Configurations in abelian categories. II. Ringel–Hall algebras

By D. Joyce. 2005

157 citations recorded by the source corpus.

Abstract This is the second in a series on configurations in an abelian category A . Given a finite poset ( I , ≼ ) , an ( I , ≼ ) -configuration ( σ , ι , π ) is a finite collection of objects σ ( J ) and morphisms ι ( J , K ) or π ( J , K ) : σ ( J ) → σ ( K ) in A satisfying some axioms, where J , K ⊆ I . Configurations describe how an object X in A decomposes into subobjects. The first paper defined configurations and studied moduli spaces of ( I , ≼ ) -configurations in A , using the theory of Artin stacks. It showed well-behaved moduli stacks Obj A , M ( I , ≼ ) A of objects and configurations in A exist when A is the abelian category coh ( P ) of coherent sheaves on a projective scheme P, or mod- K Q of representations of a quiver Q. Write CF ( Obj A ) for the vector space of Q -valued constructible functions on the stack Obj A . Motivated by the idea of Ringel–Hall algebras, we define an associative multiplication ∗ on CF ( Obj A ) using pushforwards and pullbacks along 1-morphisms between configuration moduli stacks, so that CF ( Obj A ) is a Q -algebra. We also study representations of CF ( Obj A ) , the Lie subalgebra CF ind ( Obj A ) of functions supported on indecomposables, and other algebraic structures on CF ( Obj A ) . Then we generalize all these ideas to stack functions SF ( Obj A ) , a universal generalization of constructible functions, containing more information. When Ext i ( X , Y ) = 0 for all X , Y ∈ A and i > 1 , or when A = coh ( P ) for P a Calabi–Yau 3-fold, we construct (Lie) algebra morphisms from stack algebras to explicit algebras, which will be important in the sequels on invariants counting τ-semistable objects in A .

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f99b6da797812ec84e6a518b5fad10f2a496f7bf

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Citations in the corpus

Stability conditions and curve counting invariants on Calabi–Yau 3-folds (2011, unknown): “The following result is proved in [17].”
Moduli stacks and invariants of semistable objects on K3 surfaces (2007, background): “Joyce’s recent works [19], [20], [16], [17], [21], especially [17, Conjecture 6.”
Constructible Functions on Artin Stacks (2004, background): “We highlight three interesting cases, the näıve pushforward CFna, which ignores stabilizer groups, the stack pushforward CFstk, which is most natural in many stack problems, and the orbifold pushforward CForb, related to Deligne–Mumford stacks and their crepant resolutions.”
Dirichlet Branes and Mirror Symmetry (2009, background): “Without making further technical points, let us just mention that the theory also has connections to topological strings [381, 354, 386, 437], modular forms [111], Ringel-Hall algebras [269, 270, 271, 273, 59] as well as the combinatorics of mutations on quivers [313].”
Hilbert schemes and stable pairs: GIT and derived category wall crossings (2009, unknown): “The more fundamental ingredient in our proof is Joyce’s motivic Ringel-Hall algebra [10].”
MOTIVIC INVARIANTS OF ARTIN STACKS AND ‘STACK FUNCTIONS’ (2005, methodology): “They will be important tools in the author’s series on ‘Configurations in abelian categories’.”
Hall algebras and curve-counting invariants (2010, methodology): “Joyce’s proof of this crucial result uses the full force of the technology developed in the long series of papers [11, 12, 13], including moduli spaces of configurations and the virtual projection operators Πk.”
A theory of generalized Donaldson–Thomas invariants (2008, methodology): “The proof that ˜ � is a Lie algebra morphism combines the proof in [ 12 ] that � is a Lie algebra morphism with the two Behrend function identities”