Tensor complexes: Multilinear free resolutions constructed from higher tensors

Christine Berkesch Zamaere, Daniel Erman, Manoj Kummini, Steven V. Sam

Research output: Contribution to journalArticlepeer-review

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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.

Original languageEnglish (US)
Pages (from-to)2257-2295
Number of pages39
JournalJournal of the European Mathematical Society
Issue number6
StatePublished - 2013


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