Fonctorial Construction of Frobenius Categories

Let $\Ascr,\Bscr$ be exact categories with $\Ascr$ karoubian and $M$ be an exact functor. Under suitable adjonction hypotheses for $M$, we are able to show that the direct factors of the objects of $\Ascr$ of the form $MY$ with $Y \in \Bscr$ make up a Frobenius category which allow us to define an $M$-stable category for $\Ascr$ only by quotienting. In addition, we propose a construction of an $M$-stable category for $\Ascr,\Bscr$ triangulated categories and $M$ a triangulated functor. We illustrate this notion with a theorem of Keller and Vossieck which links the two notions of $M$-stable category.

💡 Research Summary

The paper investigates how a single exact (or triangulated) functor (M) between two exact categories (\Ascr) and (\Bscr) can be used to produce a Frobenius structure and, consequently, an (M)-stable category. The authors begin by assuming that (\Ascr) is Karoubian (all idempotents split) and that (M:\Bscr\to\Ascr) is exact and admits both a left adjoint (L) and a right adjoint (R), i.e. a triple adjunction ((L\dashv M\dashv R)). This hypothesis guarantees that (M) preserves exact sequences and that the adjoints provide enough “free” and “co‑free” objects to control projective and injective behavior inside (\Ascr).

From these data the authors define a full subcategory
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