Algebraic geometry and representation theory in the Verlinde category

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Algebraic Geometry and Representation Theory in the

Verlinde Category

by

Siddharth Venkatesh

B.A, University of California, Berkeley (2014)

Submitted to the Department of Mathematics

in partial fulfillment of the requirements for the degree of

Doctor of Philosophy

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

June 2019

@

Massachusetts Institute of Technology 2019. All rights reserved.

A uth or ...

Signature redacted

Department of Mathematics

May 3, 2019

C ertified by ....

...

Signature redacted

Professor Pavel Etingof

Professor of Mathematics

Thesisfupervisor

Signature redacted

A

ccepted by ...

JUN 0

5 2019

LIBRARIES

Davesh Maulik

Co-Chair, Graduate Committee, Mathematics

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Abstract

This thesis studies algebraic geometry and the representation theory of group schemes in the setting of symmetric tensor categories over algebraically closed fields of positive char-acteristic. A specific focus is paid to the Verlinde category, a symmetric fusion category in characteristic p that serves as a universal base for all such categories.

Symmetric tensor categories provide a natural setting in which it makes sense to discuss the notion of a commutative, associative unital algebra. In the first third of the thesis, we prove some fundamental facts about these algebras, showing that, in the Verlinde category and any category built out of it, finitely generated algebras are Noetherian, have finitely generated invariants and are finite as a module over their invariants. Subsequently, we use this result to extend some fundamental properties of commutative algebras from the original setting of vector spaces to the more general setting of symmetric tensor categories.

The middle portion of the thesis focuses on some other applications of commutative algebra in the Verlinde category to elaborate on some work of Ostrik and to obtain important combinatorial decomposition formulas. This is a presentation of joint work by the author

with Etingof and Ostrik.

Symmetric tensor categories also provide a natural setting in which to discuss affine group schemes and their representations, namely the commutative Hopf algebras and their comodules. The last part of this thesis focuses on the structure and representation theory of affine group schemes of finite type in the Verlinde category. Using some of the basic commutative algebra properties proved in the thesis, we extend some results of Masuoka from the setting of super vector spaces in positive characteristic to that of the Verlinde category. In particular, we define the notion of a Harish-Chandra pair and show that the category of affine group schemes and their representations in the Verlinde category is equivalent to the category of Harish-Chandra pairs and their representations in the Verlinde category. Thesis Supervisor: Pavel Etingof

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Acknowledgements

I would first like to express my deep gratitude towards my advisor Pavel Etingof, for

all the guidance he has provided to me during my time at MIT. He has taught me a vast amount of mathematics and has been a near endless source of interesting research problems. In particular, all of the problems considered in this thesis have been suggested by him. I am also extremely grateful for his kindness and patience during periods in which I have struggled. I could not have asked for a better advisor and would not have been able to complete my degree without his help.

I would also like to thank Roman Bezrukavnikov, Ivan Losev, George Lusztig, Andrei

Negut and Davesh Maulik for the help they have provided in learning the mathematics necessary for my research while at MIT. I greatly appreciate their taking time out of their work to explain math to me and my interactions with them have significantly broadened my mathematical horizons.

The content of this thesis is built upon the work of several other mathematicians who I must acknowledge. The most obvious contributions come from Pavel Etingof, who suggested all the problems studied in this thesis as part of a larger program to study symmetric tensor categories in positive characteristic. Motivation for these problems have also come from contributions of Dave Benson ([BE]), Victor Ostrik ([Ost1, [Ost2], [Ost3j, [EO] among other works) and Nate Harman ([EHOI) to this program. I am especially grateful to Victor for several helpful discussions that led to the results in [Ven], along with his collaboration in the writing of [EOVI. Chapter 2, 3 and 4 of this thesis are built upon these two papers. In addition to this program on symmetric tensor categories, this thesis owes a large debt to Akira Masuoka for the work he has done on studying Lie superalgebras and supergroup schemes in positive characteristic ([Mas2], [Masi]). Chapter 5 of this thesis is heavily modeled on his work.

Last, but not least, I am deeply grateful to the love and support provided by my parents Meenu and Venki Iyer. Without them, I would never have been able to make it to MIT, let alone make it through my Ph.D.

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5.2.1 Coradical of a cocommutative coalgebra and irreducibility . . . . 5.2.2 Coradical filtration . . . . 60 62 69 . . . . 69 . . . . 69 . . . . 71 . . . . 72 . . . . 74 . . . . 74 . . . ... . . . 7 5 . . . . 76 . . . . 76 . . . . 77 . . . . 77 . . . . 78 79 81 81 83 84 86 87 91 92 95 95 97

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5.3 The Dual Coalgebra . . . . 102

5.4 Lie algebras in symmetric tensor categories in characteristic p . . . . 108

5.5 Lie algebra of an affine group scheme in Verp and the underlying ordinary affine group schem e . . . . 110

5.6 PBW theorem for Lie algebras in Ver. . ... 113

5.7 Dual Harish-Chandra pairs and Harish-Chandra pairs . . . . 114

5.8 Tensor algebras and coalgebras . . . . 117

5.9 Construction of an inverse to the functor DHC . . . . 121

5.9.1 PBW filtrations for dual Harish-Chandra pairs . . . . 122

5.9.2 PBW property for cocommutative ind-Hopf C algebras in Vera with A -1(CO CO) = 1. . . . 126

5.9.3 PBW property for the coradical filtration on cocommutative Hopf al-gebras . . . . 128

5.9.4 Proof of equivalence between the categories of cocommutative Hopf algebras in Verind and dual Harish-Chandra pairs in Ver ... 130

5.10 Inverse Functor for Harish-Chandra pairs: construction via duality . . . . 130

5.11 Representations of affine group schemes of finite type in Ver, ... 140

5.12 Affine group schemes in Verp with trivial underlying ordinary group . . . . . 144

5.13 Representation theory of GL(X) . . . . 145

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Chapter 1 Introduction

Broadly speaking, the goal of this thesis is to study algebraic geometry and the represen-tation theory of group schemes in the setting of symmetric tensor categories in positive characteristic. More specifically, this thesis will illuminate some fundamental properties of commutative algebras and Hopf algebras in the Verlinde category associated to SL2, an

im-portant and somewhat universal symmetric tensor category in positive characteristic, and then apply these results to study some associated combinatorics and representation theory. Symmetric tensor categories are generalizations of the category of comodules of commu-tative Hopf algebras. They are what you obtain if you take all the internal structure of such categories of comodules and just forget that there was an underlying Hopf algebra involved in the definition. Abstractly, a symmetric tensor category is an abelian category equipped with an associative and unital multiplication structure called the tensor product, along with an explicit symmetric swap isomorphism that enforces commutativity of the tensor prod-uct. This is essentially the minimal structure required to be able to define a commutative, associative, unital algebra and hence the minimal structure needed to talk about algebraic geometry.

In characteristic 0, it turns out that the notion of a symmetric tensor categories isn't much more general than that of a commutative Hopf algebra. A theorem of Deligne states that, over an algebraically closed field of characteristic 0, any symmetric tensor category that is suitably moderate is simply the category of comodules of some Hopf superalgebra. However, in characteristic p, this fails to be true, with the aforementioned Verlinde category

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being the simplest counterexample.

The Verlinde category associated to SL2, henceforth called simply the Verlinde category

and denoted Verp, is a finite semisimple symmetric tensor category in characteristic p > 0.

Originally, this was constructed as a modular (and not symmetric) tensor category in char-acteristic 0, by Andersen and Paradowski [AP], as the semisimplifcation of a representation category of a Lusztig quantum group associated to SL₂ (see [BKJ] for details). In char-acteristic p, this construction can be replicated via the category of tilting modules of SL₂ instead, thus resulting in a symmetric tensor category, rather than a modular one. However, the easiest and most useful construction of Very, and the one presented in this thesis, is as the semisimplifcation of the category of finite dimensional representations of Z/pZ. This construction gives a very explicit description of the additive and monoidal structure on Verp. The Verlinde category is interesting for several reasons. It acquires a lot of structure from its relationship with the representations of SL2 and Z/pZ. It is also an example of Deligne's

theorem failing in characteristic p > 0, something that is an easy combinatorial exercise

to show. However, the biggest reason why this category is worth studying is a theorem of Ostrik that shows that Verp is universal among all semisimple symmetric tensor categories in positive characteristic.

Theorem 1.0.1 (Ostrik). Let C be a finite semisimple symmetric tensor category over an algebraically closed field k of characteristic p. Then, there exists a faithful exact symmetric tensor functor F : C -+ Verp.

A consequence of this theorem is that any finite semisimple symmetric tensor category,

and more generally any category built on top of such a category, is just the category of comodules over a commutative Hopf algebra object in Verp. Hence, these Hopf algebras are inherently worth studying. Moreover, it turns out that such Hopf algebras have some very nice algebro geometric properties that allow us to relate them to ordinary Hopf algebras. These properties form the main results of this thesis.

The organization of the thesis is as follows. For the rest of this thesis, fix an algebraically closed field k of characteristic p > 0 and all symmetric tensor categories will be assumed to

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In the interest of keeping the thesis relatively self-contained, Chapter 2 will provide background necessary for the rest of the writeup. In particular, this chapter will include def-initions of symmetric tensor categories and the formal construction of the Verlinde category, along with an explicit description of the simple objects and their tensor products; definitions of commutative, associative, unital algebras in these categories; definitions of finitely gener-ated commutative ind-algebras and their invariants; definitions of Noetherianity and a proof of equivalence between the ascending chain condition and finite generation of submodules. This chapter will also include the formal statement of the theorems of Deligne and Ostrik.

Chapter 3, based almost entirely on [Ven], will prove some fundamental theorems re-garding finitely generated commutative ind-algebras in Ver,. Specifically, we will show the following results:

Theorem 1.0.2. Let C be a symmetric tensor category fibered over Verp. If A is a finitely

generated commutative ind-algebra in C, then A is Noetherian.

Theorem 1.0.3. Let C be a symmetric tensor category fibered over Very. If A is a finitely

generated commutative ind-algerba in C, then A "y is a finitely generated commutative k-algebra and A is a finite extension of Am".

Subsequently, this chapter will analyze some consequences of these two theorems to com-mutative algebra in Verp. In particular, we will prove analogs of the Nakayama Lemma, the Artin-Rees lemma and the Krull-Intersection theorem. All of these facts follow from Theorem 1.0.3 and a key result proved as part of the proof of Theorem 1.0.2

Lemma 1.0.4. Let L be a simple object in Verp not isomorphic to the monoidal unit. Then,

the symmetric power SN(L) = 0 for N > p.

This bound on N is not sharp, and a sharp bound will be described during the chapter. The moral of this lemma and the second theorem above is that commutative algebra in Verp is nilpotent and finite over ordinary commutative algebra, so long as we stick to finitely generated algebras. This moral governs all of commutative algebra in Verp and is key in proving the geometric facts we need to later study commutative Hopf algebras.

Chapters 4 and 5 take the results from Chapter 3 and then apply them in two different directions.

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Chapter 4, based largely on joint work of the author with Etingof and Ostrik in [EOV], focuses on some different consequences of results proved in chapter 3 and independent work of Etingof and Ostrik on symmetric fusion (i.e. finite semisimple tensor) categories. The chapter will have a mix of several results, some interconnected and some not: a proof of uniquness of Ostrik's fiber functor from Theorem 1.0.1; the definition of the ordinary and the super Frobenius-Perron dimension of an object in a symmetric fusion categories and an application to classifying symmetric fusion categories of rank two; formulas F(X) E Very, with X E C and F : C -+ Very Ostrik's Verlinde fiber functor, in terms of the ordinary and super Frobenius-Perron dimension of X; formulas for symmetric powers of objects in Verp and their invariants; definitions and computations of p-adic dimensions; and finally, classification of semisimple triangular Hopf algebras in any characteristic.

Chapter 5, on the other hand, moves towards finitely generated commutative ind-Hopf algebras in Verp, i.e., functions on an affine group scheme in Verp. This chapter defines commutative Hopf algebras in Verp and presents the example of the algebra of functions on GL(X), the general linear group of an object X E Verp. It also introduces the notion of a dual coalgebra to a commutative ind-algebra in Very, which is a coalgebra equipped with a non-degenerate pairing with the original algebra. Using this definition, Chapter 5 then defines Harish-Chandra pairs in Verp, which is roughly speaking the data of an affine group scheme Go over k, a Lie algebra g in Verp, along with an isomorphism between go and Lie(Go) and an extension of the adjoint action of Go on go to an action of Go on g. The key result of the chapter is the following theorem:

Theorem 1.0.5. The category of finitely generated commutative ind-Hopf algebras in Verp

is equivalent to the category of Harish-Chandra pairs in Very.The same holds for the corre-sponding category of comodules in Verp.

The rest of the chapter studies some consequences of this result to the representation theory of affine group schemes in Very, with an emphasis on the representation theory of GL(X) for X

C

Verp. The techniques used in this chapter are based heavily on work by Masuoka in [Mas2].

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Chapter 2 Background

2.1 Notation and Conventions

These notations and conventions will often be brought up in each chapter as a reminder but are all stated here for convenience of reader.

1. Unless specified otherwise, k will be an algebraically closed field of characteristic p > 0.

2. By a category over k, we mean a k-linear, locally finite and Artinian category.

3. Throughout the thesis, unit and associativity isomorphisms inside the tensor category

will be suppressed. Additionally, we will also suppress the isomorphism inherent in the structure of a tensor functor.

4. If C is a symmetric tensor category, we will always use c to denote the braiding on C. When the objects on which the braiding is acting need to be specified, we will explicitly write cxy instead of c.

5. In comparison between a symmetric tensor category C and it's ind-completion Cind, we

will us the word "object" to mean an object in C, i.e., one of finite length, and we will use the phrase "ind-object" to refer more generally to an object in Cind, one that may possibly be of infinite length. Sometimes, for emphasis, we may use the phrase "actual object" to refer to an object in C of finite length. We will also use a similar dichotomy

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to differentiate between algebras in C and ind-algebras in C (the latter being algebras in Cind), and the same for Hopf algebras.

6. For objects X inside Verpd, we will use Xo to denote the isoptypic component

cor-responding to the monoidal unit 1, and X,₀ to denote the sum of all other isotypic components.

2.2 Tensor Category Technicalities

All of the definitions and constructions in this section are taken from [EGNO], which is an

excellent introductory text for the theory of tensor categories. Further details regarding all the constructions here can be looked up there.

Definition 2.2.1. Let C be a category over k. We say that C is monoidal category if it is

equipped with:

1. A bilinear bifunctor 0 : C x C - C, called the tensor product.

2. An object 1 called the unit object (that serves as the unit for the tensor product multiplication)

3. Natural associativity isomorphisms

axy,z: (X&Y)0Z-+XO(Y0Z) for X,Y, Z E C.

4. A unit isomorphism

t : 10@1 -+1.

such that the following two axioms hold:

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((WOX)OY)®Z WXY 0 idz (W® (X®Y)) Z awox,y, (W0X)g(Y0Z) W awxoyz

W 0((X 0Y) 0Z) idw 0 ax,y,z

aw,x,y _gz

W 0 (X 0 (Y 0 Z))

commutes.

2. The unit axiom: The functors

L1 : X -4 10 X and R₁ : X i-+ X

0

1

are autoequivalences of C.

Definition 2.2.2. A braided monoidal category is a monoidal category C over k equipped with a natural isomorphism cx,y : X 0 Y - Y 0 X satisfying the hexagonal diagrams

ax,y,z (X)Y) Z CXY 0 idz ayzx YO(ZOX) idy

0

cx,z X (Y Z) cxz (Y (Z) X

(Y O X)O Z ayx,z YO(XOZ)

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(X ®

Y) OZ CX&yZ

Z ®X(X®Y)

--1

XO(Y®Z) idx 0 cyz X®(Z®Y) (Z X) O Y cxz idy (XOZ)®Y 'X,Z,Y

A symmetric monoidal category is a braided monoidal category in which cyx = cxy. Definition 2.2.3. A monoidal category C over k is rigid if every object X E C has a dual

object X* E C. Specifically this means that for every object X

C

C, we have an object

X* E C and evaluation and coevaluation maps

evx : X* 9 X -+ 1, coevx : 1 -+ X 9 X* such that the compositions

coevx ® idx

X) (Xaxxx X0 (X* ®X)

id* 0 coevx a- ,x,xx idx

x0X* (3(X 0X*) (X* (&X) 0X* ev- ix X *

are the identity.

Definition 2.2.4. A tensor category C is an abelian, rigid monoidal category in which and

idx 0 evx

x

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Endc(1) a k. A tensor category is said to be braided (resp. symmetric) if it is braided (resp. symmetric) as a monoidal category. A fusion category is a finite semisimple tensor category.

Let us now look at some examples:

Example 2.2.5. (a) The simplest examples of symmetric tensor categories are Vec and sVec which are, respectively, the categories of finite dimensional k-vector spaces and finite dimensional k-vector superspaces (the latter only existing if p > 2). Here, the braiding is just the swap map and the signed swap map, respectively.

(b) Similarly, the category of finite dimensional representations over k of a finite group G is a symmetric finite tensor category over k with braiding given by the swap map.

More generally, the category of finite dimensional comodules over a commutative Hopf algebra H is a symmetric tensor category as well.

(c) Analogous to the relationship between Vec and sVec, the category of finite dimensional super co-modules of a supercommuative Hopf superalgebra also provides an example of a symmetric tensor category, if p > 2.

(d) A slightly more complicated category is the universal Verjinde category in characteristic p > 0, which we denote as Ver,. This is constructed as a quotient of the category of

finite dimensional representations of Z/pZ over k of characteristic p. The full details regarding the construction is given in a later section of this chapter as the construction is slightly technical. This category is the central focus of the rest of this thesis.

Having defined the notion of tensor categories, we also need the notion of a tensor functor and monoidal natural transformations between tensor functors

Definition 2.2.6. A tensor functor between two tensor categories C, C' is a pair (F, J) where

F : C -+ C' is an additive functor and

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is a natural isomorphism such that F(1) is isomorphic to 1 E C' and the following diagram commutes

(F(X) 0 F(Y)) 0 F(Z)

aF(X),F(Y),F(Z)

Jxy

0 idF(Z) idF(X) 0

JY,Z

F(X OY)O F(Z) JXOY,z F((X 9 Y) F(Z) 0 F(Y 0 Z) 0 Z) F(axy,z) - F(X Jxygz 0(Y 9 Z))

A tensor functor between braided tensor categories (C, c), (C', c') is symmetric if F also

satisfies the following diagram

F(X) 0 F(Y) F(X),F(Y) F(Y) 0 F(X)

JxY

F(X 0Y)0 F(Z) F(cxy)

Some special types of tensor functors will warrant additional terminology.

Definition 2.2.7. Let F : C -+ C' be a tensor functor.

1. We say that a tensor functor F is a fiber functor if it is faithful and exact.

2. We say that F is injective if it is fully faithful.

3. We say that F is surjective if any simple object of C' is a subquotient of F(X) for some

X C C.

F(X) 0 (F(Y) 0 F(Z))

JY,x

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Definition 2.2.8. A natural transformation or morphism of between monoidal functors

(F, J), (F', J') : C -+ C' is a natural transformation q : F --+ F' such that q1 is an isomorphism

and the diagram

F(X) 0 F(Y) F(X®Y)

?7x 0 W7 rxOy

F'(X) ® F'(Y) F'(X ® Y)

Let us again look at some examples:

Example 2.2.9. 1. If C is the category of finite dimensional comodules over a commuta-tive Hopf algebra H, then we have a fiber functor F : C --+ Vec of taking the underlying vector space.

2. Similarly, for p > 2, we have a fiber functor F : C -+ sVec if C is the category of finite dimensional supercomodules of a supercommutative Hopf algebra H.

Remark (Tannakian Reconstruction). These two examples actually get at the general

picture of a fiber functor. If F : C -+ C' is a fiber functor, then we can recover C as a category

of comodules of some commutative Hopf algebra in C' (along with some compatible action of the fundamental group of C'). We won't elaborate here on the notion of a commutative Hopf algebra in a category (though this will be defined later) and we will completely ignore the technicalities of categorical fundamental groups. Details on this can be looked up in

[EGNO][5.2, 5.4] and references therein.

The last technical construction in this section that we need encapsulates the notion of infinite-dimensionality. We will be working with algebras inside symmetric tensor category that are not necessarily "finite dimensional", since finite dimensional algebras tend to be a fairly limited class. Hence, we need the notion of the ind-completion of a category.

Definition 2.2.10. Let C be a symmetric tensor category. By Cind, we denote the ind-completion of C, i.e., the closure of C under taking filtered colimits of objects in C.

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The tensor product in C is exact due to rigidity of C. Hence, it commutes with taking filtered colimits and hence extends to an exact tensor product on Cind. Additionally, nat-urality of the braiding implies that the braiding extends to a symmetric structure on Cind. Cind is thus a symmetric k-linear abelian monoidal category in which the tensor product structure 0 is exact (but it is neither rigid nor locally finite). A specific example of Cind that we will repeatedly use in the rest of this thesis is in the case where C is a symmetric fusion category, i.e., when C is finite and semisimple. In this case, the objects of Cind are precisely the (possibly infinite) direct sums of the simple objects in C.

As stated in the convention section, if C is a symmetric finite tensor category, when we use the word "object", we will mean an object in C, i.e., an object of finite length in Cind and we will use the term ind-object whenever referring to objects in Cind that may have infinite

length. Sometimes, for emphasis, we will use the phrase "actual object" to refer to the finite length objects.

A few more words on conventions: despite being important structure in the definitions

above, there is little harm in suppressing associativity and unit isomorphisms and the J-isomorphism involved in the definition of a symmetric tensor functor. We will routinely do so. We will also consistently use c, c' to denote the braiding on categories C, C' and suppress the notation whenever it is obviously present. This is all to enhance clarity since any movement of parentheses usually involves these maps in fairly obvious ways.

One important property of a symmetric tensor category is the following proposition.

Proposition 2.2.11. Let C be a symmetric tensor category over k. The tensor subcategory

generated by 1 is symmetric tensor equivalent to the category of finite-dimensional vector spaces Vec over k, with the unit object representing the one-dimensional vector space k itself.

This proposition follows from the requirement that Endc(1) = k, as this implies that 1 is simple, which is all the proposition needs. We will view this proposition as giving us

a canonical inclusion of Vec into any symmetric tensor categories, and hence use this to view vector spaces as objects in any symmetric tensor category. Objects that are inside this subcategory will be called trivial objects.

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2.3 Algebras, Hopf algebras and modules

Symmetric tensor categories are naturally equipped with notions of multiplication, associa-tivity, unitality and commutativity. Hence, we can define the notion of an algebra or Hopf algebra fairly naturally inside such a category or its ind-completion, and also examine several important properties such as associativity or unitality.

Definition 2.3.1. Let (C, c) be a symmetric tensor category over k, with ind-completion Cind. An associative, unital algebra in Cind (also called an ind-algebra in C) is an object

A E Cind equipped with multiplications maps m : A 0 A -+ A, p 1 -+ A such that the

following diagrams commute:

A&A9A m idA AA

idA m m

A®A m A

1oA p idA A®A idAOp A 1

idA m idA

A

The algebra is commutative if m o CA,A - m.

This is our first instance of suppressing unit and associativity isomorphisms and it is already evident how much cleaner the notation is, while losing little precision.

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homomorphism of algebras if

f

O PA = B andf 0 MA - MB 0

(f

f

)-We can similarly define coalgebras, bialgebras and Hopf algebras.

Definition 2.3.3. Let C be a symmetric tensor category and Cind the ind-completion. A coassociative, counital coalgebra H in Cind (also called an ind-coalgebra in C) is object in

Cind equipped with morphisms A : H -+ H 0 H, E : H -+ 1 that satisfy dual diagrams to

the ones satisfied by m and y. A coalgebra C is cocommutative if ccgc o A = A.

We can also analogously define a homomorphism of coalgebras.

Definition 2.3.4. A bialgebra in Cind (also called an ind-bialgebra in C) is an object B E Cind

equipped with the structure of both an associative, unital ind-algebra and a coassociative, counital ind-coalgebra such that the comultiplication and counit maps are algebra homomor-phisms (or equivalently, the multiplication and unit maps are coalgebra homomorhomomor-phisms). Here B 0 B is given the algebra structure by multiplying independently in each tensor

component.

Definition 2.3.5. A Hopf algebra in Cind (also called an ind-Hopf algebra) is an object

H E Cind equipped with the structure of a bialgebra and an antipode map S : H -+ H that

is an anti-involution such that the diagram

A S&idH m

H

-

HH>

HoH

H

idH®S

1

commutes

Given an algebra or Hopf algebra, we have some important constructions. For the rest of this section, fix a symmetric tensor category C over k.

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Definition 2.3.6. 1. If A is a commutative ind-algebra in C, then a subalgebra B of A

is a subobject such that m(B & B) = B. An ideal I in A is a subobject of A such that m(A 0 1) = I. If X is a subobject, the ideal generated by X is the image m(A 0 X) under the multiplication map.

2. Similarly, if H is a Hopf algebra, then a Hopf subalgebra is a subalgebra H' such that

A(H') C H' 0 H', and a Hopf ideal is an ideal I such that A(I) E H 0 I E I 0 H.

3. Let A be a commutative ind-algebra in C, with C semisimple. The underlying ordinary

commutative algebra is the quotient A := A/I, where I is the ideal generated by all simple subobjects of A not isomorphic to 1. This is an ideal due to semisimplicity of C and A is an ordinary commutative k-algebra (viewed as an ind-algebra in C via the canonical inclusion of Vec). We can similarly define an underlying ordinary

commutative Hopf algebra if A is a commutative ind-Hopf algebra.

4. The invariants Am" of A is the sum of all the simple subobjects of A isomorphic to 1. This is also the same as the algebra Homc(1, A) with pointwise multiplication. It is an ordinary commutative algebra over k and is a subalgebra of A under the canonical inclusion of Vec. Note that this is not necessarily a Hopf subalgebra, if A is a Hopf algebra.

5. Let H be an ind-Hopf algebra in C. The space of primitives inside H is the kernel of

A-idH 0P-- p9dH :H-+H0H.

A subobject X e 1 C H is grouplike if A(X) C X 0 X.

6. Finally, an important notion is that of a module. A left module for an ind-algebra A

in C is an object M C Cind equipped with a map

a:A0M-+M

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AOeAO m (9 nidM AOM idA a a

AoM

a

M

and

10M

id

>

A o M

idM a

M

commute. Note that A is a left module over itself and left ideals are simply left submodules of A. If M, N are left A-modules, a homomorphism of left A-modules from M to N is a morphism

f

: M

--+

N

E Cind

such that

f

o am

=

aN o

f-We can analogously define right comodules over an ind-coalgebra H via a coaction map

p : M -+ M 0 H and analogously define homomorphisms of right comodules.

The last definition gives the structure of an abelian category to the category of modules over a fixed algebra in Cind (or a category of comodules over a fixed coalgebra in Cind) and these categories naturally come equipped with faithful, exact, additive functors to Cind. Moreover, if H is an ind-Hopf algebra then we have the following:

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modules (resp. comodules) over H in Cind. This category is also equipped with a symmetric structure if H is cocommutative (resp. commutative)

Proof. Details can be looked up in [EGNO]. As a brief description of the constructions involved in the proof, the tensor product on the category of modules over H is acquired via the following maps:

aM®N : H®(M®N) -+ MON

is just

(aM 0 aN) 0 (idH 0 cH,M 0 idN) 0 (A 0 idM®N).

Let us now construct some algebras, Hopf algebras and modules that will be extremely important to the rest of the thesis.

Definition 2.3.8. For any object X e C, the tensor algebra T(X) E Cind is the algebra

00

T(X) = X

n=O

with the unit being the inclusion of X0 = 1 and multiplication given by concatenatation of tensors

m: X*m 0 X** -+ Xm+n

the natural identification. We can also give T(X) a Hopf algebra structure by setting X to be primitive. Note that T(X) is also graded by N in a natural manner.

Proposition 2.3.9. T(X) is the free associative ind-algebra on X. More precisely, for any

associative, unital algebra A E Ci"d

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The proof of this proposition is exactly the same as the proof when C = Vec. The important fact needed is that m : X 0 0 X -+ X®n is an isomorphism.

Definition 2.3.10. For any object X C C, the symmetric algebra S(X) is the quotient of T(X) by the ideal generated by the image of idxOx - cxx in X 0 X. This is an algebra in

Cind. It is a graded quotient of T(X) and also inherits the Hopf algebra structure in which X is primitive.

Analogous to the above proposition, we have

Proposition 2.3.11. For any X E C, S(X) is the free commutative ind-algebra on X.

Homcomm-Alg, cind(S(X), A) - Homcind (X, A).

We can also define the notion of a free module.

Definition 2.3.12. Given an object X E C and an associative, unital algebra A

c

Cind, the free left module generated by X over A is A 0 X with module map

p: A G A 9 X -+ A 9 X

simply mA 0 idx.

To justify the notion of a free module, we again have the following universal property:

Proposition 2.3.13. Given X

e

C, A an associative, unital algebra in Cind and M a left A-module in Cind

HomA-mod,cind(A O X, M)

=

Homcind (X, M).

Lastly, in the setting of commutative algebras, we also want to define a commutative algebra structure on the tensor product of two commutative algebras.

Definition 2.3.14. Let A and B be commutative ind-algebras in C. Then, we define a

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A&BOA0B-*A0B

as (mA 0 Ms) o (idA 0 CB,A 0 idB).

It is clear that A 0B satisfies the same universal property in the category of commutative ind-algebras in C as it does in the standard case when C = Vec.

We end this section by proving some important features of the symmetric algebra of an object X

c

C. This algebra will be fundamental to the rest of Chapter 2 and 3 and these

results provide some useful structure theory.

Let F : C -- C' be a symmetric tensor functor between symmetric tensor categories.

Since this is exact, it extends to a functor F : Cind _+ C.

Lemma 2.3.15. Let X E Cind. Then,

F(S(X)) -- S(F(X)) as commutative algebras in (CI)ind.

Proof. This follows from the fact that symmetric tensor functors preserve tensor products

and the commutativity isomorphisms. LI

Lemma 2.3.16. Let X, Y E Cind. Then, we have a natural isomorphism

S(X EY) C S(X)o S(Y)

of commutative ind-algebras in C.

Proof. The proof follows by showing that both objects satisfy the same universal property. Let C' be the category of commutative ind-algebras in C and let A E C'. Then, we have natural isomorphisms,

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Homc,(S(X G Y), A) Homc(X (DY, A)

Y Homc(X, A) D Homc (Y, A)

2 Homc, (S(X), A) D Homc, (S(Y), A)

Y Homc'(S(X) 0 S(Y), A).

2.4 Finite generation, Noetherianity

With the constructions of the previous section, we can finally begin discussing the notion of finite generation and Noetherianity. Algebras inside symmetric tensor categories can be fairly wild, these are important finiteness properties that potentially allow algebras to be tractable. We will soon see that for categories fibered over Verp, finite generation implies Noetherianity and a whole lot of other useful structural rigidity. But first, let us begin with a definition. In this section, we will fix a symmetric tensor category C over k and only look at commutative ind-algebras in C and their modules.

Definition 2.4.1. We say that a commutative algebra A in Cind is finitely generated if there exists some object X

E

C and a surjective homomorphism of algebras

S(X) -+ A.

For an arbitrary commutative algebra A E Cind, we say that an A-module M E Cind is finitely generated if there exists an object X E C and a surjective homomorphism of A-modules

AO X -+ M.

Definition 2.4.2. For a commutative ind-algebra A, we say that an A-module M is

Noethe-rian if its A-submodules satisfy the ascending chain condition, i.e., that for any sequence of

submodules

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MO - M1 -- M2

--in which the morphisms are monomorphisms, there exists some n such that for all N > n, the map MN -+ MN+1 is an isomorphism. We say that A is a Noetherian algebra if all of its finitely generated modules are Noetherian.

In the case where C = Vec, a fairly easy, classical result states that ascending chain condition for submodules is equivalent to finite generation of submodules. This holds true for general C as well.

Lemma 2.4.3. Let A be a commutative ind-algebra in C. Let M be an A-module. Then,

M is Noetherian if and only if every submodule N of M is finitely generated.

Proof. We first prove the forward direction. Suppose M is Noetherian and assume for contradiction that there exists a submodule N of M that is not finitely generated. We inductively create an infinite ascending chain of finitely generated submodules {Ni} of N that does not terminate.

Since N is an ind-object in C, it must contain an actual object Xo. Let No be the submodule generated by Xo, i.e., let it be the submodule given by the image of A 9 X -+ M under the action map. Then, as Xo is a subobject of N, No is a submodule of N and is finitely generated. This is the first step of the construction. Now, suppose N, has been defined as a finitely generated submodule of N. Then, the inclusion of N, into N is not an isomorphism as N would then have been finitely generated. Hence, the cokernel of this inclusion is not 0 and thus contains a nonzero object Y E C. Take the submodule generated by this object and call it Nr+i, which is a submodule of N/Nr. Hence, we can find a submodule N,+1 of N such

that the inclusion of N, into N factors properly through N,+,, and the quotient Nr+1/N, is

N,+1.

We need to show that Nr+i is finitely generated. We know that N, is finitely generated and so is N,+I/Nr. Let X, and Y be actual objects in C with X, g N, and Y C N,+1 such that the natural maps

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and

A 0 Y -+ Nr+i

are epimorphisms. We claim that there exists some subobject Z of Nr+i, with Z E C, such that the projection from Z to N,+1 contains Y. This is because the direct limit of the images

of subobjects of N,+1 under the projection map is Nr+i and hence any object of finite length

must be contained in the direct limit of some finite subset of these images. So, if Z1,..., Z,

are subobjects of Nr+i, the sum of whose images contains Y, then we take Z =

Z'

Zi. We now claim that Xr and Z together generate Nr+i. Let X,+1 = Z

+

Xr. Then, the

submodule generated by Xr+i contains N, (as it contains Xr) and surjects under projection to Nr+1 (as it contains Z). Hence, X,+1 generates Nr,+, which is therefore finitely generated. We now prove the reverse direction. Suppose every submodule of M is finitely generated. Let

MO - 1 -4

--be a sequence of monomorphisms of A-submodules of M. Then, since Cind is closed under filtered colimits, we can take the colimit (in this case the union) of this sequence to get an ind-subobject M' of M. Additionally, since the A-module structure on Mi commutes with the morphisms that we take the colimit of, M' acquires a natural structure of an A-module. Hence, by the assumption that A-submodules of M are finitely generated, there exists an object X E C and an epimorphism A 0 X -+ M' of A-modules.

But now, such a morphism has to come from a morphism in Cind from X to M'. As X is an actual object in C, its image in M' has finite length and hence must lie in some

M. (as otherwise, taking the intersection of the image with Mi gives an infinite ascending

chain of subobjects of X that does not stabilize, which cannot exist). Hence, as Mi is an A-module, the image of A 0 X -+ M' lies in Mi and hence the inclusion of Mi into M' is an isomorphism. Hence, for all N > i, the inclusion MN -+ MN+1 is an isomorphism. This proves that M is Noetherian.

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Recall that for any n, k[x1, ... , x,] can be viewed as an ind-algebra in Cind via this canonical embedding of Vec into C. Our final goal of this section is to prove that tensoring with this polynomial algebra preserves Noetherianity. The proof of this assertion is very similar to the proof of the classical Hilbert basis theorem but is stated in a categorical manner. We give the statement and proof below. If the proof seems complicated, just translate the statements to the case when everything is actually a vector space and it will make sense.

Proposition 2.4.4. Let A be a commutative ind-algebra in C. Let M be a Noetherian A-module. Then, k[x1,... , xJ](0 M has a natural structure of a Noetherian k[xi,. . ., x9] A-module.

Proof. By induction on r, we may assume r = 1. We can then write

k[x] 0 M = M E xM Ex2M D ...

Here xM can be viewed as x 0 M or as the image of M under the action of x. This clearly has a natural structure of a k[x] 0 A-module defined as idk[xl 0 CA,k[x] 0 idM followed by componentwise action. Using Lemma 2.4.3, we will show that this is Noetherian by showing that any k[x] 0 A-submodule N of k[x] 0 M is finitely generated.

For n > 0, define M = MD .. -ex'M and let rr be the projection k[x}0M -+ xnM 2 M.

For each finitely generated A-submodule X of N, we define the associated object of leading coefficients LC(X) , which will be an A-submodule of M. Any such finitely generated submodule must be contained in M, for some n. Let n(X) be the minimal such n for X

and define LC(X) =7rn(X)(X) ; xn(X)M, which we identify with M by the multiplication

isomorphism xf(X) : M -+ xn(X)M.

Define LCN to be the sum of LC(X) over all finitely generated A-submodules X of

N. This is an A-submodule of M and is hence finitely generated by Noetherianity of M. If

AOZ -+ LCN is a surjection of A-modules with Z

c

C, then this comes from a morphism Z -+ LCN in C. Let X1, . .., Xm be finitely generated A-submodules of N such that K'=_₁LC(Xi)

contains the image of the morphism from Z (some such finite list must exist as Z has finite

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Let di = n(Xi) and let d be the maximum of the di. Finally, define

B:= k[x](N n Me).

Clearly B C N and by choice of d, we also have

>Zk[x]Xi C B.

i= 1

We claim that B = N. Suppose for contradiction that B

#

N. Since N E Cind, it is the sum of all the objects it contains. Hence, we can find some object Y E C that is a subobject of N but is not contained in B. Taking the A-submodule of N generated by Y, we see that there exist finitely generated A-submodules of N that are not contained in B. Let N' be such a submodule such that h = _{n(N') is minimal amongst all such submodules. Note that h > d} as otherwise N' C NnMd C B.

Consider now the finitely generated A-submodule B

n Mh of B. Note that

m

xhd'X, C B

n Mh

i=1

and hence, LCN 7 rh(B n Mh). From this, it follows that

7h(N') LC(N') C LC(B n Mh) = 7rh(B n Mh) = LC(X,) = LCN.

We show this implies the existence of a finitely generated A-submodule N" of (B

n

Mh) + N' that is not contained in B n Mh and has n(N") < n(N'). For this purpose, consider the

inclusion of B

n

Mh into (B n Mh) + N'. Since B n Mh, N' C Mh, we have a commutative diagram

0 >BnMh-1 l Bnmh >rh (B) > 0

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where the rows are exact and the vertical maps are induced by the inclusion of B

n

Mh into

(B

n

Mh) + N'. All three vertical maps are monomorphisms. -y is an epimorphism based on the argument above. But, by choice of N', 3 is not an epimorphism. Hence, a cannot be an epimorphism by the five lemma. Hence, we can find a finitely generated nonzero A-submodule N" of ((Bn M)+N')nMhi 9 NnfMhi that is not contained in Bn Mh. Since

N" is contained in Mh_1 g Mh, this implies that N" is not contained in B. This contradicts the minimality of n(N'). Hence, N = B is hence finitely generated as a k[x] 0 A-module.

2.5 The Verlinde Category Verp: construction

The simplest construction of Very is as the semisimplifcation of the category of finite dimen-sional Z/pZ representations over k. Semisimplification of categories is a general process by which we can start with any symmetric tensor category and obtain a semisimple one that is somewhat universal (see [EO] for details). To define this semisimplifcation process, we need to define the notion of traces.

Definition 2.5.1. Let C be a locally finite, rigid, symmetric monoidal additive category in which Endc(1)

2

k (so a symmetric tensor category is a special example). If

f

: X -+ X is

a morphism in C, then the trace of

f

is the scalar given by the morphism

coevx

f

0 idx. CXX* evx

1 OX

-~XX*

**~X*0X**

~

1

in Endc(1) a k. We use Tr(f) to denote the trace of

f.

Definition 2.5.2. If C is a locally finite, rigid, symmetric monoidal additive category as above, then for any X, Y E C, the space of negligible morphisms .A(X, Y) C Homc(X, Y)

consists of those morphisms

f

: X -+ Y such that for all g : Y -+ X, Tr(g o

f)

0.

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Definition 2.5.3. Let C be a category as above. The categorical dimension of X E C is

dim(X) := Tr(idx).

We say that X is negligible if dim(X) = 0, or equivalently, idx is a negligible morphism. Proposition 2.5.4. AJ(X, Y) is a tensor ideal, i.e., the following properties hold:

1. If f, f'

C K(X,

Y), and r E k, then rf + f'

E

(X, Y).

2. If

f

E Af(X, Y), f' E Homc(Y, Z), f"

C

Homc(Z, X), then

f o f" E K(Z, Y), f' o f E .,(X, Z).

3. If

f

E Af(X, Y), f' E Homc (X', Y'), then

f 0 f' E A(X 0 X',Y 0Y'), f' 0 f C J(X' 0 X,Y'

0

Y).

Proof. See [EOI[Lemma 2.31.

Hence, we can construct a quotient category.

Definition 2.5.5. Given a locally finite, rigid, symmetric monoidal additive category C in

which Endc(1) a k, the quotient category C which has the same objects as C but in which

Homj(X, Y) 2 Homc (X, Y)/A((X, Y)

is called the semisimplification of C.

Here are some important properties of C that can be looked up in [EO]:

1. C is semisimple and hence abelian. Thus, it is a symmetric tensor category that is

semisimple (it need not be finite a priori). The monoidal structure on C is induced from that on C as N(X, Y) is a tensor ideal.

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2. The simple objects of C are the images under the quotient functor of the indecomposable objects of C that are not negligible.

With this in hand, we can now define the Verlinde category.

Definition 2.5.6. The Verlinde category is the semisimplification of the category of finite

dimensional k-representations of Z/pZ.

Let us now describe the additive and monoidal structure of Verp and give some other useful representation theoretic constructions associated to it. Proofs of these facts are omit-ted here. They can be looked up in [Osti] and [GK, GM]. This description is more about building some intuition for Verp.

Example 2.5.7. 1. A representation of Z/pZ is simply a matrix whose pth power is

the identity. The indecomposable representations of Z/pZ are the indecomposable Jordan blocks of eigenvalue 1 and size 1 through p. Let us call these representations M1, ... , M. The dimension of Mi is simply its dimension mod p. Hence, Mp is the

only negligible indecomposable. Thus, the simple objects of Ver, are: L1, ... , LP,1,

where Li is the image under semisimplification of the indecomposable Jordan block of dimension i.

2. To describe the monoidal structure, we need to describe the decomposition of Li 0 Lj into direct sum of simples:

min(i,j,p-is-i)

Li

0

L, ~

LIj-i+2k-1-k=1

This rule seems somewhat complicated but is very cleanly understood in terms of representations of SL2(C). Let V be the irreducible representation of SL2(C) of

di-mension i. Let p be the map from the simple objects in Verp to the simple objects

in Rep(SL2(C)) defined by sending Li to V. Then p(Li 0 Lj) is obtained by taking

Vi

o

V, removing any representations of dimension > p, and then also removing V,

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3. This relationship between the representations of SL2 is not accidental, it comes from a

relationship between Verp and tilting modules for SL₂(k) described in [Ostl][3.2, 4.3]

and the additional references [GK, GM] contained within. Consider the category of rational k-representations of a simple algebraic group G of Coxeter number less than

p. This has a full subcategory consisting of tilting modules, which are those

represen-tations T such that T and its contragredient both have filtrations whose composition factors are Weyl modules V, corresponding to dominant integral weights A. This is a rigid, locally finite, Karoubian symmetric monoidal category (one closed under direct summands) and hence, we can still take its quotient by negligible morphisms. This gives us a symmetric fusion category which we denote Verp(G), the Verlinde category corresponding to G. For G = SL2, Ostrik showed in [Osti, 4.3] that Verp(SL2) e Ver,,

with the functor being induced by restriction to the generator E E SL2(k). This

rela-tionship between Verp(SL2) and Verp will prove useful in analyzing a very important

piece of the structure theory of commutative algebras in Verp.

4. The subcategory additively generated by Li for i odd is a fusion subcategory, which we denote by Very. Additionally, the subcategory additively generated by L1 and L,_ 1 is

also a fusion subcategory and is isomorphic to sVec. Ostrik shows in [Ostl] that these are the only tensor subcategories of Verp and, as symmetric fusion categories,

VerpCa Ver+ M sVec the Deligne tensor product.

5. As some simple examples: Ver2 = Vec, Vcr3 = sVec and Vert has simple objects L1, L3 with L 2 _{= L}

3

D

L1.

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2.6 Deligne's Theorem in characteristic 0, Ostrik's

coun-terpart in characteristic p > 0

For this section alone, we will not assume that k has characteristic p > 0. The reason

this thesis begins already in characteristic p > 0 is that a large class of symmetric tensor

categories in characteristic 0 are nothing more than super comodules of Hopf superalgebras. This class of categories is indicated by a moderateness property introduced by Deligne in

[Del]

Definition 2.6.1. Let C be a symmetric tensor category of an algebraically closed field k (of any characteristic). We say that C has subexponential growth if for any object X E C, there exists a positive number cx such that length(XO') < c' for all n > 0.

A theorem of Deligne (see [Del], [DM] and also [EGNO, Theorem 9.9.261) then classifies

all symmetric tensor categories of subexponential growth in characteristic 0.

Theorem 2.6.2. Let k be an algebraically closed field of characteristic 0. Let C be a

symmetric tensor category over k of subexponential growth. Then, there exists a fiber functor

F : C -+ sVeck.

A consequence of this result and Tannakian reconstruction is that any symmetric tensor

category of subexponential growth in characteristic 0 is simply the category of supercomod-ules over a supercommutative Hopf superalgebra. Thus, geometry in this setting is simply supergeometry, which is a fairly well studied setting.

In characteristic p > _{0, this theorem of Deligne fails to hold. In fact for p}_>_{3, Verp} provides a counterexample. Since it is a fusion category, it is automatically subexponential.

However, it is not fibered over sVec. If such a fiber functor did exist, say F : Ver5 -+ sVec,

then the dimension of F(L3) would be a tensor invariant, but there is no natural number d

that satisfied d2

= d + 1. Similar dimension arguments also allow us to show that Verp does

not fiber over sVec for p > 5 as well. Hence, for p > 3 at least, symmetric tensor categories

Algebraic geometry and representation theory in the Verlinde category