This article is cited in 1 paper

RESEARCH ARTICLE

Dg algebras with enough idempotents, their $\mathrm{dg}$ modules and their derived categories

Manuel Saorín

Departemento de Matemáticas, Universidad de Murcia, Aptdo. 4021, 30100 Espinardo, Murcia, Spain

Abstract: We develop the theory $\mathrm{dg}$ algebras with enough idempotents and their $\mathrm{dg}$ modules and show their equivalence with that of small $\mathrm{dg}$ categories and their $\mathrm{dg}$ modules. We introduce the concept of $\mathrm{dg}$ adjunction and show that the classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules over associative unital algebras are extended as $\mathrm{dg}$ adjunctions between categories of $\mathrm{dg}$ bimodules. The corresponding adjunctions of the associated triangulated functors are studied, and we investigate when they are one-sided parts of bifunctors which are triangulated on both variables. We finally show that, for a $\mathrm{dg}$ algebra with enough idempotents, the perfect left and right derived categories are dual to each other.

Keywords: $\mathrm{dg}$ algebra, $\mathrm{dg}$ module, $\mathrm{dg}$ category, $\mathrm{dg}$ functor, $\mathrm{dg}$ adjunction, homotopy category, derived category, derived functor.

MSC: Primary 16E45, 18E30; Secondary 16E35, 18E25

Received: 14.12.2016

Language: English

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