# horizontalStrand -- extracts one horizontal strand from an Eagon double complex

## Synopsis

• Usage:
F = horizontalStrand(E,i)
• Inputs:
• E, an instance of the type EagonData, produced by eagon(R,b)
• i, an integer, which strand
• Outputs:
• F, , beginning of the free resolution of the i-th boundary module of the Koszul complex

## Description

The 0-th strand is a possibly non-minimal resolution of the residuce field. More generally, the i-th strand resolves the i-th boundary module in the Koszul complex of R. These resolutions are all minimal iff R is Golod.

 i1 : S = ZZ/101[x,y,z] o1 = S o1 : PolynomialRing i2 : R = S/((ideal(x,y))^2+ideal(z^3)) o2 = R o2 : QuotientRing i3 : E = eagon(R,5); i4 : F = horizontalStrand(E,2) 3 6 17 41 104 o4 = R <-- R <-- R <-- R <-- R 0 1 2 3 4 o4 : ChainComplex i5 : picture F +-------------------------------------------------------------------+ |+-------+-------+--------+ | o5 = || |(3, {})|(0, {2})| | |+-------+-------+--------+ | ||(2, {})| * | * | | |+-------+-------+--------+ | +-------------------------------------------------------------------+ |+--------+--------+--------+ | || |(0, {3})|(1, {2})| | |+--------+--------+--------+ | || (3, {})| * | * | | |+--------+--------+--------+ | ||(0, {2})| . | * | | |+--------+--------+--------+ | +-------------------------------------------------------------------+ |+--------+--------+--------+-----------+ | || |(1, {3})|(2, {2})|(0, {1, 2})| | |+--------+--------+--------+-----------+ | ||(0, {3})| * | . | 2,2 | | |+--------+--------+--------+-----------+ | ||(1, {2})| . | * | * | | |+--------+--------+--------+-----------+ | +-------------------------------------------------------------------+ |+-----------+--------+-----------+--------+-----------+-----------+| || |(2, {3})|(0, {1, 3})|(3, {2})|(0, {2, 2})|(1, {1, 2})|| |+-----------+--------+-----------+--------+-----------+-----------+| || (1, {3}) | * | * | . | . | 6,6 || |+-----------+--------+-----------+--------+-----------+-----------+| || (2, {2}) | . | . | * | * | * || |+-----------+--------+-----------+--------+-----------+-----------+| ||(0, {1, 2})| . | . | . | . | * || |+-----------+--------+-----------+--------+-----------+-----------+| +-------------------------------------------------------------------+