-- -*- M2-comint -*- {* hash: 420933298 *}

i1 : kk=ZZ/32003

o1 = kk

o1 : QuotientRing

i2 : S=kk[a..d]

o2 = S

o2 : PolynomialRing

i3 : M=matrix{{a,0,0,0},{0,b,0,0},{0,0,c,0},{0,0,0,d}}

o3 = | a 0 0 0 |
     | 0 b 0 0 |
     | 0 0 c 0 |
     | 0 0 0 d |

             4       4
o3 : Matrix S  <--- S

i4 : elementary(M,0,1)-- since k=0, this command simply eliminates the last row of M.

o4 = | a 0 0 0 |
     | 0 b 0 0 |
     | 0 0 c 0 |

             3       4
o4 : Matrix S  <--- S

i5 : kk=ZZ/32003

o5 = kk

o5 : QuotientRing

i6 : S=kk[a..d]

o6 = S

o6 : PolynomialRing

i7 : I=ideal(a^2,b^3,c^4, d^5)

             2   3   4   5
o7 = ideal (a , b , c , d )

o7 : Ideal of S

i8 : F=res I

      1      4      6      4      1
o8 = S  <-- S  <-- S  <-- S  <-- S  <-- 0
                                         
     0      1      2      3      4      5

o8 : ChainComplex

i9 : M=image F.dd_3

o9 = image {5} | c4  d5  0   0   |
           {6} | -b3 0   d5  0   |
           {7} | a2  0   0   d5  |
           {7} | 0   -b3 -c4 0   |
           {8} | 0   a2  0   -c4 |
           {9} | 0   0   a2  b3  |

                             6
o9 : S-module, submodule of S

i10 : f=matrix gens M

o10 = {5} | c4  d5  0   0   |
      {6} | -b3 0   d5  0   |
      {7} | a2  0   0   d5  |
      {7} | 0   -b3 -c4 0   |
      {8} | 0   a2  0   -c4 |
      {9} | 0   0   a2  b3  |

              6       4
o10 : Matrix S  <--- S

i11 : fascending=transpose sort(transpose f, DegreeOrder=>Descending) -- this is f with rows sorted so that the degrees are ascending.

o11 = {5} | c4  d5  0   0   |
      {6} | -b3 0   d5  0   |
      {7} | 0   -b3 -c4 0   |
      {7} | a2  0   0   d5  |
      {8} | 0   a2  0   -c4 |
      {9} | 0   0   a2  b3  |

              6       4
o11 : Matrix S  <--- S

i12 : g=elementary(fascending,1,1) --k=1, so add random multiples of the last row to the preceding row

o12 = {5} | c4  d5  0       0           |
      {6} | -b3 0   d5      0           |
      {7} | 0   -b3 -c4     0           |
      {7} | a2  0   0       d5          |
      {8} | 0   a2  10875a3 10875ab3-c4 |

              5       4
o12 : Matrix S  <--- S

i13 : g1=elementary(fascending,1,3)

o13 = {5} | c4  d5  0                       0                          |
      {6} | -b3 0   d5                      0                          |
      {7} | 0   -b3 -c4                     0                          |
      {7} | a2  0   0                       d5                         |
      {8} | 0   a2  15770a3-9599a2b-5559a2c 15770ab3-9599b4-5559b3c-c4 |

              5       4
o13 : Matrix S  <--- S

i14 :