Author: Eiko
Tags: stability, abelian category, moduli
Time: 2024-11-13 16:43:46 - 2024-11-13 16:43:46 (UTC)
Reference: Stability for an Abelian Category by Alexei Rudakov
Stability In Abelian Category
Let be an abelian category with a preorder, (which does not have the anti-symmetry property).
A stability structure on is a preorder such that for any exact sequence we have one of the following:
is stable if and for all non-trivial subobjects of .
is semistable if and for all non-trivial subobjects of .
Elementary Properties
If is semistable, then whenever , we have .
An obvious corollary is, the set , , are closed under extensions, which generalizes easily to filtrations:
Lemma Given nonzero objects and a filtration
whose factors are , then the set , , are closed under composing filtrations form factors, for example if for all then .
Stability In Terms Of Quotients
Because of the elementary properties, the definition of stability can be rephrased in terms of quotients, since they sit in the middle of the exact sequence .
is stable iff for all nontrivial quotients , .
is semistable iff for all nontrivial quotients , .
Stable Objects Are Similar To Irreducible Objects
Let be semistables with a nonzero morphism , then
is stable is an epimorphism.
is stable is a monomorphism.
Some consequences: (all objects are semistable)