tensorComplex1(f,w)tensorComplex1 fLet $X := Proj(Spec S \times \mathbb P^{b_1-1}\times \cdots \times \mathbb P^{b_n-1})$, and let $$ \mathbf K := \cdots \wedge^2 \oplus_1^a O_X(-1,\dots, -1) \to O_X \to 0 $$ be the Koszul complex of the multilinear forms corresponding to f, on $X$. The output of tensorComplex1(f,w) is the first map of the complex obtained by pushing $\mathbf K \otimes {\mathcal O}_X(w_1,\dots,w_n)$ down to $Spec S$.
This script implements the construction of tensor complexes from the paper ``Tensor Complexes: Multilinear free resolutions constructed from higher tensors'' of Berkesch, Erman, Kummini and Sam (BEKS).
The program requires that $f$ is a flattened tensor, that is, a map $A \to B_1\otimes\cdots\otimes B_n$. Returns the first map in the tensor complex $F(f,w)$ of BEKS, requiring that $w$ satisfies: $$ w_0 = 0, w_1 \geq 0, w_2 \geq w_1+b_1, \ {\rm and }\ w_i>w_{i-1} \ {\rm for }\ i\geq 2. $$
When $rank A=\sum rank B_i$, that is, $L_0 = \sum_{i=1}^n L_i$ then we are in the ``balanced case'' discussed in Section 3 of BEKS. In this case giving a weight vector is unnecessary, and one can use the format tensorComplex1 f.
The example from section 12 of BEKS appears below.
|
|
|
|
|
|
|
We can recover the Eagon-Northcott complex as follows.
|
|
|
|
|
|
The following example is taken from the introduction to BEKS.
|
|
|
|
The input map need not be generic.
|
|
|
|
|
|
Unlike BEKS, this method does not work with arbitrary weight vectors w.
The object tensorComplex1 is a method function.