A $b_1\times \cdots\times b_n$ tensor with coefficients in a ring S may be thought of as a multilinear linear form on $X := Proj(Spec S \times \mathbb P^{b_1-1}\times \cdots \times \mathbb P^{b_n-1})$. (If $S$ is graded, we may replace $Spec S$ by $Proj S$.)
This package provides a family of definitions around the notion of LabeledModule that makes it convenient to manipulate complicated multilinear constructions with tensors. We implement one such construction, that of Tensor Complexes, from the paper ``Tensor Complexes: Multilinear free resolutions constructed from higher tensors'' of Berkesch, Erman, Kummini and Sam (BEKS), which extends the construction of pure resolutions in the paper ``Betti numbers of graded modules and cohomology of vector bundles'' of Eisenbud and Schreyer. This itself is an instance of the technique of ``collapsing homogeneous vector bundles'' developed by Kempf and described, for example, in the book ``Cohomology of vector bundles and syzygies'' of Weyman.
Tensor complexes specialize to several well-known constructions including: the Eagon-Northcott and Buchsbaum-Rim complexes, and the others in this family described by Eisenbud and Buchsbaum (see Eisenbud ``Commutative algebra with a view towards algebraic geometry'', A2.6), and the hyperdeterminants of Weyman and Zelevinsky.
A collection of $a$ tensors of type $b_1\times \dots \times b_n$ may be regarded as a map $E := \mathcal O_X^a(-1,-1,\dots,-1) \to \mathcal O_X$ (with $X$ as above). Equivalently, we may think of this as a single $a \times b_1 \times \cdots \times b_n$ tensor.
One important construction made from such a collection of tensors is the Koszul complex $$ \mathbf K := \cdots \to \wedge^2 (\oplus_1^a O_X(-1,\dots, -1)) \to \oplus_1^a O_X(-1,\dots, -1)\to O_X \to 0. $$ Let $\mathcal O_X(d, e_1,\dots e_n)$ be the tensor product of the pull-backs to $X$ of the line bundles $\mathcal O_{\mathbb P^n}(d)$ and $\mathcal O_{\mathbb P^{b_i-1}}(-1)$. If we twist the Koszul complex by $O_X(0, -w_1, \dots -w_n)$ and then push it forward to $Spec S$ we get the tensor complex $F(\phi,w)$ of BEKS.
Each map $\partial_i$ in the tensor complex can be defined by a rather involved construct in multilinear algebra. This package implements the construction of $\partial_1$ in the range of cases described explicitly in BEKS (Sections 4 and 12). This range includes the hyperdeterminants of boundary format, the construction of the first map of the pure resolutions of Eisenbud-Schreyer, and the first map in most of the much larger family of generic pure resolutions of BEKS.
This documentation describes version 1.0 of TensorComplexes.
The source code from which this documentation is derived is in the file TensorComplexes.m2.
The object TensorComplexes is a package.