Verdier quotients of Calabi-Yau categories from quivers with potential

Abstract

We study a class of triangulated categories obtained as Verdier quotients of 3-Calabi-Yau categories combinatorially described by quivers with potential from (decorated) marked surfaces. We study their bounded t-structures and consider in particular the exchange graphs of hearts and silting objects, and show that the Koszul isomorphism between these graphs is preserved under Verdier quotient.

Figures (4)

• Figure 1: The choice of $I=\{2\}$ for the quivers $Q=A_3$ and $Q=\mu_2 A_3$ produces the same quiver $Q^{eJe}$ with trivial potential. The abelian module category does not change.
• Figure 2: A collapse of a subsurface with four components, which is in fact a collision.
• Figure 3: On the left, an example of $P_k\cup P_\ell$ with $k=8$ and $\ell=6$. On the right, a polygon with two edges identified.
• Figure 4: (Part of) the exchange graphs $\operatorname{EG}^\bullet(\mathcal{D})$ and $\operatorname{EG}^\bullet(\mathcal{D}/\mathcal{V})$ for $\mathcal{D}=\operatorname{pvd}(A_3)=\langle X_1,Y_1,Z_1\rangle$ and $\mathcal{V}=\langle X_1\rangle$ (blue) or $\mathcal{V}=\langle Y_1\rangle$ (red). The dotted lines stand for taking the quotient of hearts $\mathcal{A}/(\mathcal{A}\cap\mathcal{V})$ when the t-structure $\mathcal{A}$ is compatible with $\mathcal{V}$.

Theorems & Definitions (53)

• Theorem 1: Theorems \ref{['thm_eg_eGe']} and \ref{['quot_graph']}
• Theorem 2: Theorem \ref{['any_heart_quot']}
• Theorem 3: Theorem \ref{['thm:SPdual']}
• Theorem 4
• Definition 1
• Lemma 1: neeman, gabriel
• Remark
• Remark
• Proposition 1: antieau
• Definition 2