpotent endomorphism in $\cat{C}$ (e.g. when $\cat{C}$ admits a monoidal functor to an abelian STC), then $\bar{\cat{C}}$ is a [semisimple category](semisimple%20category) STC, and in particular is abelian and every object is a direct sum of simple objects. - True if $\cat{C}$ is abelian: take any nilpotent endomorphism, filter by kernels of powers and take the associated graded. Trace of original equals trace of associated graded, but the latter is zero. - Can compute trace *after* pushing through an abelian functor. - $\bar{\cat{C}}$ is the [semisimplification](semisimplification) of $\cat{C}$ - [simple objects of a category](simple%20objects%20of%20a%20category.md) in $\bar{\cat{C}}$ are [indecomposable objects of a category](indecomposable%20objects%20of%20a%20category) in $\cat{C}$ of nonzero dimension. - This procedure is forcing [Schur's Lemma](Schur's%20Lemma.md) to be true! - Run into functors that aren't exact on either side, e.g. the Frobenius. - But still additive, and every additive functor on a *semisimple* category is exact. - Frobenius functors: take $X^{\tensor p}$, allow cyclic permutations $c$ with $c^p = 1$, get [equivariance](equivariant.md) with respect to $\ZZ/p$. - Thus $X^{\tensor p} \in \cat{C} \boxprod \Rep_{/k} \ZZ/p$, the [Deligne tensor product](Deligne%20tensor%20product). - Can take $\id_{\cat C} \boxprod \ss$, i.e. [semisimplification](semisimplification) on the 2nd component, to get an additive monoidal twisted-linear functor $\Fr: \cat{C} \to \cat{C} \boxprod \Ver_p$ - Here $\Ver_p$ is a **Verlinde category**: just denotes $\bar{\cat C}$ with a modified tensor product formula...? - Can [filtration](filtration) $X^{\tensor p}$ by $\Fr_i \da \ker (1-c)^i$.