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extractBlocks -- displays components of a map in a labeled complex

Synopsis

Description

The terms of the burkeResolution resolution are direct sums of labeled modules. the function picture shows the symbols associated to the summands, while the function extractBlocks provides the submatrix associated with the summands specified.

i1 : R = ZZ/101[a,b,c,d]/ideal(a^2, b^2, c^3, d^4)

o1 = R

o1 : QuotientRing
 
i2 : M = R^1/ideal(a*b,c*d)

o2 = cokernel | ab cd |

                            1
o2 : R-module, quotient of R
 
i3 : F = burkeResolution(M,5)

      1      6      17      42      112      295
o3 = R  <-- R  <-- R   <-- R   <-- R    <-- R
                                             
     0      1      2       3       4        5

o3 : Complex
 
i4 : picture F.dd_3

     +------+---+------+------+
o4 = |      |{3}|{3, 0}|{2, 1}|
     +------+---+------+------+
     |  {2} | * |   8  |   8  |
     +------+---+------+------+
     |{2, 0}| . |   .  |   *  |
     +------+---+------+------+
 
i5 : extractBlocks(F.dd_3, {2,1})

o5 = {3} | 0 a  b 0  0 0 -b 0  0 0  0 0 0  0  0  0  0 0   0  0  0  0  0   0 |
     {3} | 0 0  a 0  0 0 -a -b 0 0  0 0 0  0  0  0  0 0   0  0  0  0  0   0 |
     {4} | 0 0  0 1  0 0 0  0  0 0  0 0 0  0  0  0  0 0   0  0  0  0  0   0 |
     {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 |
     {4} | 0 0  0 0  0 0 0  0  0 1  0 0 0  0  0  0  0 0   0  0  0  0  0   0 |
     {4} | 0 0  0 0  0 0 0  0  0 0  0 0 0  0  0  -c 0 -d3 0  0  0  0  d3  0 |
     {5} | 0 0  0 0  1 0 0  0  0 0  0 0 -1 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  -1 0  0  0 0   0  0  0  0  0   0 |
     {5} | 0 0  0 0  0 0 0  0  0 0  1 0 0  0  -1 0  0 0   0  0  0  0  0   0 |
     {5} | 0 0  0 0  0 0 0  0  0 0  0 0 0  0  0  0  0 c2  0  0  0  -d -c2 0 |
     {6} | 0 0  0 0  0 1 0  0  0 0  0 0 0  0  0  0  0 0   -1 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  -1 0  0  0   0 |
     {6} | 0 0  0 0  0 0 0  0  0 0  0 1 0  0  0  0  0 0   0  0  -1 0  0   0 |
     {2} | 0 ab 0 cd 0 0 0  0  0 0  0 0 0  0  0  0  0 0   0  0  0  0  0   0 |
     {2} | 0 0  0 0  0 0 0  ab 0 cd 0 0 0  0  0  0  0 0   0  0  0  0  0   0 |
     {3} | 0 0  0 0  0 0 0  0  0 0  0 0 0  ab 0  cd 0 0   0  0  0  0  0   0 |
     {4} | 0 0  0 0  0 0 0  0  0 0  0 0 0  0  0  0  0 0   0  ab 0  cd 0   0 |

             17      24
o5 : Matrix R   <-- R
 
i6 : extractBlocks(F.dd_3,{2,0}, {2,1})

o6 = {2} | 0 ab 0 cd 0 0 0 0  0 0  0 0 0 0  0 0  0 0 0 0  0 0  0 0 |
     {2} | 0 0  0 0  0 0 0 ab 0 cd 0 0 0 0  0 0  0 0 0 0  0 0  0 0 |
     {3} | 0 0  0 0  0 0 0 0  0 0  0 0 0 ab 0 cd 0 0 0 0  0 0  0 0 |
     {4} | 0 0  0 0  0 0 0 0  0 0  0 0 0 0  0 0  0 0 0 ab 0 cd 0 0 |

             4      24
o6 : Matrix R  <-- R
 
i7 : extractBlocks(F.dd_3,{2,0}, {{3,0},{2,1}})

o7 = {2} | 0 0 0 0 0 0 0 ab 0 cd 0 0 0 0  0 0  0 0 0 0  0 0  0 0 0 0  0 0  0
     {2} | 0 0 0 0 0 0 0 0  0 0  0 0 0 ab 0 cd 0 0 0 0  0 0  0 0 0 0  0 0  0
     {3} | 0 0 0 0 0 0 0 0  0 0  0 0 0 0  0 0  0 0 0 ab 0 cd 0 0 0 0  0 0  0
     {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 ab 0 cd 0
     ------------------------------------------------------------------------
     0 |
     0 |
     0 |
     0 |

             4      30
o7 : Matrix R  <-- R

See also

Ways to use extractBlocks :

For the programmer

The object extractBlocks is a method function.