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SpectralSequences :: Examples of change of rings Spectral Sequences

Examples of change of rings Spectral Sequences

Here are some examples of change of rings spectral sequences.

Given a ring map f: R -> S, an R-module M and an R-module S, there is a spectral sequence E with E2p,q = TorSp(TorRq(M,S),N) that abuts to TorRp+q(M,N).

i1 : k=QQ;
i2 : R=k[a,b,c];
i3 : S=k[s,t];
i4 : f = map(S,R,{s^2,s*t,t^2});

o4 : RingMap S <--- R
i5 : N = coker vars S;
i6 : M = coker vars R --;

o6 = cokernel | a b c |

                            1
o6 : R-module, quotient of R
i7 : F := complete res N;
i8 : pushFwdF := pushFwd(f,F);
i9 : G := complete res M;
i10 : E := spectralSequence(filteredComplex(G) ** pushFwdF);
i11 : EE := spectralSequence(G ** (filteredComplex pushFwdF));
i12 : e = prune E;
i13 : ee = prune EE;
i14 : e^0

      +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
o14 = |cokernel {2} | b2-ac 0  0  |            |cokernel {3} | b2-ac 0  0  0     0  0  0     0  0  |                                    |cokernel {4} | b2-ac 0  0  0     0  0  0     0  0  |                                    |cokernel {5} | b2-ac 0  0  |            |
      |         {3} | 0     -c -b |            |         {4} | 0     -c -b 0     0  0  0     0  0  |                                    |         {5} | 0     -c -b 0     0  0  0     0  0  |                                    |         {6} | 0     -c -b |            |
      |         {3} | 0     b  a  |            |         {4} | 0     b  a  0     0  0  0     0  0  |                                    |         {5} | 0     b  a  0     0  0  0     0  0  |                                    |         {6} | 0     b  a  |            |
      |                                        |         {3} | 0     0  0  b2-ac 0  0  0     0  0  |                                    |         {4} | 0     0  0  b2-ac 0  0  0     0  0  |                                    |                                        |
      |{0, 2}                                  |         {4} | 0     0  0  0     -c -b 0     0  0  |                                    |         {5} | 0     0  0  0     -c -b 0     0  0  |                                    |{3, 2}                                  |
      |                                        |         {4} | 0     0  0  0     b  a  0     0  0  |                                    |         {5} | 0     0  0  0     b  a  0     0  0  |                                    |                                        |
      |                                        |         {3} | 0     0  0  0     0  0  b2-ac 0  0  |                                    |         {4} | 0     0  0  0     0  0  b2-ac 0  0  |                                    |                                        |
      |                                        |         {4} | 0     0  0  0     0  0  0     -c -b |                                    |         {5} | 0     0  0  0     0  0  0     -c -b |                                    |                                        |
      |                                        |         {4} | 0     0  0  0     0  0  0     b  a  |                                    |         {5} | 0     0  0  0     0  0  0     b  a  |                                    |                                        |
      |                                        |                                                                                        |                                                                                        |                                        |
      |                                        |{1, 2}                                                                                  |{2, 2}                                                                                  |                                        |
      +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
      |cokernel {1} | b2-ac 0     0  0  0  0  ||cokernel {2} | b2-ac 0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||cokernel {3} | b2-ac 0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||cokernel {4} | b2-ac 0     0  0  0  0  ||
      |         {1} | 0     b2-ac 0  0  0  0  ||         {2} | 0     b2-ac 0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {3} | 0     b2-ac 0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     b2-ac 0  0  0  0  ||
      |         {2} | 0     0     -c -b 0  0  ||         {3} | 0     0     -c -b 0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     0     -c -b 0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0     -c -b 0  0  ||
      |         {2} | 0     0     0  0  -c -b ||         {3} | 0     0     0  0  -c -b 0     0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     0     0  0  -c -b 0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0     0  0  -c -b ||
      |         {2} | 0     0     b  a  0  0  ||         {3} | 0     0     b  a  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     0     b  a  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0     b  a  0  0  ||
      |         {2} | 0     0     0  0  b  a  ||         {3} | 0     0     0  0  b  a  0     0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     0     0  0  b  a  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0     0  0  b  a  ||
      |                                        |         {2} | 0     0     0  0  0  0  b2-ac 0     0  0  0  0  0     0     0  0  0  0  ||         {3} | 0     0     0  0  0  0  b2-ac 0     0  0  0  0  0     0     0  0  0  0  ||                                        |
      |{0, 1}                                  |         {2} | 0     0     0  0  0  0  0     b2-ac 0  0  0  0  0     0     0  0  0  0  ||         {3} | 0     0     0  0  0  0  0     b2-ac 0  0  0  0  0     0     0  0  0  0  ||{3, 1}                                  |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     -c -b 0  0  0     0     0  0  0  0  ||         {4} | 0     0     0  0  0  0  0     0     -c -b 0  0  0     0     0  0  0  0  ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     0  0  -c -b 0     0     0  0  0  0  ||         {4} | 0     0     0  0  0  0  0     0     0  0  -c -b 0     0     0  0  0  0  ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     b  a  0  0  0     0     0  0  0  0  ||         {4} | 0     0     0  0  0  0  0     0     b  a  0  0  0     0     0  0  0  0  ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     0  0  b  a  0     0     0  0  0  0  ||         {4} | 0     0     0  0  0  0  0     0     0  0  b  a  0     0     0  0  0  0  ||                                        |
      |                                        |         {2} | 0     0     0  0  0  0  0     0     0  0  0  0  b2-ac 0     0  0  0  0  ||         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  b2-ac 0     0  0  0  0  ||                                        |
      |                                        |         {2} | 0     0     0  0  0  0  0     0     0  0  0  0  0     b2-ac 0  0  0  0  ||         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     b2-ac 0  0  0  0  ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     -c -b 0  0  ||         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     -c -b 0  0  ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  -c -b ||         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  -c -b ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     b  a  0  0  ||         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     b  a  0  0  ||                                        |
      |                                        |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  b  a  ||         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  b  a  ||                                        |
      |                                        |                                                                                        |                                                                                        |                                        |
      |                                        |{1, 1}                                                                                  |{2, 1}                                                                                  |                                        |
      +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
      |cokernel {0} | b2-ac 0  0  |            |cokernel | b2-ac 0  0  0     0  0  0     0  0  |                                        |cokernel | b2-ac 0  0  0     0  0  0     0  0  |                                        |cokernel | b2-ac 0  0  |                |
      |         {1} | 0     -c -b |            |         | 0     -c -b 0     0  0  0     0  0  |                                        |         | 0     -c -b 0     0  0  0     0  0  |                                        |         | 0     -c -b |                |
      |         {1} | 0     b  a  |            |         | 0     b  a  0     0  0  0     0  0  |                                        |         | 0     b  a  0     0  0  0     0  0  |                                        |         | 0     b  a  |                |
      |                                        |         | 0     0  0  b2-ac 0  0  0     0  0  |                                        |         | 0     0  0  b2-ac 0  0  0     0  0  |                                        |                                        |
      |{0, 0}                                  |         | 0     0  0  0     -c -b 0     0  0  |                                        |         | 0     0  0  0     -c -b 0     0  0  |                                        |{3, 0}                                  |
      |                                        |         | 0     0  0  0     b  a  0     0  0  |                                        |         | 0     0  0  0     b  a  0     0  0  |                                        |                                        |
      |                                        |         | 0     0  0  0     0  0  b2-ac 0  0  |                                        |         | 0     0  0  0     0  0  b2-ac 0  0  |                                        |                                        |
      |                                        |         | 0     0  0  0     0  0  0     -c -b |                                        |         | 0     0  0  0     0  0  0     -c -b |                                        |                                        |
      |                                        |         | 0     0  0  0     0  0  0     b  a  |                                        |         | 0     0  0  0     0  0  0     b  a  |                                        |                                        |
      |                                        |                                                                                        |                                                                                        |                                        |
      |                                        |{1, 0}                                                                                  |{2, 0}                                                                                  |                                        |
      +----------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+

o14 : SpectralSequencePage
i15 : e^1

      +------------------+------------------------------+------------------------------+------------------+
o15 = |cokernel | c b a ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a ||
      |                  |         | 0 0 0 c b a 0 0 0 ||         | 0 0 0 c b a 0 0 0 ||                  |
      |{0, 0}            |         | 0 0 0 0 0 0 c b a ||         | 0 0 0 0 0 0 c b a ||{3, 0}            |
      |                  |                              |                              |                  |
      |                  |{1, 0}                        |{2, 0}                        |                  |
      +------------------+------------------------------+------------------------------+------------------+

o15 : SpectralSequencePage
i16 : e^2

      +------------------+------------------------------+------------------------------+------------------+
o16 = |cokernel | c b a ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a ||
      |                  |         | 0 0 0 c b a 0 0 0 ||         | 0 0 0 c b a 0 0 0 ||                  |
      |{0, 0}            |         | 0 0 0 0 0 0 c b a ||         | 0 0 0 0 0 0 c b a ||{3, 0}            |
      |                  |                              |                              |                  |
      |                  |{1, 0}                        |{2, 0}                        |                  |
      +------------------+------------------------------+------------------------------+------------------+

o16 : SpectralSequencePage
i17 : e^infinity

      +------------------+------------------------------+------------------------------+------------------+
o17 = |cokernel | c b a ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a 0 0 0 0 0 0 ||cokernel | c b a ||
      |                  |         | 0 0 0 c b a 0 0 0 ||         | 0 0 0 c b a 0 0 0 ||                  |
      |{0, 0}            |         | 0 0 0 0 0 0 c b a ||         | 0 0 0 0 0 0 c b a ||{3, 0}            |
      |                  |                              |                              |                  |
      |                  |{1, 0}                        |{2, 0}                        |                  |
      +------------------+------------------------------+------------------------------+------------------+

o17 : Page
i18 : ee^0

      +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
o18 = |cokernel {3} | b2-ac 0  0  |                        |cokernel {4} | b2-ac 0     0  0  0  0  |                                                |cokernel {5} | b2-ac 0  0  |                        |
      |         {4} | 0     -c -b |                        |         {4} | 0     b2-ac 0  0  0  0  |                                                |         {6} | 0     -c -b |                        |
      |         {4} | 0     b  a  |                        |         {5} | 0     0     -c -b 0  0  |                                                |         {6} | 0     b  a  |                        |
      |                                                    |         {5} | 0     0     0  0  -c -b |                                                |                                                    |
      |{0, 3}                                              |         {5} | 0     0     b  a  0  0  |                                                |{2, 3}                                              |
      |                                                    |         {5} | 0     0     0  0  b  a  |                                                |                                                    |
      |                                                    |                                                                                        |                                                    |
      |                                                    |{1, 3}                                                                                  |                                                    |
      +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
      |cokernel {2} | b2-ac 0  0  0     0  0  0     0  0  ||cokernel {3} | b2-ac 0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||cokernel {4} | b2-ac 0  0  0     0  0  0     0  0  ||
      |         {3} | 0     -c -b 0     0  0  0     0  0  ||         {3} | 0     b2-ac 0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     -c -b 0     0  0  0     0  0  ||
      |         {3} | 0     b  a  0     0  0  0     0  0  ||         {4} | 0     0     -c -b 0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     b  a  0     0  0  0     0  0  ||
      |         {2} | 0     0  0  b2-ac 0  0  0     0  0  ||         {4} | 0     0     0  0  -c -b 0     0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     0  0  b2-ac 0  0  0     0  0  ||
      |         {3} | 0     0  0  0     -c -b 0     0  0  ||         {4} | 0     0     b  a  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0  0  0     -c -b 0     0  0  ||
      |         {3} | 0     0  0  0     b  a  0     0  0  ||         {4} | 0     0     0  0  b  a  0     0     0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0  0  0     b  a  0     0  0  ||
      |         {2} | 0     0  0  0     0  0  b2-ac 0  0  ||         {3} | 0     0     0  0  0  0  b2-ac 0     0  0  0  0  0     0     0  0  0  0  ||         {4} | 0     0  0  0     0  0  b2-ac 0  0  ||
      |         {3} | 0     0  0  0     0  0  0     -c -b ||         {3} | 0     0     0  0  0  0  0     b2-ac 0  0  0  0  0     0     0  0  0  0  ||         {5} | 0     0  0  0     0  0  0     -c -b ||
      |         {3} | 0     0  0  0     0  0  0     b  a  ||         {4} | 0     0     0  0  0  0  0     0     -c -b 0  0  0     0     0  0  0  0  ||         {5} | 0     0  0  0     0  0  0     b  a  ||
      |                                                    |         {4} | 0     0     0  0  0  0  0     0     0  0  -c -b 0     0     0  0  0  0  ||                                                    |
      |{0, 2}                                              |         {4} | 0     0     0  0  0  0  0     0     b  a  0  0  0     0     0  0  0  0  ||{2, 2}                                              |
      |                                                    |         {4} | 0     0     0  0  0  0  0     0     0  0  b  a  0     0     0  0  0  0  ||                                                    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  b2-ac 0     0  0  0  0  ||                                                    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     b2-ac 0  0  0  0  ||                                                    |
      |                                                    |         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     -c -b 0  0  ||                                                    |
      |                                                    |         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  -c -b ||                                                    |
      |                                                    |         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     b  a  0  0  ||                                                    |
      |                                                    |         {4} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  b  a  ||                                                    |
      |                                                    |                                                                                        |                                                    |
      |                                                    |{1, 2}                                                                                  |                                                    |
      +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
      |cokernel {1} | b2-ac 0  0  0     0  0  0     0  0  ||cokernel {2} | b2-ac 0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||cokernel | b2-ac 0  0  0     0  0  0     0  0  |    |
      |         {2} | 0     -c -b 0     0  0  0     0  0  ||         {2} | 0     b2-ac 0  0  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         | 0     -c -b 0     0  0  0     0  0  |    |
      |         {2} | 0     b  a  0     0  0  0     0  0  ||         {3} | 0     0     -c -b 0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         | 0     b  a  0     0  0  0     0  0  |    |
      |         {1} | 0     0  0  b2-ac 0  0  0     0  0  ||         {3} | 0     0     0  0  -c -b 0     0     0  0  0  0  0     0     0  0  0  0  ||         | 0     0  0  b2-ac 0  0  0     0  0  |    |
      |         {2} | 0     0  0  0     -c -b 0     0  0  ||         {3} | 0     0     b  a  0  0  0     0     0  0  0  0  0     0     0  0  0  0  ||         | 0     0  0  0     -c -b 0     0  0  |    |
      |         {2} | 0     0  0  0     b  a  0     0  0  ||         {3} | 0     0     0  0  b  a  0     0     0  0  0  0  0     0     0  0  0  0  ||         | 0     0  0  0     b  a  0     0  0  |    |
      |         {1} | 0     0  0  0     0  0  b2-ac 0  0  ||         {2} | 0     0     0  0  0  0  b2-ac 0     0  0  0  0  0     0     0  0  0  0  ||         | 0     0  0  0     0  0  b2-ac 0  0  |    |
      |         {2} | 0     0  0  0     0  0  0     -c -b ||         {2} | 0     0     0  0  0  0  0     b2-ac 0  0  0  0  0     0     0  0  0  0  ||         | 0     0  0  0     0  0  0     -c -b |    |
      |         {2} | 0     0  0  0     0  0  0     b  a  ||         {3} | 0     0     0  0  0  0  0     0     -c -b 0  0  0     0     0  0  0  0  ||         | 0     0  0  0     0  0  0     b  a  |    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  -c -b 0     0     0  0  0  0  ||                                                    |
      |{0, 1}                                              |         {3} | 0     0     0  0  0  0  0     0     b  a  0  0  0     0     0  0  0  0  ||{2, 1}                                              |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  b  a  0     0     0  0  0  0  ||                                                    |
      |                                                    |         {2} | 0     0     0  0  0  0  0     0     0  0  0  0  b2-ac 0     0  0  0  0  ||                                                    |
      |                                                    |         {2} | 0     0     0  0  0  0  0     0     0  0  0  0  0     b2-ac 0  0  0  0  ||                                                    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     -c -b 0  0  ||                                                    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  -c -b ||                                                    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     b  a  0  0  ||                                                    |
      |                                                    |         {3} | 0     0     0  0  0  0  0     0     0  0  0  0  0     0     0  0  b  a  ||                                                    |
      |                                                    |                                                                                        |                                                    |
      |                                                    |{1, 1}                                                                                  |                                                    |
      +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
      |cokernel {0} | b2-ac 0  0  |                        |cokernel {1} | b2-ac 0     0  0  0  0  |                                                |cokernel | b2-ac 0  0  |                            |
      |         {1} | 0     -c -b |                        |         {1} | 0     b2-ac 0  0  0  0  |                                                |         | 0     -c -b |                            |
      |         {1} | 0     b  a  |                        |         {2} | 0     0     -c -b 0  0  |                                                |         | 0     b  a  |                            |
      |                                                    |         {2} | 0     0     0  0  -c -b |                                                |                                                    |
      |{0, 0}                                              |         {2} | 0     0     b  a  0  0  |                                                |{2, 0}                                              |
      |                                                    |         {2} | 0     0     0  0  b  a  |                                                |                                                    |
      |                                                    |                                                                                        |                                                    |
      |                                                    |{1, 0}                                                                                  |                                                    |
      +----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+

o18 : SpectralSequencePage

See also