symmetricQuotient -- computes the image of the surjection from the simplicial tensor product onto the second symmetric power of a simplicial module

i1 : Q = ZZ/2[a,b];

i2 : K = koszulComplex vars Q;

i3 : phi = prune symmetricQuotient(K,4)

          1             1
o3 = 0 : Q  <--------- Q  : 0
               | 1 |

          5                               8
     1 : Q  <--------------------------- Q  : 1
               {1} | 1 0 1 0 0 0 0 0 |
               {1} | 0 1 0 1 0 0 0 0 |
               {2} | 0 0 0 0 1 0 0 0 |
               {2} | 0 0 0 0 0 1 1 0 |
               {2} | 0 0 0 0 0 0 0 1 |

          10                                                     19
     2 : Q   <------------------------------------------------- Q   : 2
                {2} | 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                {2} | 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 |
                {3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 |
                {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

          9                                                   18
     3 : Q  <----------------------------------------------- Q   : 3
               {3} | 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
               {3} | 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
               {3} | 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 |
               {3} | 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
               {3} | 0 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 |
               {3} | 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 |
               {4} | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 |
               {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 |
               {4} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

          3                           6
     4 : Q  <----------------------- Q  : 4
               {4} | 0 1 0 0 0 1 |
               {4} | 1 0 0 0 1 0 |
               {4} | 0 0 1 0 0 1 |

o3 : ComplexMap

i4 : isWellDefined phi

o4 = true

i5 : isCommutative phi

o5 = true

i6 : prune coker phi == 0

o6 = true

i7 : prune HH phi

o7 = 0 : cokernel | b a | <--------- cokernel | b a | : 0
                             | 1 |

o7 : ComplexMap

i8 : prune coker oo --the induced map on homology is NOT surjective, in contrast to the case when 2 is a unit

o8 = cokernel {2} | b a |
      
     2

o8 : Complex