Decorated Marked Surfaces III: The Derived Category of a Decorated Marked Surface

@article{Buan2019DecoratedMS,
  title={Decorated Marked Surfaces III: The Derived Category of a Decorated Marked Surface},
  author={Aslak Bakke Buan and Yu Qiu and Yu Zhou},
  journal={International Mathematics Research Notices},
  year={2019}
}
We study the Ginzburg dg algebra $\Gamma _{\mathbf {T}}$ associated with the quiver with potential arising from a triangulation $\mathbf {T}$ of a decorated marked surface ${\mathbf {S}}_\bigtriangleup$, in the sense of [22]. We show that there is a canonical way to identify all finite-dimensional derived categories $\operatorname {\mathcal {D}}_{fd}(\Gamma _{\mathbf {T}})$, denoted by $\operatorname {\mathcal {D}}_{fd}({\mathbf {S}}_\bigtriangleup )$. As an application, we show that the… Expand

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References

SHOWING 1-10 OF 33 REFERENCES
Decorated marked surfaces (part B): topological realizations
We study categories associated to a decorated marked surface $${\mathbf {S}}_\bigtriangleup $$S△, which is obtained from an unpunctured marked surface $$\mathbf {S}$$S by adding a set of decoratingExpand
Decorated marked surfaces: spherical twists versus braid twists
We are interested in the 3-Calabi-Yau categories $${\mathcal {D}}$$D arising from quivers with potential associated to a triangulated marked surface $$\mathbf {S}$$S (without punctures). We proveExpand
Decorated marked surfaces II: Intersection numbers and dimensions of Homs
  • Y. Qiu, Yu Zhou
  • Mathematics
  • Transactions of the American Mathematical Society
  • 2018
We study the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a decorated marked surface $\mathbf{S}_\bigtriangleup$ introduced by the first author. We proveExpand
Stability conditions on triangulated categories
This paper introduces the notion of a stability condition on a triangulated category. The motivation comes from the study of Dirichlet branes in string theory, and especially from M.R. Douglas'sExpand
Deformed Calabi–Yau completions
Abstract We define and investigate deformed n-Calabi–Yau completions of homologically smooth differential graded (= dg) categories. Important examples are: deformed preprojective algebras ofExpand
Weight structures and simple dg modules for positive dg algebras
Using techniques due to Dwyer-Greenlees-Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whoseExpand
Exchange graphs and Ext quivers
Abstract We study the oriented exchange graph EG ∘ ( Γ N Q ) of reachable hearts in the finite-dimensional derived category D ( Γ N Q ) of the CY-N Ginzburg algebra Γ N Q associated to an acyclicExpand
Deriving DG categories
— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5],Expand
Quivers with potentials associated to triangulated surfaces
We attempt to relate two recent developments: cluster algebras associated to triangulations of surfaces by Fomin-Shapiro-Thurston, and quivers with potentials and their mutations introduced byExpand
Calabi-Yau algebras
We introduce some new algebraic structures arising naturally in the geometry of Calabi-Yau manifolds and mirror symmetry. We give a universal construction of Calabi-Yau algebras in terms of aExpand
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