burkeResolution -- compute a resolution from A-infinity structures

Synopsis

Usage:

F = burkeResolution(M,len)
Inputs:
- M, a module, over a factor ring R/I
- len, an integer, length of resolution desired
Optional inputs:
- Check => a Boolean value, default value false
Outputs:
- F, a complex, resolution of M over R, of length len

Description

The construction follows the recipe in Jesse Burke's paper. The resolution produced is minimal iff M is a Golod module. if the optional argument Check => true then the program checks that the differential produced squares to 0 and that the complex is acyclic. The default is Check => false.

i1 : S = ZZ/101[x_1..x_4]

o1 = S

o1 : PolynomialRing

i2 : I = x_1^2*ideal(vars S)

             3   2     2     2
o2 = ideal (x , x x , x x , x x )
             1   1 2   1 3   1 4

o2 : Ideal of S

i3 : R = S/I

o3 = R

o3 : QuotientRing

i4 : M = R^1/ideal(x_1..x_3)

o4 = cokernel | x_1 x_2 x_3 |

                            1
o4 : R-module, quotient of R

i5 : F = burkeResolution(M, 4, Check =>true)

      1      3      7      19      50
o5 = R  <-- R  <-- R  <-- R   <-- R
                                   
     0      1      2      3       4

o5 : Complex

If one only wants a single differential, use burkeDifferential instead, but one must construct the Ainfinity structures separately. The same syntax also works for burkeResolution.

i6 : mR = aInfinity R

o6 = HashTable{{2, 2, 2} => 0                                                                                                                }
               {2, 2, 3} => 0
               {2, 2} => {4} | 0 -x_1^2 0      0      x_1^2 0 0      0      0     -x_1x_3 0 0      0     -x_1x_4 0     0 |
                         {4} | 0 0      -x_1^2 0      0     0 0      0      x_1^2 x_1x_2  0 0      0     0       0     0 |
                         {4} | 0 0      0      0      0     0 -x_1^2 0      0     0       0 0      0     0       0     0 |
                         {4} | 0 0      0      -x_1^2 0     0 0      0      0     0       0 0      x_1^2 x_1x_2  0     0 |
                         {4} | 0 0      0      0      0     0 0      -x_1^2 0     0       0 0      0     0       0     0 |
                         {4} | 0 0      0      0      0     0 0      0      0     0       0 -x_1^2 0     0       x_1^2 0 |
               {2, 3, 2} => 0
               {2, 3} => {5} | 0 0 -x_1^2 0 0      0      0 x_1^2 0 0     0 0      0 0 -x_1x_3 0     0       0 0 0      -x_1x_4 0 0       0 |
                         {5} | 0 0 0      0 -x_1^2 0      0 0     0 x_1^2 0 0      0 0 0       0     -x_1x_3 0 0 0      0       0 -x_1x_4 0 |
                         {5} | 0 0 0      0 0      -x_1^2 0 0     0 0     0 0      0 0 0       x_1^2 x_1x_2  0 0 -x_1^2 -x_1x_2 0 0       0 |
                         {5} | 0 0 0      0 0      0      0 0     0 0     0 -x_1^2 0 0 0       0     0       0 0 0      0       0 0       0 |
               {2, 4} => {6} | 0 0 0 -x_1^2 0 0 x_1^2 0 0 0 0 -x_1x_3 0 0 0 -x_1x_4 |
               {2} => 0
               {3, 2, 2} => 0
               {3, 2} => {5} | 0 0 x_1^2 0     0 0 0 0     x_1^2 x_1x_2 0 0     0 0 0      0 0     0      0      0 0     0      0 0 |
                         {5} | 0 0 0     x_1^2 0 0 0 0     0     0      0 0     0 0 0      0 x_1^2 x_1x_2 0      0 0     0      0 0 |
                         {5} | 0 0 0     0     0 0 0 x_1^2 0     0      0 0     0 0 -x_1^2 0 0     0      0      0 x_1^2 x_1x_2 0 0 |
                         {5} | 0 0 0     0     0 0 0 0     0     0      0 x_1^2 0 0 0      0 0     0      -x_1^2 0 0     0      0 0 |
               {3, 3} => {6} | 0 0 0 0 0 x_1^2 0 0 0 0 0 0 0 0 0 x_1^2 x_1x_2 0 0 0 0 0 0 0 0 -x_1^2 -x_1x_2 0 0 0 0 0 0 0 0 0 |
               {3} => {3} | x_2  x_3  0    x_4  0    0    |
                      {3} | -x_1 0    x_3  0    x_4  0    |
                      {3} | 0    -x_1 -x_2 0    0    x_4  |
                      {3} | 0    0    0    -x_1 -x_2 -x_3 |
               {4, 2} => {6} | 0 0 0 -x_1^2 0 0 x_1^2 0 0 0 0 0 x_1^2 x_1x_2 0 0 |
               {4} => {4} | -x_3 -x_4 0    0    |
                      {4} | x_2  0    -x_4 0    |
                      {4} | -x_1 0    0    -x_4 |
                      {4} | 0    x_2  x_3  0    |
                      {4} | 0    -x_1 0    x_3  |
                      {4} | 0    0    -x_1 -x_2 |
               {5} => {5} | x_4  |
                      {5} | -x_3 |
                      {5} | x_2  |
                      {5} | -x_1 |
                                4      6      4      1
               "resolution" => S  <-- S  <-- S  <-- S
                                                     
                               2      3      4      5
               "ring" => R

o6 : HashTable

i7 : mG = aInfinity(mR,M)

o7 = HashTable{{1} => | x_1 x_2 x_3 |                                                         }
               {2, 0} => {1} | x_1^2 x_1x_2 x_1x_3 x_1x_4 |
                         {1} | 0     0      0      0      |
                         {1} | 0     0      0      0      |
               {2, 1} => {2} | 0 x_1^2 0     0 x_1x_2 x_1x_3 0 x_1x_3 0      0 x_1x_4 0      |
                         {2} | 0 0     x_1^2 0 0      0      0 0      x_1x_3 0 0      x_1x_4 |
                         {2} | 0 0     0     0 0      x_1^2  0 0      0      0 0      0      |
               {2, 2, 0} => 0
               {2, 2, 1} => 0
               {2, 2} => {3} | 0 0 x_1^2 0 -x_1^2 0 0 0 x_1x_3 0 0 x_1x_4 |
               {2, 3, 0} => 0
               {2} => {1} | -x_2 -x_3 0    |
                      {1} | x_1  0    -x_3 |
                      {1} | 0    x_1  x_2  |
               {3, 0} => 0
               {3, 1} => {3} | 0 0 x_1^2 0 0 0 0 0 -x_1x_3 0 0 0 0 0 -x_1x_4 0 0 0 |
               {3, 2, 0} => 0
               {3} => {2} | x_3  |
                      {2} | -x_2 |
                      {2} | x_1  |
               {4, 0} => 0
               "module" => cokernel | x_1 x_2 x_3 |
                                1      3      3      1
               "resolution" => S  <-- S  <-- S  <-- S
                                                     
                               0      1      2      3

o7 : HashTable

i8 : burkeDifferential(mR,mG,5)

o8 = {5} | x_4  x_1  x_2  x_3  0    0    0    0    0    0    0    0    0   
     {5} | -x_3 0    0    0    x_1  x_2  x_3  0    0    0    0    0    0   
     {5} | x_2  0    0    0    0    0    0    x_1  x_2  x_3  0    0    0   
     {5} | -x_1 0    0    0    0    0    0    0    0    0    x_1  x_2  x_3 
     {5} | 0    -x_3 0    0    -x_4 0    0    0    0    0    0    0    0   
     {5} | 0    0    -x_3 0    0    -x_4 0    0    0    0    0    0    0   
     {5} | 0    0    0    -x_3 0    0    -x_4 0    0    0    0    0    0   
     {5} | 0    x_2  0    0    0    0    0    -x_4 0    0    0    0    0   
     {5} | 0    0    x_2  0    0    0    0    0    -x_4 0    0    0    0   
     {5} | 0    0    0    x_2  0    0    0    0    0    -x_4 0    0    0   
     {5} | 0    -x_1 0    0    0    0    0    0    0    0    -x_4 0    0   
     {5} | 0    0    -x_1 0    0    0    0    0    0    0    0    -x_4 0   
     {5} | 0    0    0    -x_1 0    0    0    0    0    0    0    0    -x_4
     {5} | 0    0    0    0    x_2  0    0    x_3  0    0    0    0    0   
     {5} | 0    0    0    0    0    x_2  0    0    x_3  0    0    0    0   
     {5} | 0    0    0    0    0    0    x_2  0    0    x_3  0    0    0   
     {5} | 0    0    0    0    -x_1 0    0    0    0    0    x_3  0    0   
     {5} | 0    0    0    0    0    -x_1 0    0    0    0    0    x_3  0   
     {5} | 0    0    0    0    0    0    -x_1 0    0    0    0    0    x_3 
     {5} | 0    0    0    0    0    0    0    -x_1 0    0    -x_2 0    0   
     {5} | 0    0    0    0    0    0    0    0    -x_1 0    0    -x_2 0   
     {5} | 0    0    0    0    0    0    0    0    0    -x_1 0    0    -x_2
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {5} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     {6} | 0    0    0    0    0    0    0    0    0    0    0    0    0   
     ------------------------------------------------------------------------
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     x_2  x_3  0    0    0    0    0    0    0    0    0    0    0    0   
     -x_1 0    x_3  0    0    0    0    0    0    0    0    0    0    0   
     0    -x_1 -x_2 0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    x_2  x_3  0    0    0    0    0    0    0    0    0   
     0    0    0    -x_1 0    x_3  0    0    0    0    0    0    0    0   
     0    0    0    0    -x_1 -x_2 0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    x_2  x_3  0    0    0    0    0    0   
     0    0    0    0    0    0    -x_1 0    x_3  0    0    0    0    0   
     0    0    0    0    0    0    0    -x_1 -x_2 0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    x_2  x_3  0    0    0   
     0    0    0    0    0    0    0    0    0    -x_1 0    x_3  0    0   
     0    0    0    0    0    0    0    0    0    0    -x_1 -x_2 0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    x_2  x_3 
     0    0    0    0    0    0    0    0    0    0    0    0    -x_1 0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    -x_1
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     x_2  0    0    x_3  0    0    0    0    0    x_4  0    0    0    0   
     0    x_2  0    0    x_3  0    0    0    0    0    x_4  0    0    0   
     0    0    x_2  0    0    x_3  0    0    0    0    0    x_4  0    0   
     -x_1 0    0    0    0    0    x_3  0    0    0    0    0    x_4  0   
     0    -x_1 0    0    0    0    0    x_3  0    0    0    0    0    x_4 
     0    0    -x_1 0    0    0    0    0    x_3  0    0    0    0    0   
     0    0    0    -x_1 0    0    -x_2 0    0    0    0    0    0    0   
     0    0    0    0    -x_1 0    0    -x_2 0    0    0    0    0    0   
     0    0    0    0    0    -x_1 0    0    -x_2 0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    -x_1 0    0    -x_2 0   
     0    0    0    0    0    0    0    0    0    0    -x_1 0    0    -x_2
     0    0    0    0    0    0    0    0    0    0    0    -x_1 0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     0    0    0    0    0    0    0    0    0    0    0    0    0    0   
     ------------------------------------------------------------------------
     0    0    0    0    0    0    0    0    0      0       x_1^2   0      
     0    0    0    0    0    0    0    0    0      0       0       x_1^2  
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    -x_1^2 -x_1x_2 -x_1x_3 -x_1x_4
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     x_3  0    0    0    0    0    0    0    0      0       0       0      
     -x_2 0    0    0    0    0    0    0    0      0       0       0      
     0    x_2  x_3  0    0    0    0    0    0      0       0       0      
     0    -x_1 0    x_3  0    0    0    0    0      0       0       0      
     0    0    -x_1 -x_2 0    0    0    0    0      0       0       0      
     0    0    0    0    x_3  0    0    0    0      0       0       0      
     0    0    0    0    -x_2 0    0    0    0      0       0       0      
     0    0    0    0    x_1  0    0    0    0      0       0       0      
     0    0    0    0    0    x_3  0    0    0      0       0       0      
     0    0    0    0    0    -x_2 0    0    0      0       0       0      
     x_4  0    0    0    0    x_1  0    0    0      0       0       0      
     0    x_4  0    0    0    0    x_3  0    0      0       0       0      
     0    0    x_4  0    0    0    -x_2 0    0      0       0       0      
     0    0    0    x_4  0    0    x_1  0    0      0       0       0      
     0    -x_3 0    0    0    0    0    x_3  0      0       0       0      
     0    0    -x_3 0    0    0    0    -x_2 0      0       0       0      
     -x_2 0    0    -x_3 0    0    0    x_1  0      0       0       0      
     0    0    0    0    0    0    0    0    x_2    0       0       0      
     0    0    0    0    0    0    0    0    0      x_2     0       0      
     0    0    0    0    0    0    0    0    0      0       x_2     0      
     0    0    0    0    0    0    0    0    0      0       0       x_2    
     0    0    0    0    0    0    0    0    -x_1   0       0       0      
     0    0    0    0    0    0    0    0    0      -x_1    0       0      
     0    0    0    0    0    0    0    0    0      0       -x_1    0      
     0    0    0    0    0    0    0    0    0      0       0       -x_1   
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     0    0    0    0    0    0    0    0    0      0       0       0      
     ------------------------------------------------------------------------
     0      0       0       0       x_1^2  x_1x_2  0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       x_1^2   0      0       0       0       0     
     0      0       0       0       0      0       0       x_1^2   0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     -x_1^2 -x_1x_2 -x_1x_3 -x_1x_4 0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       -x_1^2 -x_1x_2 -x_1x_3 -x_1x_4 0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       -x_1^2
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     x_3    0       0       0       0      0       0       0       x_4   
     0      x_3     0       0       0      0       0       0       0     
     0      0       x_3     0       0      0       0       0       0     
     0      0       0       x_3     0      0       0       0       0     
     0      0       0       0       x_3    0       0       0       0     
     0      0       0       0       0      x_3     0       0       0     
     0      0       0       0       0      0       x_3     0       0     
     0      0       0       0       0      0       0       x_3     0     
     -x_1   0       0       0       -x_2   0       0       0       0     
     0      -x_1    0       0       0      -x_2    0       0       0     
     0      0       -x_1    0       0      0       -x_2    0       0     
     0      0       0       -x_1    0      0       0       -x_2    0     
     0      0       0       0       0      0       0       0       -x_1  
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     0      0       0       0       0      0       0       0       0     
     ------------------------------------------------------------------------
     0       0       0       0      0       0       0       0      0      
     0       0       0       x_1^2  x_1x_2  0       0       0      0      
     0       -x_1^2  0       0      0       0       0       x_1^2  x_1x_2 
     0       0       0       0      0       -x_1^2  0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     -x_1x_2 -x_1x_3 -x_1x_4 0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       -x_1^2 -x_1x_2 -x_1x_3 -x_1x_4 0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       -x_1^2 -x_1x_2
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     x_4     0       0       0      0       0       0       0      0      
     0       x_4     0       0      0       0       0       0      0      
     0       0       x_4     0      0       0       0       0      0      
     0       0       0       x_4    0       0       0       0      0      
     0       0       0       0      x_4     0       0       0      0      
     0       0       0       0      0       x_4     0       0      0      
     0       0       0       0      0       0       x_4     0      0      
     0       0       0       0      0       0       0       x_4    0      
     0       0       0       0      0       0       0       0      x_4    
     0       0       0       0      0       0       0       0      0      
     0       0       0       0      0       0       0       0      0      
     0       0       0       -x_2   0       0       0       -x_3   0      
     -x_1    0       0       0      -x_2    0       0       0      -x_3   
     0       -x_1    0       0      0       -x_2    0       0      0      
     0       0       -x_1    0      0       0       -x_2    0      0      
     ------------------------------------------------------------------------
     0       0       0    0    -x_1^2 0    0      0      0    x_1^2 0   
     0       0       0    0    0      0    -x_1^2 0      0    0     0   
     0       0       0    0    0      0    0      -x_1^2 0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     -x_1x_3 -x_1x_4 0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       x_2  x_3  0      x_4  0      0      0    0     0   
     0       0       -x_1 0    x_3    0    x_4    0      0    0     0   
     0       0       0    -x_1 -x_2   0    0      x_4    0    0     0   
     0       0       0    0    0      -x_1 -x_2   -x_3   0    0     0   
     0       0       0    0    0      0    0      0      x_2  x_3   0   
     0       0       0    0    0      0    0      0      -x_1 0     x_3 
     0       0       0    0    0      0    0      0      0    -x_1  -x_2
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     x_4     0       0    0    0      0    0      0      0    0     0   
     0       x_4     0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     0       0       0    0    0      0    0      0      0    0     0   
     -x_3    0       0    0    0      0    0      0      0    0     0   
     0       -x_3    0    0    0      0    0      0      0    0     0   
     ------------------------------------------------------------------------
     0     0    0      0    0    -x_1x_3 0     0       0    0    0     
     x_1^2 0    0      0    0    0       0     -x_1x_3 0    0    0     
     0     0    0      0    0    0       x_1^2 x_1x_2  0    0    -x_1^2
     0     0    -x_1^2 0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     0     0    0      0    0    0       0     0       0    0    0     
     x_4   0    0      0    0    0       0     0       0    0    0     
     0     x_4  0      0    0    0       0     0       0    0    0     
     0     0    x_4    0    0    0       0     0       0    0    0     
     -x_1  -x_2 -x_3   0    0    0       0     0       0    0    0     
     0     0    0      x_2  x_3  0       x_4   0       0    0    0     
     0     0    0      -x_1 0    x_3     0     x_4     0    0    0     
     0     0    0      0    -x_1 -x_2    0     0       x_4  0    0     
     0     0    0      0    0    0       -x_1  -x_2    -x_3 0    0     
     0     0    0      0    0    0       0     0       0    x_2  x_3   
     0     0    0      0    0    0       0     0       0    -x_1 0     
     0     0    0      0    0    0       0     0       0    0    -x_1  
     0     0    0      0    0    0       0     0       0    0    0     
     ------------------------------------------------------------------------
     -x_1x_4 0    0       0    0   0     0     0      0      0      0     
     0       0    -x_1x_4 0    0   0     0     0      0      0      0     
     -x_1x_2 0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     -x_1^2 0      0      0     
     0       0    0       0    0   0     0     0      -x_1^2 0      0     
     0       0    0       0    0   0     0     0      0      -x_1^2 0     
     0       0    0       0    0   0     0     0      0      0      -x_1^2
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   x_1^2 0     0      x_1x_2 x_1x_3 0     
     0       0    0       0    0   0     x_1^2 0      0      0      0     
     0       0    0       0    0   0     0     0      0      x_1^2  0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    x_1 x_2   x_3   0      0      0      0     
     0       0    0       0    0   0     0     x_1    x_2    x_3    0     
     0       0    0       0    0   0     0     0      0      0      x_1   
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       0    0       0    0   0     0     0      0      0      0     
     0       x_4  0       0    0   0     0     0      0      0      0     
     x_3     0    x_4     0    0   0     0     0      0      0      0     
     -x_2    0    0       x_4  0   0     0     0      0      0      0     
     0       -x_1 -x_2    -x_3 0   0     0     0      0      0      0     
     ------------------------------------------------------------------------
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      x_1^2 0     0     0   0      0     
     0      0      0      0      0      0     x_1^2 0     0   0      0     
     0      0      0      0      0      0     0     x_1^2 0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     -x_1^2 0      0      0      0      0     0     0     0   0      0     
     0      -x_1^2 0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      -x_1^2 0      0      0     0     0     0   0      0     
     0      0      0      -x_1^2 0      0     0     0     0   0      0     
     0      0      0      0      -x_1^2 0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     x_1x_3 0      0      x_1x_4 0      0     0     0     0   0      0     
     0      x_1x_3 0      0      x_1x_4 0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     x_1^2 0     0   x_1x_2 x_1x_3
     0      0      0      0      0      0     0     x_1^2 0   0      0     
     0      0      0      0      0      0     0     0     0   0      x_1^2 
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     x_2    x_3    0      0      0      0     0     0     0   0      0     
     0      0      x_1    x_2    x_3    0     0     0     0   0      0     
     0      0      0      0      0      x_1   x_2   x_3   0   0      0     
     0      0      0      0      0      0     0     0     x_1 x_2    x_3   
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     0      0      0      0      0      0     0     0     0   0      0     
     ------------------------------------------------------------------------
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     -x_1x_3
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      x_1^2 0     0     x_1x_2 
     0      0      0      0      0      0      0     x_1^2 0     0      
     0      0      0      0      0      0      0     0     x_1^2 0      
     -x_1^2 0      0      0      0      0      0     0     0     0      
     0      -x_1^2 0      0      0      0      0     0     0     0      
     0      0      -x_1^2 0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      -x_1^2 0      0      0     0     0     0      
     0      0      0      0      -x_1^2 0      0     0     0     0      
     0      0      0      0      0      -x_1^2 0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      x_1x_3 0      0      x_1x_4 0      0     0     0     0      
     0      0      x_1x_3 0      0      x_1x_4 0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     x_1^2 0     0      
     0      0      0      0      0      0      0     0     x_1^2 0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     x_1    x_2    x_3    0      0      0      0     0     0     0      
     0      0      0      x_1    x_2    x_3    0     0     0     0      
     0      0      0      0      0      0      x_1   x_2   x_3   0      
     0      0      0      0      0      0      0     0     0     x_1    
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     0      0      0      0      0      0      0     0     0     0      
     ------------------------------------------------------------------------
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     -x_1x_3 0       0   0      0      0      0      0      0     0     0    
     0       -x_1x_3 0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     x_1x_2  0       0   0      0      0      0      0      0     0     0    
     0       x_1x_2  0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      x_1^2 0     0    
     0       0       0   0      0      0      0      0      0     x_1^2 0    
     0       0       0   0      0      0      0      0      0     0     x_1^2
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      -x_1^2 0      0      0     0     0    
     0       0       0   0      0      0      -x_1^2 0      0     0     0    
     0       0       0   0      0      0      0      -x_1^2 0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     x_1x_2  x_1x_3  0   x_1x_3 0      0      x_1x_4 0      0     0     0    
     0       0       0   0      x_1x_3 0      0      x_1x_4 0     0     0    
     0       x_1^2   0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     x_1^2 0    
     0       0       0   0      0      0      0      0      0     0     x_1^2
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     x_2     x_3     0   0      0      0      0      0      0     0     0    
     0       0       x_1 x_2    x_3    0      0      0      0     0     0    
     0       0       0   0      0      x_1    x_2    x_3    0     0     0    
     0       0       0   0      0      0      0      0      x_1   x_2   x_3  
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     0       0       0   0      0      0      0      0      0     0     0    
     ------------------------------------------------------------------------
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     -x_1x_4 0       0       0     0      0      0   0      0      |
     0       -x_1x_4 0       0     0      0      0   0      0      |
     0       0       -x_1x_4 0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     x_1x_2  0       0       0     0      0      0   0      0      |
     0       x_1x_2  0       0     0      0      0   0      0      |
     0       0       x_1x_2  0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       x_1^2 0      0      0   0      0      |
     0       0       0       0     x_1^2  0      0   0      0      |
     0       0       0       0     0      x_1^2  0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       x_1x_2  x_1x_3  0     x_1x_3 0      0   x_1x_4 0      |
     0       0       0       0     0      x_1x_3 0   0      x_1x_4 |
     0       0       x_1^2   0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     0       0       0       0     0      0      0   0      0      |
     x_1     x_2     x_3     0     0      0      0   0      0      |
     0       0       0       x_1   x_2    x_3    0   0      0      |
     0       0       0       0     0      0      x_1 x_2    x_3    |

             50      131
o8 : Matrix S   <-- S

i9 : burkeResolution(M, 5, Check =>true)

      1      3      7      19      50      131
o9 = R  <-- R  <-- R  <-- R   <-- R   <-- R
                                           
     0      1      2      3       4       5

o9 : Complex

the function golodBetti displays the Betti table of the resolution that would be constructed by burkeResolution, without actually making the construction.

i10 : golodBetti (M,12)

             0 1 2  3  4   5   6   7    8    9    10    11     12
o10 = total: 1 3 7 19 50 131 345 907 2385 6272 16493 43371 114051
          0: 1 3 3  1  .   .   .   .    .    .     .     .      .
          1: . . 4 18 34  35  21   7    1    .     .     .      .
          2: . . .  . 16  96 260 420  448  328   165    55     11
          3: . . .  .  .   .  64 480 1680 3640  5448  5940   4840
          4: . . .  .  .   .   .   .  256 2304  9856 26624  50832
          5: . . .  .  .   .   .   .    .    .  1024 10752  54272
          6: . . .  .  .   .   .   .    .    .     .     .   4096

o10 : BettiTally

i11 : betti F

             0 1 2  3  4
o11 = total: 1 3 7 19 50
          0: 1 3 3  1  .
          1: . . 4 18 34
          2: . . .  . 16

o11 : BettiTally

The advantage of resolutions computed from A-infinity structures is the decomposition of the differential into blocks corresponding to tensor products of the modules in the finite resolutions. In the following display, the symbol {u_1..u_n} denotes B_(u_1)**..**B_(u_(n-1))**G_(u_n), where G is the S-free resolution of M and B is the truncated shift of the S-free resolution A of R^1: that is, B_i = A_(i-1), i = 2...length A.

i12 : picture F

      +---------------------------------------+
      |+---+---+                              |
o12 = ||   |{1}|                              |
      |+---+---+                              |
      ||{0}| * |                              |
      |+---+---+                              |
      +---------------------------------------+
      |+---+---+------+                       |
      ||   |{2}|{2, 0}|                       |
      |+---+---+------+                       |
      ||{1}| * |   *  |                       |
      |+---+---+------+                       |
      +---------------------------------------+
      |+------+---+------+------+             |
      ||      |{3}|{3, 0}|{2, 1}|             |
      |+------+---+------+------+             |
      ||  {2} | * |   .  |   *  |             |
      |+------+---+------+------+             |
      ||{2, 0}| . |   *  |   *  |             |
      |+------+---+------+------+             |
      +---------------------------------------+
      |+------+------+------+------+---------+|
      ||      |{4, 0}|{3, 1}|{2, 2}|{2, 2, 0}||
      |+------+------+------+------+---------+|
      ||  {3} |   .  |   *  |   *  |    .    ||
      |+------+------+------+------+---------+|
      ||{3, 0}|   *  |   *  |   .  |    *    ||
      |+------+------+------+------+---------+|
      ||{2, 1}|   .  |   *  |   *  |    *    ||
      |+------+------+------+------+---------+|
      +---------------------------------------+

the functions displayBlocks and extractBlocks allow the examination of these submatrices.

i13 : displayBlocks F.dd_2

      +---+----------------------+----------------------------------+
o13 = |   |          {2}         |              {2, 0}              |
      +---+----------------------+----------------------------------+
      |{1}|{1} | -x_2 -x_3 0    ||{1} | x_1^2 x_1x_2 x_1x_3 x_1x_4 ||
      |   |{1} | x_1  0    -x_3 ||{1} | 0     0      0      0      ||
      |   |{1} | 0    x_1  x_2  ||{1} | 0     0      0      0      ||
      +---+----------------------+----------------------------------+

i14 : extractBlocks(F.dd_4, {{2,1}},{{3,1},{2,2}})

o14 = {4} | x_2  0    0    x_3  0    0    0    0    0    x_4  0    0    0   
      {4} | 0    x_2  0    0    x_3  0    0    0    0    0    x_4  0    0   
      {4} | 0    0    x_2  0    0    x_3  0    0    0    0    0    x_4  0   
      {4} | -x_1 0    0    0    0    0    x_3  0    0    0    0    0    x_4 
      {4} | 0    -x_1 0    0    0    0    0    x_3  0    0    0    0    0   
      {4} | 0    0    -x_1 0    0    0    0    0    x_3  0    0    0    0   
      {4} | 0    0    0    -x_1 0    0    -x_2 0    0    0    0    0    0   
      {4} | 0    0    0    0    -x_1 0    0    -x_2 0    0    0    0    0   
      {4} | 0    0    0    0    0    -x_1 0    0    -x_2 0    0    0    0   
      {4} | 0    0    0    0    0    0    0    0    0    -x_1 0    0    -x_2
      {4} | 0    0    0    0    0    0    0    0    0    0    -x_1 0    0   
      {4} | 0    0    0    0    0    0    0    0    0    0    0    -x_1 0   
      -----------------------------------------------------------------------
      0    0    0    0    0    -x_2 -x_3 0    0    0    0    0    0    0   
      0    0    0    0    0    x_1  0    -x_3 0    0    0    0    0    0   
      0    0    0    0    0    0    x_1  x_2  0    0    0    0    0    0   
      0    0    0    0    0    0    0    0    -x_2 -x_3 0    0    0    0   
      x_4  0    0    0    0    0    0    0    x_1  0    -x_3 0    0    0   
      0    x_4  0    0    0    0    0    0    0    x_1  x_2  0    0    0   
      0    0    x_4  0    0    0    0    0    0    0    0    -x_2 -x_3 0   
      0    0    0    x_4  0    0    0    0    0    0    0    x_1  0    -x_3
      0    0    0    0    x_4  0    0    0    0    0    0    0    x_1  x_2 
      0    0    -x_3 0    0    0    0    0    0    0    0    0    0    0   
      -x_2 0    0    -x_3 0    0    0    0    0    0    0    0    0    0   
      0    -x_2 0    0    -x_3 0    0    0    0    0    0    0    0    0   
      -----------------------------------------------------------------------
      0    0    0    |
      0    0    0    |
      0    0    0    |
      0    0    0    |
      0    0    0    |
      0    0    0    |
      0    0    0    |
      0    0    0    |
      0    0    0    |
      -x_2 -x_3 0    |
      x_1  0    -x_3 |
      0    x_1  x_2  |

              12      30
o14 : Matrix R   <-- R

References

Jesse Burke, Higher Homotopies and Golod Rings. arXiv:1508.03782v2, October 2015.

Caveat

Requires standard graded ring, module. Something to fix in a future version

Ways to use burkeResolution :

burkeResolution(Module,ZZ)

For the programmer

The object burkeResolution is a method function with options.