# pureResTC -- constructs the balanced tensor complex of a given type

## Synopsis

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

## Description

Given a degree sequence $d$, this function returns a balanced tensor complex that is 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 function operates by resolving the output of pureResTC1(d,kk).

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

 i1 : d={0,2,4,5}; i2 : FF=pureResTC(d,ZZ/32003) ZZ 3 ZZ 10 ZZ 15 ZZ 8 o2 = (-----[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 ]) <-- (-----[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 ]) <-- 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 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 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 4 0 1 2 3 o2 : ChainComplex i3 : betti FF 0 1 2 3 o3 = total: 3 10 15 8 0: 3 . . . 1: . 10 . . 2: . . 15 8 o3 : BettiTally