randomChainComplex -- random chain complex over the integers with prescribed ranks of the homology group and ranks of the matrices

Synopsis

Usage:

C = randomChainComplex(h,r)
Inputs:
- h, a list, of desired ranks of the homology groups, of some length $n$
- r, a list, of desired ranks of the matrices in the complex, of length $n-1$
Optional inputs:
- Height => an integer, default value 10, the sizes of the random integers used
- WithLLL => a Boolean value, default value true, use the LLL algorithm to keep the sizes of the integers small
- ZeroMean => a Boolean value, default value true, whether to balance the random numbers around zero
Outputs:
- C, a chain complex, a random chain complex over the integers whose homology ranks match $h$, and whose matrices have ranks given by $r$

Description

i1 : h={1,4,6,5,1}

o1 = {1, 4, 6, 5, 1}

o1 : List

i2 : r={1,3,3,4}

o2 = {1, 3, 3, 4}

o2 : List

i3 : C=randomChainComplex(h,r)

       2       8       12       12       5
o3 = ZZ  <-- ZZ  <-- ZZ   <-- ZZ   <-- ZZ
                                        
     0       1       2        3        4

o3 : ChainComplex

i4 : prune HH C

           1
o4 = 0 : ZZ

           4
     1 : ZZ

           6
     2 : ZZ

           5
     3 : ZZ

           1
     4 : ZZ

o4 : GradedModule

i5 : for i from 0 to 4 list rank HH_i C

o5 = {1, 4, 6, 5, 1}

o5 : List

i6 : for i from 1 to 4 list rank(C.dd_i)

o6 = {1, 3, 3, 4}

o6 : List

The optional argument Height chooses the maximum sizes of the random numbers used. The actual numbers are somewhat larger (twice as many bits), as matrices are multiplied together.

i7 : h={1,4,0,5,1}

o7 = {1, 4, 0, 5, 1}

o7 : List

i8 : r={2,3,3,4}

o8 = {2, 3, 3, 4}

o8 : List

i9 : C=randomChainComplex(h,r, Height=>1000)

       3       9       6       12       5
o9 = ZZ  <-- ZZ  <-- ZZ  <-- ZZ   <-- ZZ
                                       
     0       1       2       3        4

o9 : ChainComplex

i10 : C.dd

            3                                                                           9
o10 = 0 : ZZ  <---------------------------------------------------------------------- ZZ  : 1
                 | 44424  -8775 -2160 -53262  -62508  -10437 -6822  -62838  1011  |
                 | 90516  14087 19368 -141214 -90936  2397   -10686 -117210 24977 |
                 | 138645 31572 37098 -226521 -127899 11070  -15363 -176148 45423 |

            9                                                              6
      1 : ZZ  <--------------------------------------------------------- ZZ  : 2
                 | 330985  16445   756805   1698     330472  348418  |
                 | 713026  -154053 -42697   1538759  531828  793108  |
                 | 1313518 -554313 -1321367 952247   994858  2800    |
                 | 1504309 -561736 -934444  734919   1193028 74986   |
                 | -566001 95813   3887     -1731404 -398918 -893718 |
                 | -940456 315736  168999   191100   -803825 96554   |
                 | -505191 256413  438512   888174   -448781 587460  |
                 | -380977 341680  1286800  724845   -281448 878246  |
                 | 1910440 -687155 -752920  2107     1588083 -178598 |

            6                                                                                                                                     12
      2 : ZZ  <-------------------------------------------------------------------------------------------------------------------------------- ZZ   : 3
                 | 41388167  -95238274 5488161   12322184  -52314315 -15854513 50425642  -29862626 -100716183 -78673592 -64535295 57729071  |
                 | 9777314   29735878  8710464   42894623  -632054   -31139424 6643135   83313808  -38773955  -56265921 3886477   -86494233 |
                 | 21143227  -17020460 17282979  975209    12792629  -27787635 -357149   9785294   -29488934  -28260935 -29357422 16504066  |
                 | 20760737  -1043822  21170415  7109264   22491307  -36115359 -6466526  30843814  -27310181  -32178816 -24665021 -4404699  |
                 | -40854113 120671920 421659    2171879   64292129  1029525   -57179939 69499514  98152576   65571655  71827928  -98294804 |
                 | -47055725 11588622  -43668471 -17943160 -39150723 76088519  6825892   -62888800 68596487   76741536  56882559  6646445   |

            12                                                         5
      3 : ZZ   <---------------------------------------------------- ZZ  : 4
                  | -2098931 -882197  -1197770 192505   -821100  |
                  | -540649  1132991  -2558060 653635   -1393888 |
                  | -1333740 3364752  -3624762 -4878996 2320266  |
                  | -1425716 -2621421 1862275  564680   -64468   |
                  | 1352058  -1470320 3544570  -96870   1495604  |
                  | -121418  1868293  -1472318 -1881704 649512   |
                  | -37268   -77123   -385029  440508   -259828  |
                  | -469251  -460737  -885687  1959819  -1865142 |
                  | -1299118 659729   -1037071 -1031938 -476574  |
                  | 28821    -1464654 -120682  2377619  -941278  |
                  | -857790  -1893725 2246803  -1115706 1519562  |
                  | -1160441 -1211038 -190488  1500701  -1537374 |

o10 : ChainComplexMap

i11 : C.dd^2 == 0

o11 = true

i12 : prune HH C

o12 = 0 : cokernel | 3 |
                   | 0 |       

            4
      1 : ZZ                   

      2 : cokernel | 64931150 |

      3 : cokernel | 6 |
                   | 0 |
                   | 0 |
                   | 0 |
                   | 0 |
                   | 0 |       

            1
      4 : ZZ                   

o12 : GradedModule

i13 : for i from 0 to 4 list rank HH_i C

o13 = {1, 4, 0, 5, 1}

o13 : List

i14 : for i from 1 to 4 list rank(C.dd_i)

o14 = {2, 3, 3, 4}

o14 : List

Caveat

This returns a chain complex over the integers. Notice that if one gives h to be a list of zeros, then that doesn't mean that the complex is exact, just that the ranks are as expected.

Ways to use randomChainComplex:

randomChainComplex(List,List)

For the programmer