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Semi-steady non-commutative crepant resolutions via regular dimer models. (English) Zbl 1419.13019
Non-commutative crepant resolutions (NCCRs) were introduced in [M. Van den Bergh, in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 749–770 (2004; Zbl 1082.14005)]. Given a Cohen-Macaulay normal domain $$R$$, an NCCR is the endomorphism ring $$\Lambda = \mathrm{End}_R(M)$$ of a reflexive $$R$$-module $$M$$, with $$\Lambda$$ satisfying some non-singularity conditions. Many NCCRs can be realized as path algebras of quivers. The paper under review investigates NCCRs where $$M$$ is called semi-steady, showing an equivalence to path algebras from square dimer models.
This paper is a sequel to [O. Iyama and Y. Nakajima, J. Noncommut. Geom. 12, No. 2, 457–471 (2018; Zbl 1419.16012)]. That previous work studied steady modules. A reflexive module $$M$$ is steady if $$M$$ is a generator (that is, $$R \in \mathrm{add}_R(M)$$) and $$\mathrm{End}_R(M) \in \mathrm{add}_R(M)$$. An NCCR is steady if $$M$$ is. Here $$\mathrm{add}_R(M)$$ is the full subcategory consisting of direct summands of finite direct sums of some copies of $$M$$. The authors found that a dimer model $$\Gamma$$ was homotopy equivalent to a dimer model of regular hexagons if and only if $$\Gamma$$ gives a steady NCCR.
The paper under review generalizes to semi-steady $$M$$. That is, when $$M$$ is again a generator, and given a decomposition $$M = \oplus M_i$$ into non-isomorphic summands, $$\mathrm{Hom}_R(M_i, M) \in \mathrm{add}_R(M)$$ or $$\mathrm{Hom}_R(M_i, M) \in \mathrm{add}_R(M^*)$$. It is shown that $$M$$ is steady if and only if $$M$$ is semi-steady and $$\mathrm{add}_R(M) = \mathrm{add}_R(M^*)$$.
The main result is that a dimer model $$\Gamma$$ is homotopy equivalent to a dimer model of squares if and only if $$\Gamma$$ gives a semi-steady, but not steady, NCCR. A tiling of equilateral triangles cannot be realized as a dimer model. This leads to a satisfying conclusion, that a dimer model of regular polygons is in some sense equivalent to semi-steady NCCRs, with regular hexagons and squares corresponding to steady and non-steady NCCRs, respectively.
##### MSC:
 13C14 Cohen-Macaulay modules 05B45 Combinatorial aspects of tessellation and tiling problems 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 16S38 Rings arising from noncommutative algebraic geometry
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