Exact sequences of rt-categories

Abstract

Our aim is to consider what the exact sequence for rt-categories is. For this, we introduce the notion of exact sequence of rt-categories, modeled on exact sequences of finite tensor categories. Our central result explores the relationship of exactness at different levels. Specifically, let $H_1\stackrel{f}{\to }{H}_2\stackrel{g}{\to }{H}_3$ be a sequence of finite-dimensional Hopf algebras. We prove that $H_1\stackrel{f}{\to }{H}_2\stackrel{g}{\to }{H}_3$ is strictly exact if and only if $H_1\text{-}\mathrm{comod}\stackrel{f_{\ast }}{\to }{H}_2\text{-}\mathrm{comod}\stackrel{g_{\ast }}{\to }{H}_3\text{-}\mathrm{comod}$ is an exact sequence of finite tensor categories and $g_{\ast }$ admits an exact left adjoint, if and only if $D_{H_1\text{-}\mathrm{comod}}^b(H_2\text{-}\mathrm{comod})\to D^b(H_2\text{-}\mathrm{comod})\to D^b(H_3\text{-}\mathrm{comod})$ is an exact sequence of rt-categories and $f_{\ast }$ is fully faithful.

11 pages