||Sheaf cohomology and direct images are fundamental objects in algebraic geometry. However, they are defined in an abstract way (as right derived functors), and thus they are often hard to compute in explicit examples. In this talk, we briefly review Bernstein-Gel'fand-Gel'fand (BGG) correspondence and resolutions over an exterior algebra. Then, we review Tate resolutions and how it can be used to understand a given coherent sheaf and its cohomology groups in terms of Beilinson monad. Finally, we discuss an algorithm to compute direct images using Eisenbud-Erman-Schreyer's generalization on products of projective spaces. A part of the talk is a joint work in progress with J. Barrott and F.-O. Schreyer.