  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.
