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The most fundamental complexes of free modules over a commutative ring arethe Koszul complex, which is constructed from a vector (i.e., a 1-tensor), andthe Eagon-Northcott and the Buchsbaum-Rim complexes, which are constructed froma matrix (i.e., a 2-tensor). The subject of this paper is a multilinearanalogue of these complexes, which we construct from an arbitrary highertensor. Our construction provides detailed new examples of minimal freeresolutions, as well as a unifying view on a wide variety of complexesincluding: the Eagon-Northcott, Buchsbaum-Rim and similar complexes, theEisenbud-Schreyer pure resolutions, and the complexes used byGelfand-Kapranov-Zelevinsky and Weyman to compute hyperdeterminants. Inaddition, we provide applications to the study of pure resolutions andBoij-Soederberg theory, including the construction of infinitely many newfamilies of pure resolutions and the first explicit description of thedifferentials of the Eisenbud-Schreyer pure resolutions.

Tensor complexes: Multilinear free resolutions constructed from higher tensors