Abstract: We introduce non-commutative resolutions of singular Calabi-Yau double covers to investigate their derived category of coherent sheaves. The non-commutative geometry is studied via a GLSM description, and B-branes in these GLSMs provide the non-commutative analogues of sheaves on smooth Calabi-Yau manifolds. We compute the central charges of the B-branes, which are known to be annihilated by the GKZ system of the mirror Calabi-Yau. We show that 
(i) There always exists a smooth Calabi-Yau complete intersection which satisfies the same GKZ system;
(ii) The B-branes on the non-commutative resolution form the invariant sub-category of a certain equivariant category of coherent sheaves on the smooth complete intersection.
This is a universal phenomenon, which allows us to find the GKZ system for the mirror Calabi-Yau, even when the mirror geometry is not known.
Based on joint work with Tsung-Ju Lee, Bong Lian and Mauricio Romo.