bibkey: berkesch2013tensorcomplexes authors: Christine Berkesch Zamaere, Daniel Erman, Manoj Kummini, Steven V. Sam year: 2013 title: “Tensor complexes: Multilinear free resolutions constructed from higher tensors” doi: 10.4171/JEMS/421 url: https://arxiv.org/abs/1101.4604v5 claim: “Proposition 9.1 identifies the boundary-format integral hyperdeterminant with the determinant of the two-term tensor complex, up to sign.” strata_touched: [] license: citation-only triage: anchor

Integral boundary-format determinant

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DOI: 10.4171/JEMS/421.

Immutable version: https://arxiv.org/abs/1101.4604v5. The primary text is https://arxiv.org/html/1101.4604v5, Section 9, Proposition 9.1 (S9.E1). The article appears in Journal of the European Mathematical Society 15 (2013), 2257–2295.

Immediately before Proposition 9.1 the boundary format is $a=1+{\sum }_i(b_i-1)$. Its hyperdeterminant is regarded as an integral polynomial not divisible by any prime, determined up to sign. Proposition 9.1 states that for every pinching weight the tensor complex has length one and its square differential has determinant equal to this hyperdeterminant up to sign. Its proof compares that differential to Gelfand–Kapranov–Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Proposition 14.3.2.

Finite specification and scope

For faces $M_0,M_1$ with $k+1$ rows and $k$ columns, the relevant coefficient map is multiplication by the pencil of linear forms. In the binary monomial bases its matrix is indexed by rows $(j,t)$, $j<k$, $t\le k$, and columns $(s,r)$, $s<k$, $r\le k$:

$$C_{(j,t),(s,r)}={\delta }_{t,s}(M_0{)}_{r,j}+{\delta }_{t,s+1}(M_1{)}_{r,j}.$$

The source-defined polynomial is the determinant of this integral coefficient matrix with independent tensor entries. This finite specification fixes the object being counted; its nonzero locus is unchanged by the integral sign convention. Reduction of an equality up to sign is valid in every characteristic, including two.

This note cites the source determinant specification. It does not assert that a distinct geometric object, an independently chosen polynomial, or an arbitrary matrix with the same dimensions has been identified with it. The entry and basis correspondence is a separate mathematical obligation.