Derived equivalences of gentle algebras via Fukaya categories

Lekili, Yankı; Polishchuk, Alexander

doi:10.48550/arxiv.1801.06370

Cited by 18 publications

(53 citation statements)

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“…In the case that the gentle algebra is homologically smooth (and with grading concentrated in degree 0), this model coincides with the model in [13] shown to be equivalent to the partially wrapped Fukaya category of the same surface with stops [18]. The geometric models in [13,18,19] give the higher extensions between indecomposable objects, in that Ext n Λ (M, N ) ∼ = Hom D b (Λ) (M, N [n]). However, since in this set-up, the identification of D b (Λ) with the homotopy category of projectives K −,b (projΛ) is used, it is difficult in practice to concretely deduce higher extensions between indecomposable modules over gentle algebras in terms of the surface geometry.…”

Section: Introductionsupporting

confidence: 57%

“…In this context, in [19] the authors give a geometric model of the bounded derived category of a finite dimensional gentle algebra Λ. In the case that the gentle algebra is homologically smooth (and with grading concentrated in degree 0), this model coincides with the model in [13] shown to be equivalent to the partially wrapped Fukaya category of the same surface with stops [18]. The geometric models in [13,18,19] give the higher extensions between indecomposable objects, in that Ext n Λ (M, N ) ∼ = Hom D b (Λ) (M, N [n]).…”

Section: Introductionmentioning

confidence: 99%

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Higher extensions for gentle algebras

Baur¹,

Schroll²

2019

Preprint

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In this paper we determine extensions of higher degree between indecomposable modules over gentle algebras. In particular, our results show how such extensions either eventually vanish or become periodic. We give a geometric interpretation of vanishing and periodicity of higher extensions in terms of the surface underlying the gentle algebra. For gentle algebras arising from triangulations of surfaces, we give an explicit basis of higher extension spaces between indecomposable modules.

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

Higher extensions for gentle algebras

Baur¹,

Schroll²

2019

Preprint

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“…We can choose a line field to give W(Σ, Z) a Z-grading (see [16] for a recent study of this structure). There are effectively H 1 (Σ) worth of choices for the line field.…”

Section: ∂ Figure 5 Resolution Of Strand Diagrammentioning

confidence: 99%

“…Let Σ be the pair-of-pants, that is, a 3-punctured sphere, Λ be 2 stops at the outer boundary as drawn in Figure 1. We also choose a line field η on Σ which has rotation number 2 around the outer boundary and 0 along the interior boundary components (see [16] for a recent detailed study of line fields on punctured surfaces).…”

Section: Introductionmentioning

confidence: 99%

Homological mirror symmetry for higher dimensional pairs of pants

Lekili,

Polishchuk

2018

Preprint

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Using Auroux's description of Fukaya categories of symmetric products of punctured surfaces, we compute the partially wrapped Fukaya category of the complement of k + 1 generic hyperplanes in CP n , for k ≥ n, with respect to certain stops in terms of the endomorphism algebra of a generating set of objects. The stops are chosen so that the resulting algebra is formal. In the case of the complement of (n + 2)-generic hyperplanes in CP n (n-dimensional pair-of-pants), we show that our partial wrapped Fukaya category is equivalent to a certain categorical resolution of the derived category of the singular affine variety x 1 x 2 ..x n+1 = 0. By localizing, we deduce that the (fully) wrapped Fukaya category of the n-dimensional pair-of-pants is equivalent to the derived category of x 1 x 2 ...x n+1 = 0. We also prove similar equivalences for finite abelian covers of the n-dimensional pair-of-pants.

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“…This is then applied in [7] to give an explicit description of the Ext spaces between indecomposable modules over a gentle algebra. Furthermore, the computation in [6] has been applied in [9] (see also [8]) to give an interpretation of the mapping cone calculus in the context of Fukaya categories of surfaces.…”

mentioning

confidence: 99%