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TateOnProducts :: lowerCorner

lowerCorner -- compute the lower corner

Synopsis

Description

Let k = |deg| be the total degree and G ⊂Fk the summand spanned by the generators of Fk in degree d, H ⊂Fk+1 the summand spanned by generators of degree d’ with 0 ≤d-d’ ≤n. The function returns the corresponding submatrix m: H -> G of the differential.

i1 : n={1,2}; kk=ZZ/101; (S,E)=setupRings(kk,n);
i4 : F=dual res((ker transpose vars E)**E^{{ 2,3}},LengthLimit=>4)

      70      35      15      5      1
o4 = E   <-- E   <-- E   <-- E  <-- E
                                     
     -4      -3      -2      -1     0

o4 : ChainComplex
i5 : betti F

            -4 -3 -2 -1 0
o5 = total: 70 35 15  5 1
         0: 70 35 15  5 1

o5 : BettiTally
i6 : tallyDegrees F

o6 = (Tally{{-1, -3} => 20}, Tally{{-1, -2} => 12}, Tally{{-1, -1} => 6},
            {-2, -2} => 18         {-2, -1} => 9          {-2, 0} => 3   
            {-3, -1} => 12         {-3, 0} => 4           {0, -2} => 6
            {-4, 0} => 5           {0, -3} => 10
            {0, -4} => 15
     ------------------------------------------------------------------------
     Tally{{-1, 0} => 2}, Tally{{0, 0} => 1})
           {0, -1} => 3

o6 : Sequence
i7 : deg={2,1}

o7 = {2, 1}

o7 : List
i8 : m=lowerCorner(F,deg);

             9       9
o8 : Matrix E  <--- E
i9 : tally degrees target m, tally degrees source m

o9 = (Tally{{-2, -1} => 9}, Tally{{-1, -1} => 6})
                                  {-2, 0} => 3

o9 : Sequence
i10 : Fm=(res(coker m,LengthLimit=>7))[sum deg]

       9      9      8      15      32      57      91      137
o10 = E  <-- E  <-- E  <-- E   <-- E   <-- E   <-- E   <-- E
                                                            
      -3     -2     -1     0       1       2       3       4

o10 : ChainComplex
i11 : betti Fm

             -3 -2 -1  0  1  2  3   4
o11 = total:  9  9  8 15 32 57 91 137
          0:  9  9  5  1  .  .  .   .
          1:  .  .  .  3  7 11 15  19
          2:  .  .  3 11 25 45 71 103
          3:  .  .  .  .  .  1  5  15

o11 : BettiTally
i12 : cohomologyTable(Fm,-{3,3},{4,4})

o12 = | 0   0  0 0  0   0   0 0 |
      | 0   0  0 0  0   0   0 0 |
      | 0   0  0 0  0   0   0 0 |
      | 6h  3h 0 3  6   9   0 0 |
      | 2h  h  0 1  2   3   0 0 |
      | 0   0  0 0  0   0   0 0 |
      | 0   0  0 0  0   0   0 0 |
      | 2h3 h3 0 h2 2h2 3h2 0 0 |

                       8                8
o12 : Matrix (ZZ[h, k])  <--- (ZZ[h, k])

Ways to use lowerCorner :