cone(ChainComplexMap) -- mapping cone of a chain map

Synopsis

Usage:
cone f
Function: cone
Inputs:
- f, a chain complex map
Outputs:
- a chain complex, the mapping cone of a f

Description

i1 : R = ZZ/101[x,y,z]

o1 = R

o1 : PolynomialRing

i2 : m = image vars R

o2 = image | x y z |

                             1
o2 : R-module, submodule of R

i3 : m2 = image symmetricPower(2,vars R)

o3 = image | x2 xy xz y2 yz z2 |

                             1
o3 : R-module, submodule of R

i4 : M = R^1/m2

o4 = cokernel | x2 xy xz y2 yz z2 |

                            1
o4 : R-module, quotient of R

i5 : N = R^1/m

o5 = cokernel | x y z |

                            1
o5 : R-module, quotient of R

i6 : C = cone extend(resolution N,resolution M,id_(R^1))

      1      4      9      9      3
o6 = R  <-- R  <-- R  <-- R  <-- R  <-- 0
                                         
     0      1      2      3      4      5

o6 : ChainComplex

Let's check that the homology is correct; for example, HH_0 should be zero.

i7 : prune HH_0 C

o7 = 0

o7 : R-module

Let's check that HH_1 is isomorphic to m/m2.

i8 : prune HH_1 C

o8 = cokernel {1} | z y x 0 0 0 0 0 0 |
              {1} | 0 0 0 z y x 0 0 0 |
              {1} | 0 0 0 0 0 0 z y x |

                            3
o8 : R-module, quotient of R

i9 : prune (m/m2)

o9 = cokernel {1} | z y x 0 0 0 0 0 0 |
              {1} | 0 0 0 z y x 0 0 0 |
              {1} | 0 0 0 0 0 0 z y x |

                            3
o9 : R-module, quotient of R