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 A be an abelian category with a preorder, (which does not have the anti-symmetry property).

A stability structure on A is a preorder such that for any exact sequence 0ABC0 we have one of the following:

  • A<BA<CB<C

  • A>BA>CB>C

  • ABACBC

A is stable if A0 and A<A for all non-trivial subobjects A of A.

A is semistable if A0 and AA for all non-trivial subobjects A of A.

Elementary Properties

  • If A is semistable, then whenever 0AAA0, we have AAA.

  • An obvious corollary is, the set {<D}, {>D}, {D} are closed under extensions, which generalizes easily to filtrations:

    Lemma Given nonzero objects B,D and a filtration

    B=F0BF1BFmB=0

    whose factors are griB=FiB/Fi+1B, then the set {<D}, {>D}, {D} are closed under composing filtrations form factors, for example if griBD for all i then BD.

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 0AAA0.

  • A is stable iff for all nontrivial quotients Q, A<Q.

  • A is semistable iff for all nontrivial quotients Q, AQ.

Stable Objects Are Similar To Irreducible Objects

  • Given any morphism φ:AB of two semistables, we have

    0kerφAImφ0,0ImφBcokerφ0

    therefore kerφAImφBcokerφ.

Let AB be semistables with a nonzero morphism φ:AB, then

  1. AB

  2. B is stable φ is an epimorphism.

  3. A is stable φ is a monomorphism.

Some consequences: (all objects are semistable)

  • If both AB are stable, then any nonzero morphism AB is an isomorphism.

  • If A is stable, Hom(A,A)=k.

  • If A>B, then Hom(A,B)=0.