Geometric construction of toric NCRs

The Rouquier dimension of a toric variety is shown to be achieved by the Frobenius pushforward of O via homological mirror symmetry. From the perspective of noncommutative geometry, this result leads to a geometric construction of the corresponding toric NCR of the invariant ring of the Cox ring with respect to a multi-grading which also gives the information about its global dimension. From the perspective of algebraic geometry, the same construction provides a universal ``wall skeleton'' capturing VGIT wall-crossings, which contains a window for each chamber as a full subcategory. From the perspective of commutative algebra, the same construction indicates the existence of virtual resolutions of the multigraded diagonal bimodule, which agrees with a recent result of Hanlon-Hicks-Larzarev constructing one such resolution explicitly. In this talk, I will explain how these fit together. This is a joint work in progress with D. Favero.

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