| 總結: | <p>This is the last in a series on configurations in an abelian category <em>A</em>. Given a finite poset (<em>I</em>, and#10927;), an (<em>I</em>, and#10927;)-configuration (σ, ι, π) is a finite collection of objects σ(<em>J</em>) and morphisms ι(<em>J, K</em>) or andpi;(<em>J, K</em>) : σ (<em>J</em>) andrightarrow; σ (<em>K</em>) in <em>A</em> satisfying some axioms, where <em>J, K</em> are subsets of <em>I</em>. Configurations describe how an object <em>X</em> in <em>A</em> decomposes into subobjects.</p> <p>The first paper defined configurations and studied moduli spaces of configurations in <em>A</em>, using Artin stacks. It showed well-behaved moduli stacks <em>Obj<sub>A</sub>, M</em>(<em>I</em>, and#10927;)<em><sub>A</sub></em> of objects and configurations in <em>A</em> exist when <em>A</em> is the abelian category coh(<em>P</em>) of coherent sheaves on a projective scheme <em>P</em>, or mod-<em>KQ</em> of representations of a quiver <em>Q</em>. The second studied algebras of constructible functions and stack functions on <em>Obj<sub>A</sub></em>.</p> <p>The third introduced <em>stability conditions</em>(τ, <em>T</em>, and#8804;) on <em>A</em>, and showed the moduli space <em>Obj</em><sup>andalpha;</sup><sub style="position: relative; left: -.6em;">ss</sub>(τ) of τ-semistable objects in class α is a constructible subset in <em>Obj<sub>A</sub></em>, so its characteristic function <em>anddelta;</em><sup>andalpha;</sup><sub style="position: relative; left: -.6em;">ss</sub>(τ) is a constructible function. It formed algebras <em>H</em><sup>pa</sup><sub style="position: relative; left: -.8em;">τ</sub>, <em>H</em><sup>to</sup><sub style="position: relative; left: -.8em;">τ</sub>, <em>H</em><sup><sup>-</sup>pa</sup><sub style="position: relative; left: -.8em;">τ</sub>, <em>H</em><sup><sup>-</sup>to</sup><sub style="position: relative; left: -.8em;">τ</sub> of constructible and stack functions on <em>Obj<sub>A</sub></em>, and proved many identities in them.</p> <p>In this paper, if (τ, T, and#8804;) and (<sup>andtilde;</sup><sub style="position: relative; left: -.4em;">andtau;</sub>, <sup>andtilde;</sup><sub style="position: relative; left: -.5em;">T</sub>, and#8804;) are stability conditions on <em>A</em> we write <em>anddelta;</em><sup>andalpha;</sup><sub style="position: relative; left: -.6em;">ss</sub>(<sup>andtilde;</sup><sub style="position: relative; left: -.4em;">andtau;</sub>) in terms of the <em>anddelta;</em><sup>andbeta;</sup><sub style="position: relative; left: -.6em;">ss</sub>(τ), and deduce the algebras <em>H</em><sup>pa</sup><sub style="position: relative; left: -.8em;">τ</sub>,..., <em>H</em><sup><sup>-</sup>to</sup><sub style="position: relative; left: -.8em;">τ</sub> are independent of (τ, <em>T</em>, and#8804;). We study invariants <em>I</em><sup>andalpha;</sup><sub style="position: relative; left: -.6em;">ss</sub>(τ) or <em>I<sub>ss</sub></em>(<em>I</em>, and#10927;, κ, τ) ‘counting’ τ-semistable objects or configurations in <em>A</em>, which satisfy additive and multiplicative identities. We compute them completely when <em>A</em>=mod-<em>KQ</em> or <em>A</em>=coh(<em>P</em>) for <em>P</em> a smooth curve. We also find invariants with special properties when <em>A</em>=coh(<em>P</em>) for <em>P</em> a smooth surface with <em>K</em><sup>-1</sup><sub style="position: relative; left: -.8em;"><em>P</em></sub> nef, or a Calabi–Yau 3-fold.</p>
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