Computing in algebraic geometry and commutative algebra using Macaulay 2

We present recent research of Eisenbud, Fløystad, Schreyer, and others, which was discovered with the help of experimentation with Macaulay 2. In this invited, expository paper, we start by considering the exterior algebra, and the computation of Gröbner bases. We then present, in an elementary manner, the explicit form of the Bernstein–Gelfand–Gelfand relationship between graded modules over the polynomial ring and complexes over the exterior algebra, that Eisenbud, Fløystad and Schreyer found. We present two applications of these techniques: cohomology of sheaves, and the construction of determinantal formulae for (powers of) Macaulay resultants. We show how to use Macaulay 2 to perform these computations.