Tensor complexes: Multilinear free resolutions constructed from higher tensors

TL;DR

This paper introduces a multilinear analogue of classical free resolutions constructed from higher tensors, unifying various known complexes and providing new examples and applications in algebraic geometry.

Contribution

It constructs a new class of free resolutions from higher tensors, generalizing classical complexes and linking them to modern theories like Boij-Soederberg.

Findings

Provides detailed new examples of minimal free resolutions.

Unifies a wide variety of complexes including Eagon-Northcott, Buchsbaum-Rim, and Eisenbud-Schreyer resolutions.

Constructs infinitely many new pure resolutions and describes differentials explicitly.

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 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.