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Grothendieck Categories, Monoidal Categories and Frobenius Algebras

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Grothendieck Categories, Monoidal Categories and Frobenius Algebras University of Bucharest, Faculty of Mathematics and Computer Science Grothendieck Categories, Monoidal Categories and Frobenius Algebras Author: Laura Elena N˘ast˘asescu Doctoral supervisor: Prof. Dr. Sorin D˘asc˘alescu A thesis presented for the degree of Doctor of Philosophy Bucharest, 2016 ABSTRACT INTRODUCTION Grothendieck categories were introduced by Alexander Grothendieck in his paper [32] published in 1957. His aim was to use the homological alge- bra methods for the study of sheaves. A systematic study of Grothendieck categories was done by Gabriel in his thesis [28]. The study of Grothendieck categories has unified many theories in mathematics. Relevant examples of Grothendieck categories are: the category of left modules over a ring, the category of quasi-coherent sheaves on an algebraic variety, the category of sheaves of abelian groups on a topological space, the category of graded mod- ules over a graded ring, the category of comodules over a coalgebra, the cate- gory of generalized Doi-Hopf modules. At the beginning of the theory it was an open question whether any Grothendieck category is equivalent to a cate- gory of modules (over a ring). The answer turned out to be negative. For ex- ample it is possible to show that certain categories of graded modules are not equivelnt to a category of modules. However, Popescu and Gabriel proved in 1964 a famous theorem that shows that each Grothendieck category is equiv- alent to a quotient category of a module category, see [29]. More precisely, if A is a Grothendieck category with a generator G, R = EndA(G) is the endo- morphism ring of G and Mod(R) is the category of unitary right R-modules, then the functor T : A! Mod(R) defined by T (X) = HomA(G; X) on objects X of A and by T (f) = HomA(G; f): T (X) ! T (Y ) on morphisms f : X ! Y in A is fully faithful and has an exact left adjoint S. The initial proof was rather complicated; especially the part on the exactness of the functor S. In order to simplify the proof, the initial proof was revisited by several authors, among them the most elegant and short proof in chronologi- cal order where given by Takeuchi [65], Ulmer [67] and Mitchell [44]. The last one uses an ingenious lemma, that remained in literature as Mitchell Lemma and also the existence of enough injective objects in any Grothendieck cate- gory. 2 Abstract The concept of localization was first studied for commutative rings. In this case we consider P a prime ideal of a commutative ring R. The com- plement of this ideal is denoted by SP = R − P , that is a multiplicative set. Then we can consider the localizing subcategory of R − Mod denoted by −1 CP = fM 2 R − ModjSP M = 0g It is easy to see that this is the same thing with CP = fM 2 R − ModjHomR(M; E(R=P )) = 0g where E(R=P ) denotes the injective hull of R=P . Moreover, the quotient category R − Mod=CP is equivalent to the category RP − Mod of all left modules over the localized ring RP . When the notion of abelian category was introduced the notion of hereditary torsion theory appeared and it quickly became a powerful instrument for studying categories. A hereditary torsion theory in an arbitrary abelian category A is the same thing with having a localizing subcategory C in A. Moreover, for every localizing subcategory C in A we have the concept of quotient category A=C. The natural question to study here is the proximity between any localizing subcategory C of A and the localizing subcategories of CP -type. Cahen was the first who introduced a notion of "stability". He introduced this notion for a localizing subcategory of a module category over a commutative ring. In this case a localizing T subcategory is said to be stable if C = P 2X CP , for a set X ⊆ Spec(R). One of the aims of this thesis is to revisit the Gabriel-Popescu Theorem and its generalizations, and also to introduce a notion of stability in the general case of Grothendieck categories. Another aim is to look at special examples of Grothendieck categories, more precisely categories of corepre- sentations over Hopf algebras, and to investigate Frobenius algebras and symmetric algebras in such categories. Frobenius algebras originate in the work of F.G. Frobenius on representation theory of finite groups. Their study was initiated by Brauer, Nesbitt and Nakayama and then was continued by Dieudonne,Eilenberg, Azumaya etc. Frobenius algebras have played an im- portant role in Hopf algebra theory, because any finite dimensional Hopf alge- bra is Frobenius, cohomology rings of compact oriented manifolds, solutions of the quantum Yang-Baxter equation, Jones polynomials and topological quantum field theory. It is a challenging problem to understand the signifi- cance of Frobenius algebras in monoidal categories other than the categories Abstract 3 of vector spaces. We discuss this problem for the category of right comodules over a Hopf algebra H. Under additional assumptions on H, we also look at symmetric algebras in the category. A special attention is given to the case where H is a group Hopf algebra kG, where G is an arbitrary group; in this situation the category of corepresentations over H is just the category of G-graded vector spaces, and an algebra in the category is a G-graded alge- bra. One last direction of study in this thesis is also related to group graded algebras. The theory of algebras graded by an arbitrary group has been sys- tematically developed after 1970. The main two sources of inspiration were polynomial algebras with the (usual) positive grading by integers, and group algebras with the standard grading by the considered group. The study of group gradings on several classes of algebras have been of interest in commu- tative and noncommutative algebras, on Lie algebras and in representation theory. We are interested in gradings on polynomial algebras. The thesis is divided in four chapters. Chapter 1: Grothendieck categories. A generalization of Mitchell's Lemma In 1964 Pierre Gabriel and Nicolae Popescu showed that each Grothendieck category is equivalent to a quotient category of a module category. More precisely if: •A is a Grothendieck category with a generator G • R = EndA(G) is the endomorphism ring of G • Mod(R) the category of unitary right R-modules then the functor T : A! Mod(R) defined by: • T (X) = HomA(G; X), where X 2 A • T (f) = HomA(G; f): T (X) ! T (Y ) on morphisms f : X ! in A is fully faithful and has an exact left adjoint S. Afterward it was revisited by several authors; among the most elegant and short proofs in cronological order are those of Takeuchi, Ulmer and Mitchell. The latter uses an ingenious lemma, referred to as Mitchell Lemma. Our aim in this chapter is to extend the Mitchell lemma from module categories to functor categories and to show how it can be used in order 4 Abstract to obtain in an easier way the Ulmer Theorem on the exactness of S, a generalized Gabriel-Popescu Theorem and a generalized Takeuchi Lemma. We will use the following notations: •A a Grothendieck category •U the set of all objects of A Gen(U) = fA 2 Aj(9)f : L U ! A ! 0g, the full subcategory of Ui2U i A, that is a preabelian category of A (has kernels and cokernels) L •AU = fM 2 Aj(8) morfism f : U2F U ! M 2 A cu F o submul- time finita a lui U, Ker(f) 2 Gen(A)g Let (U op; Ab) be a category defined in the following way: • objects: the additive contravariant functors from U to Ab • morphisms: the natural transformations between such functors op (U ; Ab) is a Grothendieck category, whose functors (hU )U2U , where hU form a generating family of finitely generated projective objects for (U op; Ab). Consider the functor T : A! (U op; Ab) defined by: • T (X) = HomA(−;X)jU on objects X 2 A • T (f) = HomA(−; f)jU : T (X) ! T (Y ) on morphism f : X ! Y in A It is known that T has a left adjoint S :(U op; Ab) !A. In particular we have that S(hU ) = U,(8)U 2 U. The most important result of this chapter is the following: Lemma 1. (Generalized Mitchell Lemma) Consider: •U be a set of objects of A • A, B objects of A with A 2 AU • MA a subobject of T (A) op • G : MA ! T (B) a morphism in (U ; Ab) We denote: S • M = U2U MA(U) Abstract 5 • (8)m 2 M take Um 2 U for which m 2 MA(U) L • (8)m 2 M, pm : m2M Um ! Um the canonical projection L • : m2M Um ! A the unique morphism with um = m, (8)m 2 M L • φ : m2M Um ! B the unique morphism with φum = GU (m), (8)m 2 MA(U) The φ factors through Im( ). We will have the following three important applications, that follow di- rectly from Mitchell Lemma. Theorem 2. (Ulmer) The functor S :(U op; Ab) !A is exact if and only if U ⊆ AU . Theorem 3. (Generalized Gabriel Popescu) Assume that U is a family of A. Then T : A! (U op; Ab) is a full and faithful functor, and its left adjoint S :(U op; Ab) !A is exact. Corollary 4. (Takeuchi Lemma): Let A be an object of A, let YA be a subobject of T (A) and denote by i : YA ! T (A) the inclusion morphism.
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