On the finiteness of the derived equivalence classes of some stable endomorphism rings

August, Jenny

doi:10.1007/s00209-020-02475-y

Cited by 14 publications

(13 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…giving that T A ∼ = (P A ⊕ P A [3]) ⊆1 ∼ = P A . (b) It follows from (a) combined with Corollary 3.4 and Lemmas 3.5 and 3.6 that…”

Section: (E) Implies That P a ⊗ A Amentioning

confidence: 99%

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Kortegaard¹

2022

Preprint

View full text Add to dashboard Cite

Consider a k-linear Frobenius category E with a projective generator such that the corresponding stable category C is 2-Calabi-Yau, Hom-finite with split idempotents. Let l, m ∈ C be maximal rigid objects with self-injective endomorphism algebras. We will show that their endomorphism algebras C (l, l) and C (m, m) are derived equivalent. Furthermore we will give a description of the two-sided tilting complex which induces this derived equivalence.

show abstract

“…giving that T A ∼ = (P A ⊕ P A [3]) ⊆1 ∼ = P A . (b) It follows from (a) combined with Corollary 3.4 and Lemmas 3.5 and 3.6 that…”

Section: (E) Implies That P a ⊗ A Amentioning

confidence: 99%

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Kortegaard¹

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…There is, however, a much more intrinsic description of that does not rely on this larger algebra, via the two-term tilting complexes of the contraction algebra . In particular, in the language of g -vectors, , where is the g -vector of the two-term complex of associated to the rigid object via the bijection [Au1, 2.18]. We do not use this description below.…”

Section: Wall Crossing and Functorial Compositionmentioning

confidence: 99%

“…The contraction algebra is a silting-discrete symmetric algebra [Au1, 3.3, 4.12]. Being symmetric, the technical condition of being silting-discrete is equivalent [AM, 2.11] to there being only finitely many basic tilting complexes between P and (with respect to the silting order ), for every tilting complex P obtained by iterated irreducible left mutation from the free module .…”

Section: Stability and T-structuresmentioning

confidence: 99%

See 1 more Smart Citation

Stability conditions for contraction algebras

August

Wemyss

2022

Forum of Mathematics, Sigma

Self Cite

View full text Add to dashboard Cite

This paper gives a description of the full space of Bridgeland stability conditions on the bounded derived category of a contraction algebra associated to a $3$ -fold flop. The main result is that the stability manifold is the universal cover of a naturally associated hyperplane arrangement, which is known to be simplicial and in special cases is an ADE root system. There are four main corollaries: (1) a short proof of the faithfulness of pure braid group actions in both algebraic and geometric settings, the first that avoid normal forms; (2) a classification of tilting complexes in the derived category of a contraction algebra; (3) contractibility of the stability space associated to the flop; and (4) a new proof of the $K(\unicode{x3c0} \,,1)$ -theorem in various finite settings, which includes ADE braid groups.

show abstract

“…The above is remarkable: it says that not only are there just finitely many algebras in the derived equivalence class of the finite dimensional algebra A con (by [Aug20b]), furthermore the dimensions of all the other algebras can be easily obtained combinatorially from the first. The proof of Corollary 1.7 is slightly subtle, since it is not a priori clear that the GV invariants defined by Toda are the same as the GV invariants defined here, but this is all discussed in Appendix A.…”

mentioning

confidence: 99%

GV and GW invariants via the enhanced movable cone

Nabijou,

Wemyss

2021

Preprint

View full text Add to dashboard Cite

Given any smooth germ of a threefold flopping contraction, we first give a combinatorial characterisation of which Gopakumar-Vafa (GV) invariants are nonzero, by prescribing multiplicities to the walls in the movable cone. This allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of [IW], and moreover that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix which controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes, (2) a closed formula which describes the change in dimension of the contraction algebra under flop, and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

show abstract

On the finiteness of the derived equivalence classes of some stable endomorphism rings

Cited by 14 publications

References 38 publications

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Stability conditions for contraction algebras

GV and GW invariants via the enhanced movable cone

Contact Info

Product

Resources

About