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We develop an analogue of Eisenbud-Floystad-Schreyer's Tate resolutions for toric varieties. Our construction, which is given by a noncommutative analogue of a Fourier-Mukai transform, works quite generally and provides a new perspective on the relationship between Tate resolutions and Beilinson's resolution of the diagonal. We also develop a Beilinson-type resolution of the diagonal for toric varieties and use it to generalize Eisenbud-Floystad-Schreyer's computationally effective construction of Beilinson monads.
Brown et al. (Mon,) studied this question.