Applications of exact structures in abelian categories

Abstract

In an abelian category A with small Ext groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors F of Ext 1 A (−, −) such that A has enough F-projectives and enough F-injectives and Quillen exact structures E with enough E -projectives and enough E -injectives. In this case, we get a strengthened version of the translation of the Wakamatsu lemma to the exact context, and also prove that subcategories which are E -resolving and epimorphic precovering with kernels in their right E -orthogonal class and subcategories which are E -coresolving and monomorphic preenveloping with cokernels in their left E -orthogonal class are determined by each other. Then we apply these results to construct some (pre)enveloping and (pre)covering classes and complete hereditary E -cotorsion pairs in the module category.

Relative left derived functors of tensor product functors

“…In [11], we introduced cotorsion pairs relative to a given balanced pair (C , D) in an abelian category. In this section, we study cotorsion pairs induced by Tor (2) By Proposition 3.7, M ∈ ⊥ * (N + ) ⇐⇒ M ∈ * N for any right R-module M .…”

“…Then Holm provided in [7] a sufficient condition for the functor − ⊗ − being balanced with respect to Gorenstein flat modules, and he investigated the relations between the Gorenstein left derived functor Gtor(−, −) and the classical left derived functor Tor(−, −). Recently, we introduced a more general right derived functors Ext i A (−, −) induced by Hom(−, −) relative to a given balanced pair in [11], where A is an abelian category. The aim of this paper is to introduce and study the relative left derived functor Tor (F ,F )…”

Balanced pairs on triangulated categories

“…It should be noted that this subject first appeared in Enochs' work (see [6] for instance), where a pair (H, G) in A is balanced if and only if the bifunctor Hom A (−, −) is right balanced by H × G. For examples of balanced pairs, the reader may refer to [6,Chapter 8]. An interesting and deep result in [13] is that in an abelian category A with small Ext groups, there exists a one-to-one correspondence between balanced pairs and Quillen exact structures ξ with enough ξ-projectives and enough ξ-injectives. Motivated by this, it seems natural to introduce the notion of balanced pairs in a triangulated category C, and to establish certain relations connecting balanced pairs with certain classes of triangles in C. Thus, we have the following main result of this paper.…”

System-level lifetime-oriented power sharing control of paralleled DC/DC converters

“…Moving towards to the Power Electronic based Power Systems (PEPSs) [1], however, also pose new challenges to the reliable operation of the power system. Thereby, Design for Reliability (DfR) in power electronic converters has gained significant interest recently [2]- [9]. Furthermore, system-level reliability assessment during operation of a converter as the main part of a PEPS should be considered in order to manage the system risks.…”

“…Lifetime extension employing active thermal control is introduced in [8] by adapting the switching frequency, while the system efficiency is reduced. Optimal operation of parallel converters is presented in [9] extending the lifetime while increasing the thermal stress of the components.…”