# verticalStrand -- extracts one vertical strand from an Eagon double complex

## Synopsis

• Usage:
F = verticalStrand(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 vertical strand is the Koszul complex of R. The vertical strands are never resolutions unless R is regular. The key lemma in Eagon's treatment identifies the i-th homology H_i of the n-th vertical strand with H_0**X_i.

 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 = verticalStrand(E,3) 18 45 41 14 o4 = R <-- R <-- R <-- R 0 1 2 3 o4 : ChainComplex i5 : picture F +-------------------------------------------------+ |+--------+--------+--------+--------+-----------+| o5 = || |(0, {3})|(1, {2})|(2, {1})|(0, {1, 1})|| |+--------+--------+--------+--------+-----------+| || (3, {})| * | * | * | * || |+--------+--------+--------+--------+-----------+| ||(0, {2})| . | * | * | 5,3 || |+--------+--------+--------+--------+-----------+| ||(1, {1})| . | . | * | * || |+--------+--------+--------+--------+-----------+| +-------------------------------------------------+ |+-----------+--------+--------+-----------+ | || |(1, {3})|(2, {2})|(0, {1, 2})| | |+-----------+--------+--------+-----------+ | || (0, {3}) | * | . | 2,2 | | |+-----------+--------+--------+-----------+ | || (1, {2}) | . | * | * | | |+-----------+--------+--------+-----------+ | || (2, {1}) | . | . | . | | |+-----------+--------+--------+-----------+ | ||(0, {1, 1})| . | . | . | | |+-----------+--------+--------+-----------+ | +-------------------------------------------------+ |+-----------+--------+-----------+ | || |(2, {3})|(0, {1, 3})| | |+-----------+--------+-----------+ | || (1, {3}) | * | * | | |+-----------+--------+-----------+ | || (2, {2}) | . | . | | |+-----------+--------+-----------+ | ||(0, {1, 2})| . | . | | |+-----------+--------+-----------+ | +-------------------------------------------------+