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