A course on derived categories

Abstract

In the forty years of existence of derived category, it was first thought as a tool in algebraic geometry, especially in the development of duality theories that were done by Hartshorne and others. After these first moments, the theory provided a powerful homological tool for the study of linear differential equations. The basic example in the literature that can be found about this is the Riemann-Hilbert problem of associating suitable regular systems of differential equations to constructible sheaves. This studies can be found in the work of Kashiwara and Schapira. See M. Kashiwara, P. Schapira  “Sheaves on manifolds" ([15]). To understand the structure of the derived category is necessary to study the axioms of triangulated categories that were introduced in the mid 1960’s by J.L.Verdier in his thesis “Des catégories dérivées des catégories abéliennes" ([21]). The role of the triangles in the derived category is a similar role of the exact sequence in the abelian category. But it is important to remember that these axioms had their origins in algebraic geometry and algebraic topology. Nowadays there are important applications of triangulated categories in areas like algebraic geometry, algebraic topology (stable homotopy theory), commutative algebra, differential geometry and representation theory of artin algebras. See, for instance, the book of D. Happel- “Triangulated categories and the representation theory of finite dimensional algebras" ([11]). The objective of this notes is to present an introdutory material to the undergraduate and graduate students that would like to know some ideas about the derived category. These are the notes a one week series of introductory lectures which I gave in the XXIII-Escola de Algebra, in Maringá, Paraná, Brazil. Firstly we introduced the concepts of additive and abelian category to show the axioms of triangulated category that are our main objective. The triangulated category obey four axioms. We first introduced the first three axioms and their consequences on chapter one and then the octahedral axioms in various equivalent forms in a separate section of the first chapter. The objective of this section is to give a model capable of making this axiom more palatable since, in general, the form that it is presented in the literature does not remind the reader of any similar structure in other fields of mathematics. So, we make the necessary efforts here to present another form of this axiom that is similar to other tools that could be seen in the abelian categories. We present in chapter one the main example of triangulated category, the homotopy category of complexes. Secondly, to understand the morphisms in the derived category I introduced the concept of localization in chapter two. To those that are starting to study localization, we present the necessary background to understand the localization of non commutative ring. We believe that with this model in mind the student will profit more from the study of localization of categories. On chapter two, the student will find the necessary information and exercises to begin to manipulate morphisms in the derived category. So, on chapter three we introduce the definition of derived category of an abelian category and we explain how one sees the original abelian category as a subcategory of its derived category. After having done all this work, it is natural to have many questions about the behavior of the derived category or its applications. Therefore, we present here a bibliography in portuguese and in english that will help the students to make further investigations. The reader that whishes to know the history and the motivation of the begining of the derived category with many details, should read the introduction of the book "Sheaves on Manifolds - M. Kashiwara and M. Schapira ([15]). Acknowledgements: I am particularly grateful to Sônia Maria Fernandes-DMAUFV, Tanise Carnieri Pierin -DMAT-UFPR and Eduardo Nascimento Marcos IMEUSP, who carefully worked through the text and sent me detailed lists of corrections, questions and remarks. These notes were writen for the first time in 2014 and were used in a minicourse which I tough in the XXIII-Escola de Algebra in Maringá, Paraná, Brazil. The last version was written during my visit to IME-USP in 2018, where I got finantial help of Fapesp, process 2018/08104 - 3.