Macaulay 2, version 0.9.20
with packages: Classic, Elimination, LLLBases, Parsing, PrimaryDecomposition,
               SchurRings, TangentCone

i1 : R=QQ[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : I=ideal(x^2,x*y,z^5)

             2        5
o2 = ideal (x , x*y, z )

o2 : Ideal of R

i3 : M=R^1/I

o3 = cokernel | x2 xy z5 |

                            1
o3 : R-module, quotient of R

i4 : C=res M

      1      3      3      1
o4 = R  <-- R  <-- R  <-- R  <-- 0

     0      1      2      3      4

o4 : ChainComplex

i5 : C.dd

          1                    3
o5 = 0 : R  <---------------- R  : 1
               | x2 xy z5 |

          3                          3
     1 : R  <---------------------- R  : 2
               {2} | -y -z5 0   |
               {2} | x  0   -z5 |
               {5} | 0  x2  xy  |

          3                  1
     2 : R  <-------------- R  : 3
               {3} | z5 |
               {7} | -y |
               {7} | x  |

          1
     3 : R  <----- 0 : 4
               0


--From this we can see that, for example, Tor_2(M,R/<x,y,z>) = Q^3.