next | previous | forward | backward | up | top | index | toc | Macaulay2 website
Complexes :: RingMap ** ComplexMap

RingMap ** ComplexMap -- tensor a map of complexes along a ring map

Synopsis

Description

These methods implement the base change of rings. As input, one can either give a ring map $\phi$, or the ring $S$ (when there is a canonical map from $R$ to $S$).

We illustrate the tensor product of a map of complexes along a ring map.

i1 : R = QQ[a,b,c,d];
i2 : S = QQ[s,t];
i3 : phi = map(S, R, {s, s+t, t, s-t})

o3 = map (S, R, {s, s + t, t, s - t})

o3 : RingMap S <--- R
i4 : I = ideal(a*b, b*c, c*d)

o4 = ideal (a*b, b*c, c*d)

o4 : Ideal of R
i5 : J = I + ideal(a^2, b^2, c^2, d^2)

                            2   2   2   2
o5 = ideal (a*b, b*c, c*d, a , b , c , d )

o5 : Ideal of R
i6 : CI = freeResolution I

      1      3      2
o6 = R  <-- R  <-- R
                    
     0      1      2

o6 : Complex
i7 : CJ = freeResolution J

      1      7      13      10      3
o7 = R  <-- R  <-- R   <-- R   <-- R
                                    
     0      1      2       3       4

o7 : Complex
i8 : f = extend(CJ, CI, map(CJ_0, CI_0, 1))

          1             1
o8 = 0 : R  <--------- R  : 0
               | 1 |

          7                     3
     1 : R  <----------------- R  : 1
               {2} | 0 0 0 |
               {2} | 1 0 0 |
               {2} | 0 0 0 |
               {2} | 0 1 0 |
               {2} | 0 0 0 |
               {2} | 0 0 1 |
               {2} | 0 0 0 |

          13                   2
     2 : R   <--------------- R  : 2
                {3} | 0 0 |
                {3} | 0 0 |
                {3} | 1 0 |
                {3} | 0 0 |
                {3} | 0 0 |
                {3} | 0 1 |
                {3} | 0 0 |
                {3} | 0 0 |
                {4} | 0 0 |
                {4} | 0 0 |
                {4} | 0 0 |
                {4} | 0 0 |
                {4} | 0 0 |

o8 : ComplexMap
i9 : assert isWellDefined f
i10 : g = phi ** f

           1             1
o10 = 0 : S  <--------- S  : 0
                | 1 |

           7                     3
      1 : S  <----------------- S  : 1
                {2} | 0 0 0 |
                {2} | 1 0 0 |
                {2} | 0 0 0 |
                {2} | 0 1 0 |
                {2} | 0 0 0 |
                {2} | 0 0 1 |
                {2} | 0 0 0 |

           13                   2
      2 : S   <--------------- S  : 2
                 {3} | 0 0 |
                 {3} | 0 0 |
                 {3} | 1 0 |
                 {3} | 0 0 |
                 {3} | 0 0 |
                 {3} | 0 1 |
                 {3} | 0 0 |
                 {3} | 0 0 |
                 {4} | 0 0 |
                 {4} | 0 0 |
                 {4} | 0 0 |
                 {4} | 0 0 |
                 {4} | 0 0 |

o10 : ComplexMap
i11 : assert isWellDefined g
i12 : dd^(source g)

           1                             3
o12 = 0 : S  <------------------------- S  : 1
                | s2+st st+t2 st-t2 |

           3                       2
      1 : S  <------------------- S  : 2
                {2} | -t 0    |
                {2} | s  -s+t |
                {2} | 0  s+t  |

o12 : ComplexMap
i13 : dd^(target g)

           1                                                       7
o13 = 0 : S  <--------------------------------------------------- S  : 1
                | s2 s2+st s2+2st+t2 st+t2 t2 st-t2 s2-2st+t2 |

           7                                                                                               13
      1 : S  <------------------------------------------------------------------------------------------- S   : 2
                {2} | -s-t 0    0  0   0   0    0    0    -t2 -st+t2 -s2+2st-t2 0          0          |
                {2} | s    -s-t -t 0   0   0    0    0    0   0      0          -s2+2st-t2 0          |
                {2} | 0    s    0  -t  0   0    0    0    0   0      0          0          -s2+2st-t2 |
                {2} | 0    0    s  s+t -t  -s+t 0    0    0   0      0          0          0          |
                {2} | 0    0    0  0   s+t 0    -s+t 0    s2  0      0          0          0          |
                {2} | 0    0    0  0   0   s+t  t    -s+t 0   s2     0          0          0          |
                {2} | 0    0    0  0   0   0    0    t    0   0      s2         s2+st      s2+2st+t2  |

           13                                                                                       10
      2 : S   <----------------------------------------------------------------------------------- S   : 3
                 {3} | 0    0   -t2 -st+t2 0    s2-2st+t2 0         0    0          0          |
                 {3} | t    0   0   0      0    0         s2-2st+t2 0    0          0          |
                 {3} | -s-t 0   -st -s2+st 0    0         0         0    -s2+2st-t2 0          |
                 {3} | s    0   0   0      0    0         0         0    0          -s2+2st-t2 |
                 {3} | 0    s-t -s2 0      0    0         0         0    0          0          |
                 {3} | 0    -t  0   -s2    0    0         0         0    -s2+st     -s2+t2     |
                 {3} | 0    s+t 0   0      -s2  0         0         0    0          0          |
                 {3} | 0    0   0   0      0    0         0         -s2  -s2-st     -s2-2st-t2 |
                 {4} | 0    0   s+t 0      -s+t 0         0         0    0          0          |
                 {4} | 0    0   0   s+t    t    0         0         -s+t 0          0          |
                 {4} | 0    0   0   0      0    -s-t      0         t    0          0          |
                 {4} | 0    0   0   0      0    s         -s-t      0    t          0          |
                 {4} | 0    0   0   0      0    0         s         0    0          t          |

           10                                  3
      3 : S   <------------------------------ S  : 4
                 {4} | 0   0    s2-2st+t2 |
                 {4} | s2  0    0         |
                 {5} | s-t 0    0         |
                 {5} | -t  -s+t 0         |
                 {5} | s+t 0    0         |
                 {5} | 0   -t   0         |
                 {5} | 0   0    -t        |
                 {5} | 0   -s-t 0         |
                 {5} | 0   s    -s-t      |
                 {5} | 0   0    s         |

o13 : ComplexMap
i14 : prune HH g

o14 = 0 : cokernel | t2 st s2 | <--------- cokernel | t2 st s2 | : 0
                                   | 1 |

o14 : ComplexMap

See also