The decomposition methods in this package can be used to study the structure of infinite resolutions. For instance, we can look for recurring indecomposable summands in successive syzygies of the residue field.
i1 : R = ZZ/101[x,y]/ideal(x^3, x^2*y^3, y^5);
|
i2 : F = res(coker vars R, LengthLimit => 6)
1 2 4 8 16 32 64
o2 = R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6
o2 : Complex
|
i3 : netList for i from 1 to 5 list tallySummands summands coker F.dd_i
+----------------------------------------------+
o3 = |Tally{cokernel | x y | => 1} |
+----------------------------------------------+
|Tally{cokernel {1} | -y x2 0 xy3 | => 1} |
| {1} | x 0 y4 0 | |
+----------------------------------------------+
|Tally{cokernel {2} | x2 0 0 0 y4 0 | => 1} |
| {3} | y x y3 0 0 0 | |
| {5} | 0 0 0 y -x x2 | |
| cokernel {5} | y x | => 1 |
+----------------------------------------------+
|Tally{cokernel {6} | x 0 y4 0 | => 3} |
| {6} | -y x2 0 xy3 | |
| cokernel {7} | y x | => 2 |
+----------------------------------------------+
|Tally{cokernel {7} | x2 0 0 0 y4 0 | => 3}|
| {8} | 35y x y3 0 0 0 | |
| {10} | 0 0 0 y x x2 | |
| cokernel {8} | x 0 y4 0 | => 2 |
| {8} | -y x2 0 xy3 | |
| cokernel {10} | y x | => 3 |
+----------------------------------------------+
|
This computation records the indecomposable summands appearing in the early syzygies of the residue field. This family of examples is studied in Syzygies of the Residue Field over Golod Rings by Cuong, Dao, Eisenbud, Kobayashi, Polini, and Ulrich.