A b1×…×bn tensor with coefficients in a ring S may be thought of as a multilinear linear form on X := Proj(Spec S ×ℙb1-1×…×ℙbn-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 b1×…×bn may be regarded as a map E := OXa(-1,-1,…,-1) →OX (with X as above). Equivalently, we may think of this as a single a ×b1 ×…×bn tensor.
One important construction made from such a collection of tensors is the Koszul complex
K := …→∧2 (⊕1a OX(-1,…, -1)) →⊕1a OX(-1,…, -1)→OX →0.
Let OX(d, e1,…en) be the tensor product of the pull-backs to X of the line bundles Oℙn(d) and Oℙbi-1(-1). If we twist the Koszul complex by OX(0, -w1, …-wn) and then push it forward to Spec S we get the tensor complex F(φ,w) of BEKS.