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

A stability structure on \(\mathcal{A}\) is a preorder such that for any exact sequence \[0\to A\to B\to C\to 0\] we have one of the following:

  • \(A < B \Leftrightarrow A < C \Leftrightarrow B < C\)

  • \(A > B \Leftrightarrow A > C \Leftrightarrow B > C\)

  • \(A \asymp B \Leftrightarrow A \asymp C \Leftrightarrow B \asymp C\)

\(A\) is stable if \(A\neq 0\) and \(A'<A\) for all non-trivial subobjects \(A'\) of \(A\).

\(A\) is semistable if \(A\neq 0\) and \(A'\le A\) for all non-trivial subobjects \(A'\) of \(A\).

Elementary Properties

  • If \(A\) is semistable, then whenever \(0\to A'\to A\to A''\to 0\), we have \(A'\le A\le A''\).

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

    Lemma Given nonzero objects \(B,D\) and a filtration

    \[B = F^0B \supset F^1B \supset \cdots \supset F^mB = 0\]

    whose factors are \(\mathrm{gr}^iB = F^iB/F^{i+1}B\), then the set \(\{<D\}\), \(\{>D\}\), \(\{\asymp D\}\) are closed under composing filtrations form factors, for example if \(\mathrm{gr}^iB \asymp D\) for all \(i\) then \(B \asymp D\).

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 \(0\to A'\to A\to A''\to 0\).

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

  • \(A\) is semistable iff for all nontrivial quotients \(Q\), \(A\le Q\).

Stable Objects Are Similar To Irreducible Objects

  • Given any morphism \(\varphi:A\to B\) of two semistables, we have

    \[ 0\to \ker\varphi \to A \to \mathrm{Im}\varphi \to 0, \quad 0\to \mathrm{Im}\varphi \to B \to \mathrm{coker}\varphi \to 0\]

    therefore \(\ker\varphi \le A\le \mathrm{Im}\varphi\le B \le \mathrm{coker}\varphi\).

Let \(A\ge B\) be semistables with a nonzero morphism \(\varphi:A\to B\), then

  1. \(A\asymp B\)

  2. \(B\) is stable \(\Rightarrow\) \(\varphi\) is an epimorphism.

  3. \(A\) is stable \(\Rightarrow\) \(\varphi\) is a monomorphism.

Some consequences: (all objects are semistable)

  • If both \(A\ge B\) are stable, then any nonzero morphism \(A\to B\) is an isomorphism.

  • If \(A\) is stable, \(\mathrm{Hom}(A,A)=k\).

  • If \(A>B\), then \(\mathrm{Hom}(A,B)=0\).