Given a ring map f: R -> S, an R-module M and an R-module S, there is a spectral sequence E with E^2_{p,q} = Tor^S_p(Tor^R_q(M,S),N) that abuts to Tor^R_{p+q}(M,N).
i1 : k=QQ;
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i2 : R=k[a,b,c];
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i3 : S=k[s,t];
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i4 : f = map(S,R,{s^2,s*t,t^2});
o4 : RingMap S <-- R
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i5 : N = coker vars S;
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i6 : M = coker vars R --;
o6 = cokernel | a b c |
1
o6 : R-module, quotient of R
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i7 : F := complete res N;
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i8 : pushFwdF := pushFwd(f,F);
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i9 : G := complete res M;
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i10 : E := spectralSequence(filteredComplex(G) ** pushFwdF);
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i11 : EE := spectralSequence(G ** (filteredComplex pushFwdF));
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i12 : e = prune E;
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i13 : ee = prune EE;
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i14 : e^0
+------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
o14 = |cokernel | 0 0 b2-ac | |cokernel {1} | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel {2} | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel {3} | 0 0 b2-ac | |
| | -c -b 0 | | {1} | -c -b 0 0 0 0 0 0 0 | | {2} | -c -b 0 0 0 0 0 0 0 | | {3} | -c -b 0 | |
| | b a 0 | | {1} | b a 0 0 0 0 0 0 0 | | {2} | b a 0 0 0 0 0 0 0 | | {3} | b a 0 | |
| | {1} | 0 0 0 0 0 0 0 b2-ac 0 | | {2} | 0 0 0 0 0 0 0 b2-ac 0 | | |
|{0, 2} | {1} | 0 0 -c -b 0 0 0 0 0 | | {2} | 0 0 -c -b 0 0 0 0 0 | |{3, 2} |
| | {1} | 0 0 b a 0 0 0 0 0 | | {2} | 0 0 b a 0 0 0 0 0 | | |
| | {1} | 0 0 0 0 0 0 0 0 b2-ac | | {2} | 0 0 0 0 0 0 0 0 b2-ac | | |
| | {1} | 0 0 0 0 -c -b 0 0 0 | | {2} | 0 0 0 0 -c -b 0 0 0 | | |
| | {1} | 0 0 0 0 b a 0 0 0 | | {2} | 0 0 0 0 b a 0 0 0 | | |
| | | | |
| |{1, 2} |{2, 2} | |
+------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
|cokernel | 0 0 0 0 b2-ac 0 ||cokernel {1} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel {2} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel {3} | 0 0 0 0 b2-ac 0 ||
| | 0 0 0 0 0 b2-ac || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 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 b2-ac ||
| | -c -b 0 0 0 0 || {1} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | -c -b 0 0 0 0 ||
| | 0 0 -c -b 0 0 || {1} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | 0 0 -c -b 0 0 ||
| | b a 0 0 0 0 || {1} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | b a 0 0 0 0 ||
| | 0 0 b a 0 0 || {1} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {3} | 0 0 b a 0 0 ||
| | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || |
|{0, 1} | {1} | 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 0 0 0 0 0 0 0 0 b2-ac 0 0 ||{3, 1} |
| | {1} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || |
| | {1} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || |
| | {1} | 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 b a 0 0 0 0 0 0 || |
| | | | |
| |{1, 1} |{2, 1} | |
+------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
|cokernel | 0 0 b2-ac | |cokernel | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel | 0 0 0 0 0 0 b2-ac 0 0 | |cokernel | 0 0 b2-ac | |
| | -c -b 0 | | | -c -b 0 0 0 0 0 0 0 | | | -c -b 0 0 0 0 0 0 0 | | | -c -b 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 0 0 0 0 0 b2-ac 0 | | | 0 0 0 0 0 0 0 b2-ac 0 | | |
|{0, 0} | | 0 0 -c -b 0 0 0 0 0 | | | 0 0 -c -b 0 0 0 0 0 | |{3, 0} |
| | | 0 0 b a 0 0 0 0 0 | | | 0 0 b a 0 0 0 0 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 -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 0 0 0 | | |
| | | | |
| |{1, 0} |{2, 0} | |
+------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------+
o14 : SpectralSequencePage
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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
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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
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i18 : ee^0
+----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
o18 = |cokernel {3} | 0 0 b2-ac | |cokernel {3} | 0 0 0 0 b2-ac 0 | |cokernel {3} | 0 0 b2-ac | |
| {3} | -c -b 0 | | {3} | 0 0 0 0 0 b2-ac | | {3} | -c -b 0 | |
| {3} | b a 0 | | {3} | -c -b 0 0 0 0 | | {3} | b a 0 | |
| | {3} | 0 0 -c -b 0 0 | | |
|{0, 3} | {3} | b a 0 0 0 0 | |{2, 3} |
| | {3} | 0 0 b a 0 0 | | |
| | | |
| |{1, 3} | |
+----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
|cokernel {2} | 0 0 0 0 0 0 b2-ac 0 0 ||cokernel {2} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel {2} | 0 0 0 0 0 0 b2-ac 0 0 ||
| {2} | -c -b 0 0 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 || {2} | -c -b 0 0 0 0 0 0 0 ||
| {2} | b a 0 0 0 0 0 0 0 || {2} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | b a 0 0 0 0 0 0 0 ||
| {2} | 0 0 0 0 0 0 0 b2-ac 0 || {2} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 0 0 0 b2-ac 0 ||
| {2} | 0 0 -c -b 0 0 0 0 0 || {2} | b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 -c -b 0 0 0 0 0 ||
| {2} | 0 0 b a 0 0 0 0 0 || {2} | 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 b a 0 0 0 0 0 ||
| {2} | 0 0 0 0 0 0 0 0 b2-ac || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || {2} | 0 0 0 0 0 0 0 0 b2-ac ||
| {2} | 0 0 0 0 -c -b 0 0 0 || {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 || {2} | 0 0 0 0 -c -b 0 0 0 ||
| {2} | 0 0 0 0 b a 0 0 0 || {2} | 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 || {2} | 0 0 0 0 b a 0 0 0 ||
| | {2} | 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 || |
|{0, 2} | {2} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 ||{2, 2} |
| | {2} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || |
| | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || |
| | {2} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || |
| | {2} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || |
| | {2} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || |
| | {2} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || |
| | {2} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || |
| | | |
| |{1, 2} | |
+----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
|cokernel {1} | 0 0 0 0 0 0 b2-ac 0 0 ||cokernel {1} | 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 0 ||cokernel | 0 0 0 0 0 0 b2-ac 0 0 | |
| {1} | -c -b 0 0 0 0 0 0 0 || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 0 || | -c -b 0 0 0 0 0 0 0 | |
| {1} | b a 0 0 0 0 0 0 0 || {1} | -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | b a 0 0 0 0 0 0 0 | |
| {1} | 0 0 0 0 0 0 0 b2-ac 0 || {1} | 0 0 -c -b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || | 0 0 0 0 0 0 0 b2-ac 0 | |
| {1} | 0 0 -c -b 0 0 0 0 0 || {1} | 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 0 0 | |
| {1} | 0 0 b a 0 0 0 0 0 || {1} | 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 0 0 | |
| {1} | 0 0 0 0 0 0 0 0 b2-ac || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 0 || | 0 0 0 0 0 0 0 0 b2-ac | |
| {1} | 0 0 0 0 -c -b 0 0 0 || {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 0 || | 0 0 0 0 -c -b 0 0 0 | |
| {1} | 0 0 0 0 b a 0 0 0 || {1} | 0 0 0 0 -c -b 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 -c -b 0 0 0 0 0 0 0 0 0 0 || |
|{0, 1} | {1} | 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 0 0 ||{2, 1} |
| | {1} | 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 b2-ac || |
| | {1} | 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 -c -b 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 0 0 || |
| | {1} | 0 0 0 0 0 0 0 0 0 0 b a 0 0 0 0 0 0 || |
| | | |
| |{1, 1} | |
+----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
|cokernel | 0 0 b2-ac | |cokernel | 0 0 0 0 b2-ac 0 | |cokernel | 0 0 b2-ac | |
| | -c -b 0 | | | 0 0 0 0 0 b2-ac | | | -c -b 0 | |
| | b a 0 | | | -c -b 0 0 0 0 | | | b a 0 | |
| | | 0 0 -c -b 0 0 | | |
|{0, 0} | | b a 0 0 0 0 | |{2, 0} |
| | | 0 0 b a 0 0 | | |
| | | |
| |{1, 0} | |
+----------------------------------------------------+----------------------------------------------------------------------------------------+----------------------------------------------------+
o18 : SpectralSequencePage
|