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})| . | . | |
|+-----------+--------+-----------+ |
+-------------------------------------------------+
|
The object verticalStrand is a method function.