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An algebraic approach to Harder-Narasimhan filtrations. (arXiv:1810.06322v1 [math.CT])

来源于:arXiv
It is a well-known fact that a stability condition $\phi: Obj^* \mathcal{A} \to \mathcal{I}$ over an abelian length category $\mathcal{A}$ induces a chain of torsion classes $\eta_\phi$ indexed by the totally ordered set $\mathcal{I}$. Inspired by this fact, in this paper we study all chains of torsion classes $\eta$ indexed by a totally ordered set $\mathcal{I}$ in $\mathcal{A}$. Our first theorem says that every chain of torsion classes $\eta$ indexed by $\mathcal{I}$ induces a Harder-Narasimhan filtration to every nonzero object of $\mathcal{A}$. Building on this, we are able to generalise several of the results showed by Rudakov in \cite{Rudakov1997}. Moreover we adapt the definition of slicing introduced by Bridgeland in \cite{Bridgeland2007} and we characterise them in terms of indexed chain of torsion classes. Finally, we follow ideas of Bridgeland to show that all chains of torsion classes of $\mathcal{A}$ indexed by the set $[0,1]$ form a metric space with a natural wall and c 查看全文>>