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exteriorInclusion -- computes the image of the 2nd exterior power into the tensor product

Description

Given a simplicial module $S$ (or a complex), this function computes the map $$\bigwedge^2 S \to S \otimes S,$$ The cokernel of this map is by definition the second symmetric power of $S$. This method is mainly used in conjunction with the tensorLES command to compute induced maps on homology for canonical short exact sequences.

i1 : Q = ZZ/101[a,b,c]

o1 = Q

o1 : PolynomialRing
 
i2 : K = koszulComplex vars Q

      1      3      3      1
o2 = Q  <-- Q  <-- Q  <-- Q
                           
     0      1      2      3

o2 : Complex
 
i3 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
 -- .308429s elapsed
 
i4 : isWellDefined phi

o4 = true
 
i5 : isCommutative phi

o5 = true
 
i6 : prune coker phi

      1      9      36      74
o6 = Q  <-- Q  <-- Q   <-- Q
                            
     0      1      2       3

o6 : Complex

All of the homology of the second symmetric/exterior powers are guaranteed to be concentrated in degrees $0$ to $3$ in the above example, so it suffices to compute only $4$ terms to understand all of the homology.

i7 : for i to 3 list prune HH_i source phi

                                                 46
o7 = {0, cokernel {1} | c b a 0 0 0 0 0 0 |, 0, Q  }
                  {1} | 0 0 0 c b a 0 0 0 |
                  {1} | 0 0 0 0 0 0 c b a |

o7 : List
 
i8 : for i to 3 list prune HH_i (coker phi)

o8 = {cokernel | c b a |, 0, cokernel {2} | c b a 0 0 0 0 0 0 |, cokernel {4}
                                      {2} | 0 0 0 c b a 0 0 0 |           {4}
                                      {2} | 0 0 0 0 0 0 c b a |           {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
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                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
                                                                          {4}
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     ------------------------------------------------------------------------
     | 2c 0    0    |}
     | 0  0    -b   |
     | 0  a    0    |
     | -b 0    50c  |
     | b  0    -50c |
     | a  50c  0    |
     | 0  -50b -50a |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | -b 0    -50c |
     | -b 0    -50c |
     | 0  c    0    |
     | a  50c  0    |
     | a  -50c 0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  -b   0    |
     | 0  50b  50a  |
     | 0  0    a    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |
     | 0  0    0    |

o8 : List

Notice that since the tensor square splits outside of characteristic 2, the symmetric power picks up the even degree homology and the exterior square picks up the odd homology. In characteristic 2 this changes:

i9 : Q = ZZ/2[a,b,c]

o9 = Q

o9 : PolynomialRing
 
i10 : K = koszulComplex vars Q

       1      3      3      1
o10 = Q  <-- Q  <-- Q  <-- Q
                            
      0      1      2      3

o10 : Complex
 
i11 : phi = elapsedTime exteriorInclusion(K,3); --specify top degree 3
 -- .299255s elapsed
 
i12 : isWellDefined phi

o12 = true
 
i13 : isCommutative phi

o13 = true
 
i14 : for i to 2 list prune HH_i source phi

o14 = {0, cokernel {1} | a 0 b 0 0 c |, cokernel {2} | b a 0 c 0 0 0 c |}
                   {1} | 0 b a 0 c 0 |           {2} | 0 0 c b a 0 0 0 |
                   {1} | 0 0 0 c b a |           {2} | 0 0 0 0 0 c b a |

o14 : List
 
i15 : for i to 2 list prune HH_i (coker phi)

o15 = {cokernel | c b a |, 0, cokernel {2} | b a 0 c 0 0 0 c |}
                                       {2} | 0 0 c b a 0 0 0 |
                                       {2} | 0 0 0 0 0 c b a |

o15 : List

See also

Ways to use exteriorInclusion:

  • exteriorInclusion(Complex)
  • exteriorInclusion(Complex,ZZ)
  • exteriorInclusion(Module)
  • exteriorInclusion(SimplicialModule)

For the programmer

The object exteriorInclusion is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:3906:0.