splittings -- compute the splittings of a split right exact sequence

Usage:

x = splittings(a,b)
Inputs:
- a, a matrix, maps into the kernel of b
- b, a matrix, representing a surjection from target a to a free module
Outputs:
- L, a list, L = \{sigma,tau\}, splittings of a,b respectively

Description

Assuming that (a,b) are the maps of a right exact sequence

$0\to A\to B\to C \to 0$

with B, C free, the script produces a pair of maps sigma, tau with $tau: C \to B$ a splitting of b and $sigma: B \to A$ a splitting of a; that is,

$a*sigma+tau*b = 1_B$

$sigma*a = 1_A$

$b*tau = 1_C$

i1 : kk= ZZ/101

o1 = kk

o1 : QuotientRing

i2 : S = kk[x,y,z]

o2 = S

o2 : PolynomialRing

i3 : setRandomSeed 0
 -- setting random seed to 0

o3 = 0

i4 : t = random(S^{2:-1,2:-2}, S^{3:-1,4:-2})

o4 = {1} | 24  -36 -30 39x-43y+45z  21x-15y-34z 34x-28y-48z  19x-47y-47z |
     {1} | -29 19  19  -47x+38y+47z -39x+2y+19z -18x+16y-16z -13x+22y+7z |
     {2} | 0   0   0   -10          -29         -8           -22         |
     {2} | 0   0   0   -29          -24         -38          -16         |

             4      7
o4 : Matrix S  <-- S

i5 : ss = splittings(syz t, t)

o5 = {{1} | 0 0 1 0 0 0   0  |, {1} | -27 2  13x-10y+43z 50x-34y-50z |}
      {2} | 0 0 0 0 0 -31 -6 |  {1} | -4  35 22x+32y+43z -7x-8y-27z  |
      {2} | 0 0 0 0 0 29  9  |  {1} | 0   0  0           0           |
                                {2} | 0   0  -25         26          |
                                {2} | 0   0  26          -2          |
                                {2} | 0   0  0           0           |
                                {2} | 0   0  0           0           |

o5 : List

i6 : ss/betti

             0 1         0 1
o6 = {total: 3 7, total: 7 4}
          0: . 3      0: . 2
          1: 1 4      1: 3 2
          2: 2 .      2: 4 .

o6 : List

Ways to use splittings:

splittings(Matrix,Matrix)

For the programmer

The object splittings is a method function.

The source of this document is in CompleteIntersectionResolutions.m2:3925:0.