Stability Conditions on Abelian Comma Categories
Ellen de Oliveira, Guido Neulaender
TL;DR
The paper develops a general theory for comma categories $(F/G)$ built from abelian categories ${\mathcal A},{\mathcal B},{\mathcal C}$ with $F:{\mathcal A}\to{\mathcal C}$ right exact and $G:{\mathcal B}\to{\mathcal C}$ left exact, showing these categories are abelian and have Grothendieck groups decomposing as $K((F/G))\cong K({\mathcal A})\oplus K({\mathcal B})$. It then analyzes stability notions, proving that stability data lift to or descend from the comma category via suitable linear combinations of stability functions on ${\mathcal A}$ and ${\mathcal B}$, and establishes when Harder–Narasimhan and Jordan–Hölder filtrations exist in $(F/G)$. The work further characterizes when noetherian/artinian properties transfer between the comma category and its constituents, and it extends the framework to co-comma categories and other variations, showing parallel abelian-structure criteria. The results unify and generalize the treatment of decorated sheaves (e.g., coherent systems) under a single categorical umbrella, with concrete implications for stability conditions and filtrations across related constructions.
Abstract
A comma category, exemplified in algebraic geometry by coherent systems, combines two categories over a third through morphisms between their objects. We establish sufficient conditions for it to be abelian, compute its Grothendieck group, and give necessary and sufficient criteria for it to be noetherian and artinian. Finally, we define a stability condition on abelian comma categories under hypotheses on the initial categories and, conversely, induce stability conditions on the initial abelian categories from those on the comma categories.