Noetherian criteria for dimer algebras

Abstract

Let $A$ be a nondegenerate dimer (or ghor) algebra on a torus, and let $Z$ be its center. Using cyclic contractions, we show the following are equivalent: $A$ is noetherian; $Z$ is noetherian; $A$ is a noncommutative crepant resolution; each arrow of $A$ is contained in a perfect matching whose complement supports a simple module; and the vertex corner rings $e_iAe_i$ are pairwise isomorphic.

Figures (6)

• Figure 1: An example of a cyclic contraction. Each quiver is drawn on a torus, and $\psi: A \to A'$ contracts the green arrow. The arrows in the two unit 2-cycles in $Q'$ are redundant generators for $A'$ and so may be removed.
• Figure 2: Examples for Proposition \ref{['Bastuff']}. All paths shown are paths of positive length in the cover $Q^+$ (with the superscripts $^+$ omitted). In (i), the paths $p, q \in e_jAe_i$ satisfy $\operatorname{t}(p^+) = \operatorname{t}(q^+)$ and $\operatorname{h}(p^+) = \operatorname{h}(q^+)$. In (ii), the path $p = p_3p_2p_1$ is in $\mathcal{C}_i \setminus \hat{\mathcal{C}}_i$, since the cyclic permutation $(p_1p_3p_2)^+$ of $p^+$ has a nontrivial cyclic subpath. Furthermore, $p_3p_1$ is in $\mathcal{C}^0$, and therefore $\mkern 1.5mu\overline{\mkern-1.5mup_3p_1\mkern-1.5mu}\mkern 1.5mu = \sigma^{\ell}$ for some $\ell \geq 1$.
• Figure 3: Setup for Lemma \ref{['p-q in I']}. Here, $p_1,p_2,q,r_1,r_3,s \in Q_{\geq 0}$ are paths (of unspecified length); $r_2 \in Q_1$ is an arrow; $r_2s$ is a unit cycle; and $r_2r_1p_2$ lifts to a nontrivial cycle in the cover. The paths $p = p_2p_1$, $q$, $r = r_3r_2r_1$, are drawn in red, blue, and green respectively.
• Figure 4: Setup for Proposition \ref{['min length']}. Here, $p_1,p_2, q, r_1, r_3, s \in Q_{\geq 0}$ are paths; $r_2 \in Q_1 \setminus Q_1^{\mathcal{S}}$ is an arrow; and $r_2s$ is a unit cycle. The paths $p = p_2p_1$, $q$, $r = r_2r_1$, are drawn in red, blue, and green respectively.
• Figure 5: Setup for Remark \ref{['eats and sleeps and not much more']}. In case (i), the paths $p,q$, drawn in red and blue, form a non-cancellative pair that is not minimal. The green path $r$ is a minimal path satisfying $rp \equiv rq$ and lies outside of $\mathcal{R}_{p,q}$. In case (ii), the paths $p',q'$, again drawn in red and blue, form a non-cancellative pair that is minimal. The green path $r'$ is a minimal path satisfying $r'p' \equiv r'q'$ and, in contrast to case (i), necessarily lies inside of $\mathcal{R}_{p',q'}$.

Theorems & Definitions (41)

• Theorem 1.1
• Proposition 2.1
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Remark 3.4
• Corollary 3.5