Fully bounded grothendieck categories part I: locally noetherian categories

Derived Categories Part I

3 Derived Categories 11 3.1 Extending Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Truncations and Hearts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Bounded Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Plump Subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

Triangulated Categories Part I

Triangulated categories are important structures lying at the confluence of several exciting areas of mathematics (and even physics). Our notes on the subject are divided into three parts which, if named by the major construction occurring within them, would be titled “Verdier quotients”, “Thomason localisaton” and “Brown representability”. There are many places to learn about triangulated cate...

Fully Bounded Noetherian Rings

Let i : A → R be a ring morphism, and χ : R → A a right R-linear map with χ(χ(r)s) = χ(rs) and χ(1 R) = 1 A. If R is a Frobenius A-ring, then we can define a trace map tr : A → A R. If there exists an element of trace 1 in A, then A is right FBN if and only if A R is right FBN and A is right noetherian. The result can be generalized to the case where R is an I-Frobenius A-ring. We recover resul...

Atomical Grothendieck Categories

Motivated by the study of Gabriel dimension of a Grothendieck category, we introduce the concept of atomical Grothendieck category, which has only two localizing subcategories, and we give a classification of this type of Grothendieck categories. 1. Introduction. Given a Grothendieck category Ꮽ, we can associate with it the lattice of all localizing categories of Ꮽ and denote it by Tors(Ꮽ). We ...

Grothendieck Categories and Support Conditions