  The most fundamental complexes of free modules over a commutative ring are
the Koszul complex, which is constructed from a vector (i.e., a 1-tensor), and
the Eagon-Northcott and the Buchsbaum-Rim complexes, which are constructed from
a matrix (i.e., a 2-tensor). The subject of this paper is a multilinear
analogue of these complexes, which we construct from an arbitrary higher
tensor. Our construction provides detailed new examples of minimal free
resolutions, as well as a unifying view on a wide variety of complexes
including: the Eagon-Northcott, Buchsbaum-Rim and similar complexes, the
Eisenbud-Schreyer pure resolutions, and the complexes used by
Gelfand-Kapranov-Zelevinsky and Weyman to compute hyperdeterminants. In
addition, we provide applications to the study of pure resolutions and
Boij-Soederberg theory, including the construction of infinitely many new
families of pure resolutions and the first explicit description of the
differentials of the Eisenbud-Schreyer pure resolutions.
