# pureResTC1 -- computes the first map of a balanced tensor complex with pure resolution of a given type

## Synopsis

• Usage:
pureResTC1(d,kk)
• Inputs:
• Outputs:

## Description

Given a degree sequence $d\in \mathbb Z^{n+1}$ and a field $k$ of arbitrary characteristic, this produces the first map of a balanced tensor complex with a pure resolution of type d, as constructed in Section 3 of the paper Tensor Complexes: Multilinear free resolutions constructed from higher tensors by Berkesch-Erman-Kummini-Sam. The cokernel of the output is an indecomposable module of codimension $n$.

The code gives an error if d is not strictly increasing with $d_0=0$.

 i1 : d={0,2,4,5}; i2 : p=pureResTC1(d,ZZ/32003) o2 = | -x_(0,1,0,0)x_(1,0,0,0)+x_(0,0,0,0)x_(1,1,0,0) -x_(0,1,0,0)x_(2,0,0,0)+x_(0,0,0,0)x_(2,1,0,0) -x_(1,1,0,0)x_(2,0,0,0)+x_(1,0,0,0)x_(2,1,0,0) -x_(0,1,0,0)x_(3,0,0,0)+x_(0,0,0,0)x_(3,1,0,0) -x_(1,1,0,0)x_(3,0,0,0)+x_(1,0,0,0)x_(3,1,0,0) -x_(2,1,0,0)x_(3,0,0,0)+x_(2,0,0,0)x_(3,1,0,0) -x_(0,1,0,0)x_(4,0,0,0)+x_(0,0,0,0)x_(4,1,0,0) -x_(1,1,0,0)x_(4,0,0,0)+x_(1,0,0,0)x_(4,1,0,0) -x_(2,1,0,0)x_(4,0,0,0)+x_(2,0,0,0)x_(4,1,0,0) -x_(3,1,0,0)x_(4,0,0,0)+x_(3,0,0,0)x_(4,1,0,0) | | -x_(0,1,1,0)x_(1,0,0,0)-x_(0,1,0,0)x_(1,0,1,0)+x_(0,0,1,0)x_(1,1,0,0)+x_(0,0,0,0)x_(1,1,1,0) -x_(0,1,1,0)x_(2,0,0,0)-x_(0,1,0,0)x_(2,0,1,0)+x_(0,0,1,0)x_(2,1,0,0)+x_(0,0,0,0)x_(2,1,1,0) -x_(1,1,1,0)x_(2,0,0,0)-x_(1,1,0,0)x_(2,0,1,0)+x_(1,0,1,0)x_(2,1,0,0)+x_(1,0,0,0)x_(2,1,1,0) -x_(0,1,1,0)x_(3,0,0,0)-x_(0,1,0,0)x_(3,0,1,0)+x_(0,0,1,0)x_(3,1,0,0)+x_(0,0,0,0)x_(3,1,1,0) -x_(1,1,1,0)x_(3,0,0,0)-x_(1,1,0,0)x_(3,0,1,0)+x_(1,0,1,0)x_(3,1,0,0)+x_(1,0,0,0)x_(3,1,1,0) -x_(2,1,1,0)x_(3,0,0,0)-x_(2,1,0,0)x_(3,0,1,0)+x_(2,0,1,0)x_(3,1,0,0)+x_(2,0,0,0)x_(3,1,1,0) -x_(0,1,1,0)x_(4,0,0,0)-x_(0,1,0,0)x_(4,0,1,0)+x_(0,0,1,0)x_(4,1,0,0)+x_(0,0,0,0)x_(4,1,1,0) -x_(1,1,1,0)x_(4,0,0,0)-x_(1,1,0,0)x_(4,0,1,0)+x_(1,0,1,0)x_(4,1,0,0)+x_(1,0,0,0)x_(4,1,1,0) -x_(2,1,1,0)x_(4,0,0,0)-x_(2,1,0,0)x_(4,0,1,0)+x_(2,0,1,0)x_(4,1,0,0)+x_(2,0,0,0)x_(4,1,1,0) -x_(3,1,1,0)x_(4,0,0,0)-x_(3,1,0,0)x_(4,0,1,0)+x_(3,0,1,0)x_(4,1,0,0)+x_(3,0,0,0)x_(4,1,1,0) | | -x_(0,1,1,0)x_(1,0,1,0)+x_(0,0,1,0)x_(1,1,1,0) -x_(0,1,1,0)x_(2,0,1,0)+x_(0,0,1,0)x_(2,1,1,0) -x_(1,1,1,0)x_(2,0,1,0)+x_(1,0,1,0)x_(2,1,1,0) -x_(0,1,1,0)x_(3,0,1,0)+x_(0,0,1,0)x_(3,1,1,0) -x_(1,1,1,0)x_(3,0,1,0)+x_(1,0,1,0)x_(3,1,1,0) -x_(2,1,1,0)x_(3,0,1,0)+x_(2,0,1,0)x_(3,1,1,0) -x_(0,1,1,0)x_(4,0,1,0)+x_(0,0,1,0)x_(4,1,1,0) -x_(1,1,1,0)x_(4,0,1,0)+x_(1,0,1,0)x_(4,1,1,0) -x_(2,1,1,0)x_(4,0,1,0)+x_(2,0,1,0)x_(4,1,1,0) -x_(3,1,1,0)x_(4,0,1,0)+x_(3,0,1,0)x_(4,1,1,0) | ZZ 3 ZZ 10 o2 : Matrix (-----[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ]) <--- (-----[x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x , x ]) 32003 0,0,0,0 0,0,1,0 0,1,0,0 0,1,1,0 1,0,0,0 1,0,1,0 1,1,0,0 1,1,1,0 2,0,0,0 2,0,1,0 2,1,0,0 2,1,1,0 3,0,0,0 3,0,1,0 3,1,0,0 3,1,1,0 4,0,0,0 4,0,1,0 4,1,0,0 4,1,1,0 32003 0,0,0,0 0,0,1,0 0,1,0,0 0,1,1,0 1,0,0,0 1,0,1,0 1,1,0,0 1,1,1,0 2,0,0,0 2,0,1,0 2,1,0,0 2,1,1,0 3,0,0,0 3,0,1,0 3,1,0,0 3,1,1,0 4,0,0,0 4,0,1,0 4,1,0,0 4,1,1,0 i3 : betti res coker p 0 1 2 3 o3 = total: 3 10 15 8 0: 3 . . . 1: . 10 . . 2: . . 15 8 o3 : BettiTally