Tensor complexes: multilinear free resolutions constructed from higher tensors

Abstract

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 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–Söderberg 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.